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Parametric, non-parametric and statistical modeling of stony coral reef data
h [electronic resource] /
by Armando Hoare.
[Tampa, Fla] :
b University of South Florida,
Title from PDF of title page.
Document formatted into pages; contains 135 pages.
Dissertation (Ph.D.)--University of South Florida, 2008.
Includes bibliographical references.
Text (Electronic dissertation) in PDF format.
ABSTRACT: Like coral reefs worldwide, the Florida Reef Tract has dramatically declined within the past two decades. Monitoring of 40 sites throughout the Florida Keys National Marine Sanctuary has undertaken a multiple-parameter approach to assess spatial and temporal changes in the status of the ecosystem. The objectives of the present study consist of the following: In chapter one, we review past coral reef studies; emphasis is placed on recent studies on the stony corals of reefs in the lower Florida Keys. We also review the economic impact of coral reefs on the state of Florida. In chapter two, we identify the underlying probability distribution function of the stony coral cover proportions and we obtain better estimates of the statistical properties of stony coral cover proportions. Furthermore, we improve present procedures in constructing confidence intervals of the true median and mean for the underlying probability distribution.In chapter three, we investigate the applicability of the normal probability distribution assumption made on the pseudovalues obtained from the jackknife procedure for the Shannon-Wiener diversity index used in previous studies. We investigate a new and more effective approach to estimating the Shannon-Wiener and Simpson's diversity index. In chapter four, we develop the best possible estimate of the probability distribution function of the jackknifing pseudovalues, obtained from the jackknife procedure for the Shannon-Wiener diversity index used in previous studies, using the xi nonparametric kernel density estimate method. This nonparametric procedure gives very effective estimates of the statistical measures for the jackknifing pseudovalues. Lastly, the present study develops a predictive statistical model for stony coral cover.In addition to identifying the attributable variables that influence the stony coral cover data of the lower Florida Keys, we investigate the possible interactions present. The final form of the developed statistical model gives good estimates of the stony coral cover given some information of the attributable variables. Our nonparametric and parametric approach to analyzing coral reef data provides a sound basis for developing efficient ecosystem models that estimate future trends in coral reef diversity. This will give the scientists and managers another tool to help monitor and maintain a healthy ecosystem.
Mode of access: World Wide Web.
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Advisor: Chris P. Tsokos, Ph.D.
Shannon-Wiener diversity index
Simpson's diversity index
t USF Electronic Theses and Dissertations.
Parametric, Non-Parametric And Statistical Modelin g Of Stony Coral Reef Data by Armando Hoare A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics and Statistics College of Arts and Sciences University of South Florida Major Professor: Chris P. Tsokos, Ph.D. Marcus McWaters, Ph.D. Kandethody Ramachandran, Ph.D. Gangaram S. Ladde, Ph.D. Pamela Hallock Muller, Ph.D. Date of Approval: April 8, 2008 Keywords: regression model, stony coral, Shannon-Wi ener diversity index, Simpson's diversity index, jackknifing, bootstrap, three-para meter lognormal distribution, kernel density estimate Copyright 2008 Armando Hoare
Dedication This dissertation is dedicated to my wife, Ana, wh o has walked this long path by my side offering her love, encouragement and even p rofessional advice all along the way. She, above all others, has understood the demands a nd the sacrifices required of a scholar. Ana has pushed me to the greater challenge of realizing my full potential. I also dedicate this work to my beloved son, Armando. You are the reason for it all; your love, patience and alarming wisdom have been my greatest inspiration. It goes without saying that my brothers and sister s and their spouses have been a godsend through it all. They have encouraged me, i nspired me, loved me and have done everything humanly possible to raise me up througho ut my pursuit of an academic career. Martha, my second mother, words can never express t he extreme gratitude and respect I have for you. Ismael and Olda, you understood what trials and challenges must be overcome and the sacrifices that must be made to su cceed at this level. Thank you, Eduardo, for in your special way, you have helped m e keep it together. Mom and Dad, the values of perseverance and strength of mind and character that you passed on to me and all your children is your greatest legacy. You are never forgotten. To my in-laws, Maria and Fernando Coye: you have helped me through out the stormy weather. Thank you all for your love and understanding. I also th ank my brotherand sisterin-law and their spouses for their encouragement and support.
ACKNOWLEDGEMENTS I express my sincerest appreciation and gratitude to my research and dissertation advisor, Dr. Chris P. Tsokos, for his constant guid ance and support and professional advice throughout the course of this dissertation, and for giving me the opportunity to work with a project that has far extending applicat ions to a real-world problem. I also give my deepest thanks to Dr. Marcus McWaters, Dr. Kandethody Ramachandran, Dr. Gangaram S. Ladde, and Dr. Pamela Hallock-Muller fo r serving on my committee. Their insightful comments and advice were instrumental in developing a strong and successful dissertation. I am very appreciative to Dr. Hamisu Salihu for agreeing to chair my dissertation defense. I am very grateful to the faculty and staff of the University of South Florida Department of Mathematics and Statistics, for the c ountless ways in which they facilitated a successful graduate program for me. Particularly, I would like to thank Dr. Stephen Suen for his quiet and humble guidance in t he initial years of my graduate program. I also thank Dr. George Yanev for his enc ouragement in all my academic endeavors. Thank you, Dr. McWaters, for your kind m oral support and for giving me the opportunity to continue to develop not only as a sc holar but also as a teacher. Without the help of Jim Tremmel, former graduate coordinato r of the department, none of this would have been possible. Jim played a vital role in putting the wheels in motion so that I could pursue my graduate career without separatio n from my wife and son; for this I am eternally indebted. Many of my fellow graduate students were supportiv e over the years and for this I am most appreciative but I would be remiss if I do not acknowledge Gokarna Aryal, Jemal Gishe and Druba Adhikari, who were not only m y colleagues but were also a font of encouragement and support and became members of my global family.
i Table of Contents List of Tables..................................... ................................................... ...............................v List of Figures.................................... ................................................... ...........................viii Abstract........................................... ................................................... .................................x Chapter 1 Review of Coral Reef Studies............. ................................................... .............1 1.1 Introduction.................................. ................................................... .............1 1.2 Economic Impact of Coral Reefs on the State of Florida............................6 1.3 Coral Reef Evaluation and Monitoring Project (CREMP)..........................9 1.3.1 Sampling and Data Collection................ .........................................12 1.3.2 Results of Statistical Analyses............. ............................................18 1.4 Focus of Chapter 2............................ ................................................... ......28 1.5 Focus of Chapter 3............................ ................................................... ......28 1.6 Focus of Chapter 4............................ ................................................... ......29 1.7 Focus of Chapter 5............................ ................................................... ......30 Chapter 2 Parametric Analysis of Stony Coral Cover from the Florida Keys...................31 2.1 Introduction.................................. ................................................... ...........31 2.2 Descriptive Statistic: Proportion of Stony Cor al Cover.............................33 2.3 Procedure in Fitting a Three Parameter Lognor mal Probability Density Function.................................. ................................................... ...36 2.3.1 Maximum Likelihood Estimation Procedure...... ...........................37 2.3.2 Goodness-of-fit Procedure.................... .........................................40
ii 2.4 Results in Fitting a Three Parameter Lognormal Probability Density Function.................................. ................................................... ...43 2.5 Comparison of Descriptive Statistics vs. Param etric Analysis..................47 2.6 Confidence Interval for the Median............. ..............................................51 2.7 Confidence Interval for the Mean............... ...............................................55 2.8 Conclusion..................................... ................................................... .........61 Chapter 3 Statistical Modeling of the Health of the Reefs: Diversity Indices...................63 3.1 Introduction............................. ................................................... ................63 3.2 Methodology of Statistical Analysis of Shannon -Wiener Diversity Index................................... ................................................... .....67 3.3 Comparison of the Bootstrap and Normality Conf idence Intervals...........68 3.4 Probability Distribution Fit of the Species Ab undance.............................76 3.5 Shannon-Wiener and SimpsonÂ’s Diversity Index: S pecies Abundance Probability Distribution................ ..........................................79 3.5.1 Shannon-Wiener Diversity Index for the 2-Pa rameter Lognormal Probability Distribution................ ...............................79 3.5.2 SimpsonÂ’s Diversity Index for the 2-Parameter Lognormal Probability Distribution................ ...............................79 3.5.3 Diversity Indices for the Probability Distrib ution of Species Abundance................................. .......................................81 3.6 Conclusion..................................... ................................................... .........85 Chapter 4 Nonparametric Statistical Analysis of Div ersity Index.....................................87 4.1 Introduction.................................. ................................................... ...........87 4.2 Kernel Probability Density.................... ................................................... .88 4.2.1 Statistical Properties of the Kernel Density Estimator..................90
iii 4.2.2 Criteria for Quality of Fit.................. .............................................92 4.3 Procedure for Developing the Kernel Density Es timate...........................94 4.4 The Kernel Density Estimate................... ..................................................9 4 4.5 Comparison of the Nonparametric and Parametric ..................................103 4.6 Conclusion.................................... ................................................... ........106 Chapter 5 Statistical Modeling of Stony Coral Cover ................................................... ..108 5.1 Introduction................................... ................................................... ........108 5.2 Response and Attributable Variables............ ...........................................109 5.3 Data Manipulation.............................. ................................................... ..110 5.4 Multivariate Statistical Model................. .................................................11 3 5.4.1 Transformations of the Response Variable..... .............................113 5.4.2 Best Initial Statistical Model............... .........................................115 5.4.3 Interaction and High Order................... .......................................118 5.4.4 Model Predictive Capability.................. ......................................120 5.4.5 Final Model.................................. ................................................122 5.5 Conclusion..................................... ................................................... .......122 Chapter 6 Future Research.......................... ................................................... ..................124 6.1 Introduction.................................. ................................................... .........124 6.2 Non-Parametric Kernel Density.................. .............................................124 6.3 Improving the Proposed Statistical Model....... ........................................124 6.4 Surface Response Analysis...................... ................................................125 6.5 Stony Coral Cover Parametric Analysis.......... ........................................125 References......................................... ................................................... ............................126
iv About The Author................................... ................................................... ............End Page
v List of Tables Table 1.1 Number of Person-Days on all Reefs by Re creational Activity June 2000 to May 2001 (Millions) (Johns et al. 200 3)...................................8 Table 1.2 Economic Contribution of Reef Related Ex penditures June 2000 to May 2001 (Johns et al. 2003)..................... ................................................... .8 Table 1.3 CREMP Sampling Sites.................... ................................................... .........14 Table 1.4 Hypothesis Testing for Change in Mean St ony Coral Cover: 1999 to 2005........................................... ................................................... ............21 Table 1.5 Hypothesis Testing Results for Species R ichness........................................26 Table 1.6 Hypothesis Testing and Confidence Interv als for Change in Number of Stations with Incidence of Disease and B leaching.....................26 Table 2.1 Descriptive Statistics for Proportion St ony Coral Cover..............................35 Table 2.2 Parameter Estimates for the Three-Parame ter Lognormal Distribution...................................... ................................................... ..........44 Table 2.3 Goodness-of-fit Statistics for the Three -Parameter Lognormal Probability Distribution Fit...................... ................................................... ..45 Table 2.4 Shapiro-WilkÂ’s Normality Test of Transfo rmed Data..................................46 Table 2.5 Probability Distribution Statistics for Stony Coral Cover Proportion Data................................... ................................................... .......48 Table 2.6 90% and 95% Confidence Interval for the True Median: Nave Method and Proposed Method........................ ..............................................53 Table 2.7 Confidence Range: Proposed Method vs. Na ve Method.............................55 Table 2.8 90% and 95% Confidence Interval for the True Mean: CoxÂ’s Method and Proposed Method.................. .........................................58 Table 2.9 Confidence Range: Proposed Method vs. Cox Â’s Method.............................61 Table 3.1 95% Confidence Interval for the True Shan non-Wiener Diversity Index for Sanctuary Region........................ ..................................................6 9
vi Table 3.2 Confidence Range: Bootstrap Confidence In terval vs. Normality Confidence Interval for the Sanctuary Region...... ........................................72 Table 3.3 95% Confidence Interval for the True Shan non-Wiener Diversity Index for Dry Tortugas............................ ................................................... ..73 Table 3.4 Confidence Range: Bootstrap Confidence In terval vs. Normality Confidence Interval for the Dry Tortugas.......... ...........................................75 Table 3.5 Normality Test of Pseudovalues........... ................................................... .....76 Table 3.6 Descriptive Statistics for Species Abunda nce...............................................7 6 Table 3.7 Parameter Estimates for the Two-Parameter Lognormal Distribution.........77 Table 3.8 Goodness-of-fit Statistics for the Two-Pa rameter Lognormal Probability Distribution Fit...................... ................................................... ..78 Table 3.9 Shannon-Wiener Â‘s and SimpsonÂ’s Diversity Index for the Two-Parameter Lognormal Probability Distribution.. ..................................81 Table 3.10 Shannon-WienerÂ’s and SimpsonÂ’s Diversit y Index for The Species Abundance Data............................ ................................................... 82 Table 3.11 Percentage Differences of the Shannon-Wi ener Diversity Index: PDF vs. Jackknifing Procedure and Direct Procedure ..................................83 Table 4.1 Some Kernels and Their Inefficiencies... ................................................... ...89 Table 4.2 Expected Value, Variance and Cumulative D istribution Function of the Kernel Density Estimate 92 Table 4.3 Parameter Estimates for the Kernel Densit y Estimate and the Normal Probability Distribution................... ................................................95 Table 4.4-A Statistical Properties of the Gaussian Kernel Density Estimate.................103 Table 4.4-B Statistical properties of the Normal Pr obability Distribution.....................104 Table 5.1 Notation of Variables.................... ................................................... ...........110 Table 5.2 CREMP and WQMP Stations Pairing List..... ............................................111
vii Table 5.3 Contour Analysis Results................. ................................................... ........113 Table 5.4 Correlation Matrix of Response and Attrib utable Variables......................116 Table 5.5 Statistical Ranking of the Attributing Va riables to Stony Coral Cover............................................. ................................................... ...........118 Table 5.6 Statistical Ranking of the Attributing Va riables with Interactions.............119 Table 5.7 Prediction Results....................... ................................................... .............121
viii List of Figures Figure 1.1 Coral reefs serve as habitat for divers e species (Cummings).........................1 Figure 1.2 Example of bleaching on Acropora palmata ..................................................5 Figure 1.3 Location of FKNMS and sampling sites of WQPP......................................11 Figure 1.4 The Three Transects Conducted at Each S tation..........................................15 Figure 1.5 Three Mosaics of the Same Transect for 1996, 1999 and 2004....................17 Figure 1.6 Schematic for Station Species Inventory Survey..........................................18 Figure 1.7 Histogram of Percentage Stony Coral Cov er: 1996 To 2005 by Region............................................ ................................................... ............19 Figure 1.8 Histogram of Percent Stony Coral Cover: 1996 to 2005 by Habitat............20 Figure 1.9 Percentage Change in Stony Coral Cover by Station...................................22 Figure 1.10 Histogram of Shannon-Wiener Diversity Index: 1996 to 2005 by Region........................................... ................................................... .............23 Figure 1.11 Histogram of Shannon-Wiener Diversity Index: 1996 to 2005 by Habitat.......................................... ................................................... ..............24 Figure 1.12 95% Confidence Interval of the Shannon -Wiener Diversity Index for the Sanctuary Region.......................... ................................................... .25 Figure 1.13 Stations With Incidence of Disease and Bleaching, 1996 Â– 2005................27 Figure 2.1 Histogram for the Stony Coral Cover Prop ortions for 2006.........................34 Figure 2.2 Boxplots for Stony Coral Cover Proporti on Data from 1996 to 2006..........36 Figure 2.3 Cumulative Distribution Function Plot f or 1997 Stony Coral Cover Proportion Data................................... ................................................... .......40
ix Figure 2.4 Standard Deviation from the Probability Distribution and from Descriptive Statistics............................ ................................................... ......49 Figure 2.5 Mean and Median from the Probability Di stribution and from Descriptive Statistics............................ ................................................... ......50 Figure 2.6 90 % Confidence Interval for the Median: Proposed Method vs. Nave Method...................................... ................................................... .......54 Figure 2.7 The 90 % Confidence Interval for the Tr ue Mean: Nave Method, CoxÂ’s Method and Proposed Method.................. .........................................60 Figure 3.1 95% Confidence Interval from the Normali ty Assumption and Bootstrapping for Sanctuary...................... ................................................... 71 Figure 3.2 95% Confidence Interval from the Normali ty Assumption and Bootstrapping for Dry Tortugas................... .................................................74 Figure 3.3 Shannon-Wiener Diversity Indices for the Sanctuary Region......................84 Figure 4.1 MISE vs. Bandwidth for 1996............. ................................................... ......96 Figure 4.2 Kernel Density Estimate Fit: 1996....... ................................................... ......96 Figure 4.3 Normal Distribution Fit: 1996........... ................................................... ........97 Figure 4.4 Kernel Density Estimate vs. Normal Proba bility Distribution: 1997 to 2006..................................... ................................................... .........98 Figure 4.5 Cumulative Distribution Function for the Gaussian Kernel Density Estimate......................................... ................................................... ...........104 Figure 4.6 Standard Deviations: Gaussian Kernel Den sity Estimate (KDE) vs. Normal Probability Distribution (N).......... ...........................................105 Figure 4.7 95 % Confidence Interval: Gaussian Kerne l Density Estimate (KDE) vs. Normal Probability Distribution (N).... .....................................106 Figure 5.1 Contour map for total nitrogen (TN)..... ................................................... ..112
x Parametric, Non-Parametric and Statistical Modeling of Stony Coral Reef Data Armando J. Hoare ABSTRACT Like coral reefs worldwide, the Florida Reef Tract has dramatically declined within the past two decades. Monitoring of 40 site s throughout the Florida Keys National Marine Sanctuary has undertaken a multiple-paramete r approach to assess spatial and temporal changes in the status of the ecosystem. Th e objectives of the present study consist of the following: In chapter one, we review past coral reef studies; emphasis is placed on recent studies on the stony corals of reefs in the lower F lorida Keys. We also review the economic impact of coral reefs on the state of Flor ida. In chapter two, we identify the underlying proba bility distribution function of the stony coral cover proportions and we obtain better estimates of the statistical properties of stony coral cover proportions. Furthermore, we i mprove present procedures in constructing confidence intervals of the true media n and mean for the underlying probability distribution. In chapter three, we investigate the applicability of the normal probability distribution assumption made on the pseudovalues ob tained from the jackknife procedure for the Shannon-Wiener diversity index used in prev ious studies. We investigate a new and more effective approach to estimating the Shann on-Wiener and SimpsonÂ’s diversity index. In chapter four, we develop the best possible esti mate of the probability distribution function of the jackknifing pseudovalu es, obtained from the jackknife procedure for the Shannon-Wiener diversity index us ed in previous studies, using the
xi nonparametric kernel density estimate method. This nonparametric procedure gives very effective estimates of the statistical measures for the jackknifing pseudovalues. Lastly, the present study develops a predictive st atistical model for stony coral cover. In addition to identifying the a ttributable variables that influence the stony coral cover data of the lower Florida Keys, w e investigate the possible interactions present. The final form of the developed statistica l model gives good estimates of the stony coral cover given some information of the att ributable variables. Our nonparametric and parametric approach to analyzing cor al reef data provides a sound basis for developing efficient ecosystem models that esti mate future trends in coral reef diversity. This will give the scientists and manage rs another tool to help monitor and maintain a healthy ecosystem.
1 Chapter 1 Review of Coral Reef Studies 1.1 Introduction Coral reef communities are very important ecosyste ms in the world. They are home to at least 4,000 species, or almost a third o f the worldÂ’s marine fish species (Paulay 1996). Hinrichsen (1997) wrote that the G reat Barrier Reef of Australia boasts 400 species of coral providing habitat for more tha n 1500 species of fish, 4000 different kinds of mollusk, and 400 species of sponge. Figur e 1.1 shows the vibrant activities that occur within the coral reefs. Bryant, Burke, McMan us and Spalding (1998) mentioned that the coral reef habitats provide about $375 bil lion each year to humans in living resources and services. Figure 1.1 Coral Reefs Serve as Habitat for Divers e Species (Cummings).
2 Coral reefs are so important that there are countl ess studies being done. Hallock (1997) investigated the history of reef formation. She has shown how long these complicated ecosystems take to develop. If the ree f-building communities are disturbed by extinctions they take millions of years to recov er. Many are studying the history of the reef in order to understand the present reef fo rmations and to explain the present changes that are occurring (Macintyre 1988, Jackson 1992, Hunter and Jones 1996, Greenstein, Curran and Pandolfi 1998, Pandolfi and Jackson 2007, Wood 2007). These give the opportunity to study the reefs before huma n impact. Some studies have argued that the changes presently experienced are related to a long term cycle unrelated to anthropogenic disturbance (Jackson 1992, Hunter and Jones 1996, Pandolfi 1996, Hubbard 1997, Pandolfi and Jackson 1997, 2001). Me solella (1968) found similarities in species dominance and diversity from Pleistocene da ta from Barbados with those described in the living reefs of Jamaica (Goreau 19 59). Jackson (1992) using the same data suggested that the coral communities were simi lar throughout a 500 Â–kyr interval. Pandolfi (1996) tested this proposition by using da ta from Huon Peninsula, Papua New Guinea. He found similarities throughout a 95-kyr interval by applying univariate and multivariate methods. The study of the past is not without controversy. Connell, Hughes and Wallace (1997) pointed out that Davis (1982) us ed single observations a century apart and that Jackson (1992) used data values 200, 000 years apart. The difficulties experienced in obtaining the information from geolo gical and fossil record has been problematic areas that are in question (Porter et a l. 2002, Pandolfi and Jackson 2007, Wellington and Glynn 2007). Connell et al. (1997) showed that short term studi es should be used to complement longer term studies. Many such short te rm studies are also carried out to investigate the present state of the coral reefs (H ughes and Tanner 2000, Boyer and Jones 2002, Porter et al. 2002, Bellwood, Hughes, Folke a nd Nystrom 2004, Brown et al. 2004, Buddemeier, Kleypas and Aronson 2004, Pavlov et al. 2004, Wiegus, Chadwick-Furman and Dubinsky 2004, Andrews, Nall, Jeffrey and Pittm an 2005, Santavy, Summers, Engle and Harwell 2005). Many of these have reported the decline of the coral reef cover (Hughes and Tanner 2000, Porter et al. 2002, Bellwo od et al. 2004, Buddemeier et al.
3 2004, Santavy et al. 2005, Callahan et al. 2006, Ts okos, Hoare and Yanev 2006a, Pante, King and Dustan 2007). To investigate the decline o f coral reef cover, many have studied different factors they believe is the cause of the decline. Coral reefs around the world are threatened by ant hropogenic and climatic factors. An article by Loft (2008) reported that biologists estimate that about 70 percent of coral species are threatened and that 20 percent are dama ged beyond repair. He quoted Ellycia Harrould-Kolieb, a researcher with Oceana, saying ( p.4), Â“IÂ’d say things are pretty critical for corals at the moment.Â” He continued by report ing that researchers at the University of North Carolina at Chapel Hill reported in the Fe bruary 14 issue of the journal Science that Â“rising ocean temperature are the most pervasi ve threat and almost half of all the worldÂ’s coral reefs have recently experienced mediu mto highlevel impacts.Â” Several anthropogenic and climatic factors have be en attributed to the decline of coral reef cover. Corals are sensitive to changes in salinity, ultraviolent radiation and nutrient levels. They are vulnerable to temperature changes, pollution, fishing methods, ocean acidification and other man-made influences. High temperatures stress or kill the microscopic plants that live in the corals and blea ching the corals exposing the white calcium carbonate skeletons of the coral colony. Shinn et al. (2000) and Garrison et al. (2003) hav e suggested that a possible effect directly and indirectly on the coral reef is the Af rican and Asia dust. The pathogen responsible for episodic outbreaks of aspergillosis has been detected in samples of African dust. Shinn et al. (2000) suggested dust a s a source for the disease outbreaks in 1983 to 1984 that were responsible for the mass mor talities of Diadema (sea urchin) and the acroporid corals from the late 1970s through th e early 1990s. Lessios (1988) discussed in detail the wide extent of the mortalit y of Diadema across the Caribbean. Aronson and Precht (2001) discussed the effect of w hite band disease on the acroporid corals in the wider Caribbean. The effects of the African and Asia dust on the coral reef have not been conclusively proven. Human activities such as coastal development, over exploitation (TalaueMcManus and Kesner 1993, Johannes and Riepen 1995, Jackson et al. 2001) and destructive fishing practices (Birkeland 1997a, Bry ant et al. 1998, Fox, Mous, Pet,
4 Muljadi and Caddwell 2005), inland pollution and er osion, and marine pollution, and natural disasters are some of the causes for the de cline in coral reef coral. The anthropogenic factors have been studied by many sci entists (Brown 1987, Hodgson 1999, Pavlov et al. 2004, Wielgus et al. 2004), clearly d ocumenting their effects on the coral reef. An effect of overexploitation of fishing, esp ecially of urchin predators, is bioerosion caused by sea urchins (Griffin, Garcia and Weil 200 3). Coral disease has also contributed to coral cover decline globally. Peters (1997) men tioned that only since the mid-1970s have scientists realized that corals were exposed t o diseases caused by pathogens and parasites, as well as to those conditions caused or aggravated by exposures to anthropogenic pollutants and habitat degradation. Diseases may either kill the organism over varying periods of time or alter the structure or function of the individual in which it may make the organism susceptible to predation or e nvironmental stresses (Peters 1997). Santavy et al. (2001) found that in spring of 1998 white plague was seen in 92% of the stations in Key West area while patchy necrosis/whi te-pox occurred at 50%, white-band type 1 at 25% and yellow blotch disease at 25% of t he stations. They found that in summer white plague and white-band disease type 1 e ach occurred at 69% of the stations. Climatic factors are also studied to measure their impact on the coral reef. The most studied is the effect of temperature on coral bleac hing (Porter, Lewis and Porter 1999, Riegl 2007, Wellington and Glynn 2007). Coral bleaching is caused by the loss of the symbi otic algae associated with the coralÂ’s tissue or the decline in photosynthetic pig ments in the symbiotic algae. According to Westmacott, Teleki, Wells and West (20 00), the actual mechanism of coral bleaching is poorly understood. Coral bleaching has been caused by unusually high sea temperatures, high levels of ultraviolet light, low light conditions, high turbidity and sedimentation, disease, abnormality salinity and po llution. Nutrient loading is another factor that is studied, because it affects the wate r quality of the coral reef. This factor contributes to diseases and bleaching of the coral reefs. Again many have studied the concentrations of nutrients in the water and sedime nts found on the coral reef (Porter et al. 1999, Keller and Itkin 2002, Lapointe and Thack er 2002). Wielgus et al. (2004) found statistically higher partial mortality of coral col onies among sites with higher total
5 organic nitrogen. Figure 1.2 shows the bleaching o f the stony coral species Acropora palmata also known as Elkhorn coral. Figure 1.2 Example of Bleaching on Acropora palmata (Courtesy of NOAA) The most extensive and severe bleaching known occu rred in 1998. Coral bleaching was reported in 60 countries and island n ations at sites in the Pacific Ocean, Indian Ocean, Red Sea, Persian Gulf, Mediterranean and Caribbean. Indian Ocean corals were particularly severely impacted, with greater t han 70% mortality reported in the Maldives, Andamans, Lakshadweep Islands, and in Sey chelles Marine Park System. He quoted Harrould-Kolieb, who said (p.4) Â“reefs provi de homes, nurseries, feeding grounds and spawning sites to a diversity of life that is v irtually unparalleled anywhere else in the world.Â” It is critical to understand the factors th at affect coral reef ecosystems. Urquhart and Kincaid (1999) and Spellerberg (2005) stressed the importance of monitoring projects among scientific research. The importance is both ecological and economical. Urquhart and Kincaid (1999) discussed the financial investment by many citizens associated with the costs involved in regu lating the activities of industry, government, agriculture, tourism and development. The monitoring process could be used to answer the following questions: Have the re gulations achieved their intended effect? Are modifications to the regulations necess ary? Jaap (2000) stated that
6 monitoring is important in restoration projects of coral reef. Monitoring helps to determine the success of restoration and reveals wa ys to improve future projects. Monitoring should include restoration areas and ref erence areas. Spellerberg (2005) mentioned that monitoring has to be resourced and f inanced for the following ecological reasons: The process of many ecosystems has not been well re searched and monitoring programs could provide basic ecological knowledge a bout those processes. Management of ecosystems, if it is to be effective, requires a baseline, which can only come from ecosystems monitoring. Anthropogenic perturbations on the worldÂ’s ecosyste ms have long-term effects, some synergistic and some cumulative: therefore, it follows that long-term studies are required. The data from long-term studies can be a basis for early detection of potentially harmful effects on components of ecosystems. With the ever-increasing loss of species, loss of h abitats and damage to biological communities, ecological monitoring is needed to ide ntify the implications of these losses and damage. Unfortunately due to the scarcity of long term reso urces and financing, monitoring projects tend to be limited in the information that is collected. Furthermore, these monitoring projects are usually designed to minimiz e cost, a criteria that is not necessarily ideal for the efficiency of providing m eaningful statistical analysis representative of the sampled ecosystems. Thus proj ects should be planned according to a cost-benefit analysis. 1.2 Economic Impact of Coral Reefs on the State o f Florida The economic impact of the coral reef to the state of Florida lies mainly in the tourism and fishery industries. Other benefits fro m the coral reef are its protection from storm surge, medicinal and academic research. Unit ed Nations Environment Programme estimated the value of coral reefs between US$1 to US$6 hundred thousand per square
7 kilometer per year (Butler 2006). They estimated t he cost of protecting the coral reef at US$775 per square kilometer per year. The annual e conomic benefits of the coral reefs far outweigh the estimated cost of maintaining the coral reefs by 130 to 775 times per kilometer. Another study Wilkinson (2002) estimate s the value of the coral reefs at US$375 billion in 1997, while the estimated expendi ture on research, monitoring and management is probably less than US$ 100 million pe r year. This report puts the benefit of the coral reef at 3750 times the cost of researc h, monitoring and management. Thus it is wise to maintain the reefs as healthy as possibl e. Tourism is the primary industry in the state of Fl orida. In 2006 it is estimated that 83.9 million tourist brought in about $65 billion to the economy of Florida (Research 2008). The tourism industry employed about 964,700 in 2006. Johns, Leeworthy, Bell and Bohn (2003) conducted a study on the impact of tourists specifically pertaining to the coral reefs in four Florida counties (Broward, Miam i-Dade, Monroe and Palm Beach) from June 2000 to July 2001. This study included b oth artificial and natural reefs. Johns et al. (2003) found that the economic benefits of n atural reefs were two to one to the artificial reefs in these counties. Some of the to urism activities related to the coral reefs are diving, snorkeling, and fishing. Table 1.1 giv es the number of person days spent on all reefs on these recreational activities. For th e four counties the direct use of the reefs through snorkeling, scuba diving, fishing and glass bottom boats were 5.81, 7.61, 14.73 and 0.14 million person-days respectively. A person -day is defined as one person participating in an activity for a portion or all o f a day. More than half of the 28.29 million person-days spent on the reefs are due to f ishing. Broward had the highest used of the reefs at 9.44 million person-days while Palm Be ach had the least at 4.23 million person-days. Table 1.1 shows that people prefer to have an active experience of the reef as people spend 28.15 million person-days in diving snorkeling and fishing, while only 0.14 million person-days using the glass bottom boa ts.
8 Table 1.1 Number of Person-Days on all Reefs by R ecreational Activity June 2000 to May 2001 (Millions) (Johns et al. 2003 ). Activities Palm Beach Broward Miami-Dade Monroe Snorkeling 0.74 1.09 2.11 1.87 Scuba Diving 1.73 3.85 1.14 0.89 Fishing 1.76 4.45 5.90 2.62 Glass Bottom Boats 0 0.05 0.02 0.07 The four counties benefited from $4.4 billion in s ales from reef related tourism between June 2000 and May 2001, (Table 1.2). It su pported about 71,300 jobs giving a total of $2 billion in annual income. Table 1.2 Economic Contribution of Reef Related E xpenditures June 2000 to May 2001 (Johns et al. 2003). Attribute Palm Beach Broward Miami-Dade Monroe Sales ($millions) 505 2,069 1,297 504 Income ($millions) 194 1,049 614 140 Employment 6,300 36,000 19,000 10,000 Andrews et al. (2005) using the same data from John s et al. (2003) reported that the reefs of Palm Beach, Broward, Miami-Dade and Monroe had a n asset value of $1.4, $2.8, $1.6 and $1.9 billion, respectively. As previously mentioned, recreational fishing amou nted to half of the earning of the four counties (Broward, Miami-Dade, Monroe and Palm Beach) between June 200 and July 2001. Commercial and recreational fishing target reef fishes and spiny lobster for food and sport. In 2001 an estimated 6.7 milli on recreational fishers took 28.9 million fishing trips in Florida catching 172 million fish of which 89.5 million were released or discarded (DOC 2003). Hodges, Mulkey, Philippakos and Adams (2006) reported that fishing and seafood products had an impact of $1.1 billion in 2003 creating an estimated 13,900 jobs. The importance of the FloridaÂ’s coral reefs is imm easurable when it comes to the protection the reefs have provided from hurricane a nd other storms over the years. The
9 reefs have also provided Florida with the attractiv e beaches that tourists and locals enjoy so much. Coral reef plants and animals are importa nt sources of new medicines being develop to treat cancer, arthritis, human bacterial infections, heart disease, viruses, and other diseases (Birkeland 1997b). The future of Florida depends immensely on the cor al reefs. In view of the bleak future facing coral reefs, including the Florida Re ef Tract, many of the aforementioned studies were conducted to investigate the factors a ffecting coral reefs. Similarly, this dissertation bears in mind the need to better under stand reef dynamics, and is an attempt to provide a meaningful approach to utilizing monit oring data. 1.3 Coral Reef Evaluation and Monitoring Project (CREMP) The Florida Keys National Marine Sanctuary (FKNMS) Protection Act (HR5909) designated over 2,800 square nautical miles of coas tal waters (south of Miami (2517.683Â’N, 8013.145Â’W) to the Tortugas Banks (2 436.703Â’N, 8252.212Â’W)), as the Florida Keys National Marine Sanctuary (Figure 1.3) (Porter et al. 2002). This Act requires the cooperation of the U.S. Environmental Protection Agency (EPA), the State of Florida and the National Oceanographic and Atmos pheric Administration (NOAA) to implement a Water Quality Protection Program (WQPP) to monitor seagrass habitats, coral reefs, hardbottom communities, and water qual ity. The WQPP acknowledged the absence of high-quality monitoring data from the ef fort to understand the status and trend of the benthic communities in the sanctuary and the ability to measure the efficacy of any future management actions in the sanctuary. Thus, the objective of the monitoring project is to provide the relevant data to make unb iased, statistically based statements about the status and trend of the benthic marine co mmunities and if possible to identify the causes for and spatial distribution of ecosyste m change in the Florida Keys. The Florida Keys Coral Reef Monitoring Project (CR MP) is one of the components of WQPP. CRMP (Porter 2002) was establi shed with the following aims and objectives: (1) to overcome the spatial and tem poral criticisms of previous studies concerning the subject matter, (2) to rigorously de termine change in coral species
10 richness, bleaching, disease, and relative benthic cover, and (3) to provide the baseline data necessary to evaluate the success of the futur e management actions in the Florida Keys. These objectives are in accordance with the requirements of the Florida Keys National Marine Sanctuary (FKNMS). CRMP monitors t he coral communities, which include sanctuaryÂ–wide spatial and temporal coverag e through repeated sampling, in turn making statistically valid findings to document the status and trends of the coral communities (Wheaton et al. 2001).
11 Figure 1.3 Location of FKNMS and Sampling Sites o f WQPP. (Cartographic services: University of Geor gia)
12 1.3.1 Sampling and Data Collection CRMP sampling and method protocol were developed i n conjunction with EPA, FKNMS, Continental Shelf Associates and the Princip al Investigators (Phillip Dustan PhD., University of Charleston; Walter Jaap, Florid a Marine Research Institute; James Porter PhD., University of Georgia) in 1994 (Hacket t 2002). Forty sampling reef sites were selected in 1994. Thirty-seven of the forty s ites were selected using stratified random sampling EPA E-Map procedure (Overton, White and Stevens 1991). The remaining three sites, Carysfort Reef, Looe Key and Western Sambo, were selected based upon existence of previous monitoring activity. Fo ur sampling stations made up a particular site and were permanently marked in 1995 The first station for each site was located by going to a randomly generated latitude a nd longitude and choosing the closest appropriate reef type (offshore shallow reef, offsh ore deep reef, hardbottom reef, or patch reef). The remaining three stations were selected at adjacent suitable habitat at a minimum distance of 5 meters. Annual sampling for the 40 sites at 160 stations b egan in 1996. In 1999 three additional sites were installed and sampling begun at 12 more stations in the Dry Tortugas. In 2000 the statistical consultants re-e valuated the number of stations and concluded that certain stations could be eliminated while allowing the project to maintain its spatial coverage and robust data set (Wheaton e t al. 2001). Statistical analysis of similarity in stony coral cover was used in the eli mination of one or two of stations in certain sites. Table 1.3 lists the sites and stati ons sampled in the CRMP project. Hypothesis testing was used to identify differences in the proportion of stony coral cover at the four stations within each site. This exerci se reduced the original 160 stations to 111 stations. A similar analysis took place in 200 2 further reducing the number of stations to 105 stations and 37 sites of the origin al 40. The sites eliminated were Rattle Snake (hardbottom), Molasses Keys (hardbottom), and Dove Key (hardbottom). In Dry Tortugas the number of sites (3) and stations (12) remained the same. The location, type of reef, depth of the sites, the number of stations reduction per sites and the sites eliminated are given in Table 1.3. Even though sta tions were eliminated, the permanent
13 markers remain in case further sampling is resumed. In 2001 the managers of the CRMP project decided to expand the sampling strategy by collecting a more comprehensive suite of indicators at 11 of the established 40 sit es (Wheaton et al. 2001). Since there was a change of focus for CRMP, the project was renamed the Coral Reef Evaluation and Monitoring Project (CREMP). Presently, sampling co ntinues on an annual basis at 40 sites (including the 3 sites from Dry Tortugas) com prising of 4 hardbottom, 11 patch, 13 offshore deep, and 12 offshore shallow reef sites.
14 Table 1.3 CREMP Sampling Sites. = 1 station reduction, **= 2 station reduction, ***=Sampling terminated after 2000. Site Habitat Type Geographic Area Depth (ft) Longitude (W) Latitude (N) Turtle** patch Upper Keys 11 to 23 -80.23 25.28 Carysfort** deep Upper Keys 40 to 52 -80.2 25.21 Carysfort** shallow Upper Keys 3 to 11 -80.25 25.2 Rattle Snake*** hardbottom Upper Keys 5 to 6 -80.34 25.16 Grecian Rocks shallow Upper Keys 8 to 21 -80.3 25.1 Porter Patch* patch Upper Keys 13 to 17 -80.34 25.09 Admiral patch Upper Keys 9 to 11 -80.41 25.03 Molasses* shallow Upper Keys 12 to 25 -80.42 25 Molasses** deep Upper Keys 40 to 50 -80.36 24.99 Conch** deep Upper Keys 17 to 21 -80.44 24.95 Conch** shallow Upper Keys 53 to 56 -80.49 24.94 El Radabob** hardbottom Upper Keys 6 to 9 -80.39 25.11 Dove Key*** hardbottom Upper Keys 8 -80.48 25.03 Alligator** deep Middle Keys 34 to 37 -80.61 24.83 Alligator* shallow Middle Keys 12 to 17 -80.66 24.83 Tennessee** deep Middle Keys 44 to 45 -80.74 24.75 Tennessee* shallow Middle Keys 17 to 21 -80.8 24.73 West Turtle Shoal patch Middle Keys 15 to 23 -80.98 24.69 Dustan Rocks* patch Middle Keys 12 to 21 -81.04 24.68 Sombrero shallow Middle Keys 8 to 20 -81.09 24.61 Sombrero** deep Middle Keys 47 to 52 -81.16 24.61 Long Key hardbottom Middle Keys 13 to 14 -80.8 24.79 Moser Channel** hardbottom Middle Keys 12 to 13 -81 .16 24.69 Molasses Keys*** hardbottom Middle Keys 12 to 14 -8 1.21 24.67 Smith Shoal* patch Lower Keys 19 to 26 -81.93 24.71 Jaap Reef* patch Lower Keys 7 to 9 -81.6 24.58 Looe Key* shallow Lower Keys 12 to 25 -81.4 24.55 Looe Key* deep Lower Keys 38 to 43 -81.46 24.54 W. Washer Woman** patch Lower Keys 15 to 25 -81.6 24.54 Eastern Sambo* shallow Lower Keys 4 to 9 -81.65 24.5 Eastern Sambo* deep Lower Keys 43 to 48 -81.68 24.48 Cliff Green** patch Lower Keys 20 to 26 -81.77 24.49 Western Head* patch Lower Keys 26 to 35 -81.82 24.49 Western Sambo* shallow Lower Keys 11 to 17 -81.75 24.47 Western Sambo* deep Lower Keys 43 to 48 -81.71 24.45 Sand Key* deep Lower Keys 24 to 34 -81.93 24.46 Sand Key* shallow Lower Keys 11 to 21 -81.92 24.43 Rock Key shallow Lower Keys 6 to 19 -81.87 24.43 Rock Key** deep Lower Keys 37 to 42 -81.84 24.45 Content Keys* hardbottom Lower Keys 17 to 19 -81.5 24.81 Black Coral Rock deep Dry Tortugas 70 to 75 -83.01 24.69 White Shoal patch Dry Tortugas 15 to 29 -82.91 24.63 Bird Key deep Dry Tortugas 30 to 45 -82.88 24.6
15 Sampling occurs at each CREMP station. The stony coral point count data is obtained from video image analysis. A video is tak en at each of the three transects of the station (Figure 1.4). The total area video is 26.4 square meters per station. From 1996 to 1999, transects were videoed using a Sony CCD-VX3 H i 8-mm analogue video camera programmed to fully automatic settings with two 50 watt artificial lights. Then in 2000 the project upgraded to the Sony TRV 900 4-mm digit al video camera (Porter et al. 2002). Figure 1.4 The Three Transects Conducted at Each Station. Hackett (2002) concluded that both cameras gave sim ilar results with the point count procedure. In fact, the technique of monitoring of coral reef is often used to estimate coral cover through camera and video technology (Pa vlov et al. 2004, Wielgus et al. 2004). These studies use the point count method on the photos or video-frames to obtain the coral cover. Many studies have documented the effectiveness and disadvantages of
16 videography as a coral sampling method. Carleton a nd Done (1995) and Lirman et al. (2007) concluded that with the proper procedures, v ideo sampling provides a quantitative measure of spatial variability and temporal change in benthic communities on coral reefs. This provides a cost saving method in sampling larg er areas and allows for a permanent record of the sampling sites. A limitation of vide ography is taxonomic resolution; video sampling does not provide a good estimate of the ab undance of rare species. Furthermore, concluded that the video-mosaic can be problematic in exposing juvenile corals (Lirman et al. 2007). Nevertheless, for a m onitoring project whose focus is to document temporal change over a large area, videogr aphy has proven quite efficient. Artificial light is used whenever necessary to ens ure quality frames. A convergent laser light system enables the videograp her to maintain the camera at a constant distance of 40 cm above the reef surface. From prior testing the distance of 40 cm is appropriate for identification of benthos and producing approximately 25 m2 of benthos to be sampled at each station (Hackett 2002 ). The videographer films at a constant swimming speed of about 4 meters per minut e producing about 9000 video frames per transect. Camera settings are optimized with progressive scan and sport mode to maximize the quality of individual frames. Befo re filming, a chain is laid on the surface of the reef directly underneath each transe ct as seen in Figure 1.4. In addition, the videographer films a clapper board before filming e ach transect to record the date and location of each film. The video filmed a field 40 cm wide and 22 m long (length of transect). Representative images for all transects are framegrabbed. These images form a mosaic of each transects. Figure 1.5 shows three mo saics of the same transect from the years 1996, 1999 and 2004. The stony coral seen in these transects is Acropora palmata These mosaics are then used to obtain point count d ata. From 1996 to 2000, the procedure of forming the mosaic is as follows: Abou t 120 frames are digitized to cover the complete coverage of the sea floor since there is a considerable overlap of the video frames. The subset of frames is grabbed based on sw im speed. From these only about 60 are selected by a trained analyst so that there is no more than 15% overlap with the
17 previous images. Since 2001, the mosaics have been formed using the image processing tool, RavenView, which uses all of the 9000 frames. Figure 1.5 Three Mosaics of the Same Transect for 1996, 1999 and 2004. From the complete mosaic, a subset of abutting ima ges is selected and given to the analyst. The analyst opens each image in a cust om software application, PointCount, for coral reefs. The analyst overlays ten random p oints on each image using the software. These points are then identified by the analyst int o categories of benthic taxa (stony coral species, octocoral, zooanthid, sponge, seagrass and macroalgae) or substrate. Whenever a stony coral cannot be positively identified it is r ecorded as Â“ScleractinaÂ”. The data are saved under a comma-separated value file (.csv). T he analyst then opens the file and performs a quality assurance before forwarding it t o the projectÂ’s data manager. The manager in turn does another quality assurance chec k before entering the data into the master Microsoft Access database (Porter et al. 200 2). The station species inventory data are obtained fr om each station by two observers (Wheaton et al. 2001). These two observe rs conduct a 15 minutes simultaneously timed inventory within the 2 x 22 me ters area of each station (Fig. 1.6).
18 They both record the number of species present, pre sence of disease/bleaching in stony corals, and the abundance of the long-spined urchin s ( Diadema antillarum ). At the end of recording they take 5 minute to cross-check their d ata sheets to identify any discrepancies. Figure 1.6 Schematic for Station Species Inventor y Survey. 1.3.2 Results of Statistical Analyses Herein, a summary of the results from the works of Yanev and Tsokos (2003a, b, 2004), Tsokos, Hoare and Yanev (2005), Tsokos, et a l. (2006a), and Tsokos, Hoare and Yanev (2006b) for Florida Fish and Wildlife Researc h Institute (FWRI) managers of the CREMP is presented. The analysis performed on the point count data are descriptive statistics and hypothesis testing. Hypothesis testing was conduct ed at regional and habitat levels using nonparametric Friedman Repeated-Measure Two-way Ana lysis of Variance by Ranks
19 and proportion testing at the station level. Figur e 1.7 gives the histogram of the percentage stony coral from 1996 to 2006 by region. The percentage coral cover has a decreasing temporal trend across all the regions. Dry Tortugas is the area with the highest percentage coral cover while the middle key s have the lowest. Figure 1.7 Histogram of Percentage Stony Coral Co ver: 1996 to 2005 by Region. 0 5 10 15 20 25Percent Cover Sanctuary Wide 11.8911.369.607.427.537.387.357.166.626.70 Upper Keys 11.6110.648.526.796.666.666.716.686.446.16 Middle Keys 6.836.825.885.035.105.155.605.204.715.23 Lower Keys 15.2614.6912.659.349.629.268.868.717.937.98 Dry Tortugas 18.8917.3717.4514.0813.4211.7011.40 1996199719981999200020012002200320042005 Figure 1.8 is the histogram of the percentage ston y coral cover from 1996 to 2006 by habitat. Patch reefs show the highest coral cov er, while the hardbottom reefs have the lowest percentage stony coral cover. The coral cov er in the hardbottom reefs is very poor. Overall a decreasing temporal trend of perce nt stony coral cover across all habitat type is obvious and the decreasing change occurred mostly from 1996 to 1999.
20 Figure 1.8 Histogram of Percent Stony Coral Cover : 1996 to 2005 by Habitat. 0 5 10 15 20 25Percent Cover Patch Reefs 19.9818.8017.8715.4815.9615.7015.5915.3214.3215.33 Deep Reefs 6.736.784.463.743.753.713.603.512.893.00 Shallow Reefs 12.1211.559.155.485.515.195.455.174.974.30 Hardbottom 1.921.921.561.791.381.891.181.310.941.20 1996199719981999200020012002200320042005 Table 1.4 gives the result of hypothesis testing p erformed at regional and habitat levels to check if there exists any statistical cha nge in mean stony coral cover over the years from 1999 to 2005. We tested the null hypoth esis that the mean stony coral cover is the same in all years from 1999 to 2005 as oppos ed to the alternative hypothesis that the mean stony coral cover is different in at least one of the years. The results show that there was a statistical significant change in mean stony coral cover in all the different types of reefs at 0.05 a = This means that there exists statistically at least one year whose mean stony coral cover is different than the rest of the years. By region, the upper keys and middle keys showed no statistical signific ant change in mean stony coral cover at 0.05 a = This means that statistically there is no eviden ce that the mean stony coral cover is different for any of the years tested. Bu t there is statistical evidence that there exist at least one year whose mean stony coral cove r is different than the rest of the years in the lower keys and the sanctuary.
21 Table 1.4 Hypothesis Testing for Change in Mean S tony Coral Cover: 1999 to 2005. Region and Statistical Significance Habitat 0.05 a = Sanctuary Yes Upper Keys No Middle Keys No Lower Keys Yes Dry Tortugas Yes Patch Reefs Yes Deep Reefs Yes Shallow Reefs Yes Hardbottom Yes We tested for statistically significant change in stony coral cover per station for consecutive years (2003 vs. 2002, 2003 vs. 2004 and 2004 vs. 2005). The null hypothesis that the stony coral proportion in one year is the same as in the other year as oppose to the alternative hypothesis that they differ was tes ted. Figure 1.9 shows the pie chart for the percentage number of stations that showed stati stical significant change (loss, gain or no change) for the three pairs of years tested. Th e number of stations that lost stony coral cover in consecutive years have fallen in 2005 vs. 2004 as compared to the other pairs of years. The number of stations tested that showed a gain in stony coral cover fell in 2004 vs. 2003 from the pair in 2003 vs. 2002 but increas ed in 2005 vs. 2004. In the case of testing for no change in stony coral cover, the per centage was the same for pairs 2003 vs. 2002 and 2005 vs. 2004 but higher in 2004 vs. 2003.
22 Figure 1.9 Percentage Change in Stony Coral Cover by Station. 2003 vs 200267% 10% 23% 2004 vs 200323% 6% 71% 2005 vs 200417% 16% 67% Loss Gain Equal The point count data were also used in calculating the Shannon-Wiener diversity index. Shannon-Wiener diversity index is described as: lnln 1 'ln 1 s nnff ii s i Hpp ii n i = =-== = (1.1) where / ii pfn = is the proportion of points where the ith stony coral species is identified, i f is the abundance of the ith stony coral species, s is the number of species present and n is the total abundance of all species in the sampl e. The Shannon-Wiener diversity index is a measure of the how diverse a habitat is by considering both the species present and their abundance. Using equation (1.1) and the Jackknifing method (explained further in section 3.2), we estimate the Shannon-Wiener div ersity index and construct the 95%
23 confidence interval for the index under the assumpt ion of normality. Figure 1.10 and Figure 1.11 shows the histogram of the Shannon-Wien er diversity index estimates for the years 1996 to 2005 by region and by habitat respect ively. Figure 1.10 shows that the highest Shannon-Wiener diversity index estimates oc cur in the middle keys while the lowest index occurs in the upper keys. There was a decrease in the Shannon-Wiener diversity index estimates from 1996 to 1999 for all regions but since then it seems to stabilize. Figure 1.10 Histogram of Shannon-Wiener Diversity Index: 1996 to 2005 by Region. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Shannon-Wiener Index Sanctuary 2.312.302.191.972.062.042.162.082.102.08 Upper Keys 1.941.901.771.581.711.551.741.541.501.56 Middle Keys 2.602.592.4184.108.40.206.282.282.292.35 Lower Keys 2.2220.127.116.111.991.982.102.052.102.01 Dry Tortugas 1.991.881.841.891.981.771.74 1996199719981999200020012002200320042005 Figure 1.11 shows no decrease across the habitats a s was evident across regions for the yearÂ’s from 1996 to 1999. Hardbottom reefs seem to have the highest Shannon-Wiener diversity index estimates. Patch reefs had the low est Shannon-Wiener diversity index estimates.
24 Figure 1.11 Histogram of Shannon-Wiener Diversity Index: 1996 to 2005 by Habitat. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Shannon-Wiener Index Patch Reefs 1.851.771.741.671.851.831.921.851.881.89 Deep Reefs 2.372.352.221.901.952.072.082.132.132.01 Shallow Reefs 18.104.22.168.941.981.922.071.981.941.95 Hardbottom 1.932.502.492.242.652.492.042.152.452.39 1996199719981999200020012002200320042005 Figure 1.12 shows the Shannon-Wiener diversity ind ex estimate plus the 95% confidence interval of the true Shannon-Wiener dive rsity index of the Sanctuary region from 1996 to 2005. To create the confidence interv al we used the jackknifing method. From this method, we were able to obtain pseudovalu es which are used to estimate the Shannon-Wiener diversity index and to carryout infe rences on the true value of the Shannon-Wiener diversity index. In constructing th e 95% confidence interval of the true value of the Shannon-Wiener diversity index, we ass ume that the underlying probability structure of the pseudovalues is the normal distrib ution. One can see that there was a decrease in the Shannon-Wiener diversity index from 1996 to 1999; there was a slight increase from 1999 to 2002, and then it seems to st abilize.
25 Figure 1.12 95% Confidence Interval of the Shanno n-Wiener Diversity Index for the Sanctuary Region. 1.00 1.50 2.00 2.50 3.00Shannon-Wiener Index Jacknife Estimate 2.312.302.191.972.062.042.162.082.102.08 1996199719981999200020012002200320042005 For the species inventory data, we performed hypot hesis testing and constructed a 95% confidence interval for each species of stony c oral species present. The null hypothesis was that the number of stations where th e stony coral species was present didnÂ’t change for the years tested. We considered t he following time periods: 2002-03 vs. 1996-01, 2003-04 vs. 1996-02, and 2004-05 vs. 199603. Table 1.5 shows the summary of the results for the hypothesis testing for all t hree time periods. It gives the number of species that had statistical significant negative, positive or no change in the number of stations.
26 Table 1.5 Hypothesis Testing Results for Species Richness. Number of Species Species 2002-2003 vs. 1996-2001 2003-2004 vs. 19962002 2004-2005 vs. 1996-2003 Significant Decrease 14 13 11 Significant Increase 1 1 1 No Change 24 25 27 None Conclusive Test 7 7 7 Total 46 46 46 For the diseases and bleaching data, we tested hyp othesis for the change in percent stations affected by white disease, black b and, and other diseases, and bleaching for the time periods: 2002-03 vs. 1996-01, 2003-04 vs. 1996-02, and 2004-05 vs. 199603. We constructed a 95% confidence interval for t he true number of stations affected. Table 1.6 shows the result for the three time perio ds. Interestingly, white diseases, bleaching and black band had a statistically signif icant decrease in the last period tested while the category involving other diseases show no change. Table 1.6 Hypothesis Testing and Confidence Inte rvals for Change in Number of Stations with Incidence of Disease and Bleaching. Significant decrease at a = 0.05 ** Significant increase at a = 0.05. p-value Percent Presence 95% Confidence Interval 2002-2003 1996-2001 Change in Number of Stations Other** 0.00 87.14 61.59 0.193 0.318 White** 0.00 77.14 50.95 0.190 0.334 Bleach** 0.02 71.43 62.38 0.015 0.166 Black 0.64 6.19 7.46 -0.054 0.029 p-value 2003-2004 1996-2002 Other** 0.00 77.62 65.58 0.051 0.189 White 0.33 60.00 55.92 -0.038 0.119 Bleach* 0.00 44.76 66.12 -0.292 -0.135 Black 0.27 4.76 7.21 -0.062 0.013 p-value 2004-2005 1996-2003 Other 0.20 72.86 67.98 -0.022 0.120 White* 0.00 43.33 57.50 -0.220 -0.064 Bleach* 0.00 32.38 64.64 -0.397 -0.249 Black* 0.01 2.38 7.14 -0.078 -0.018
27 Figure 1.13 shows the percentage number of station s with incidence of white disease, black band, bleaching and other diseases f rom 1996 to 2005. Figure 1.13 confirms there is a decrease in the white diseases, bleaching and black band incidence per stations in the last three years. But at the same time the incidence of white diseases, bleaching and other disease remain high. Figures 1 .13 also show that there may be some cyclic trend, which may become more obvious as more data are obtained over the coming years. Figure 1.13 Stations with Incidence of Disease an d Bleaching, 19962005. 0 10 20 30 40 50 60 70 80 90 100 Other 11.4343.8164.7687.6271.4390.4889.5284.7670.4875.24 White 4.7644.7665.7164.7653.3372.3885.7168.5751.4335.24 Bleach 51.4359.0580.9578.1066.6738.1088.5754.2935.2429.52 Black 6.673.8118.104.768.572.865.716.672.861.90 1996199719981999200020012002200320042005Percent of Stations Affected The main focus of this dissertation is to contribu te to the statistical analysis of the stony coral data obtained from the coral reef evalu ation and monitoring project (CREMP), the project is described in section 1.3. It is imperative to be able to statistically investigate the true status of the st ony coral cover in a monitoring project. The analysis of the CREMP data by the works of Yane v et al. (2003a, b, 2004) and
28 Tsokos et al. (2005, 2006a, b) was limited to only descriptive and non-parametric analysis when analyzing the percent coral cover. T his being a first approximation technique, it is possible to analyze the data with more powerful analysis techniques if the underlying probability structure of the data can be identified. In the case of the ShannonWiener diversity index using the jackknifing method possesses a limitation in that the pseudovalues, whose average is used as an estimate for the Shannon-Wiener diversity index, are assumed to follow a normal distribution. This assumption is used in the construction of the 95% confidence interval of the true Shannon-Wiener diversity index. These two and other concerns addressed in this stud y. 1.4 Focus of Chapter Two To accomplish the objectives of CREMP, a thorough and complete analysis of the statistical properties of the stony coral cover pro portions is vital. We address these objectives in this chapter. We shall analyze the st ony coral cover proportions data of all the stations within the Sanctuary Region from 1996 to 2006. The main purpose is to identify the probability density function (pdf) of stony coral cover proportions in the Florida Keys and to investigate if the reported mea n values of the coral cover in the Technical Reports (Yanev et al. 2003a, b, 2004, Tso kos et al. 2005, 2006a) are good estimates of the true mean stony coral cover of the sanctuary region of CREMP over the years of the project. In addition, the 90% and 95% confidence intervals for the true median and the true mean of the population stony co ral cover are given. A comparison of the Â“a proposed methodÂ” and the Â“nave methodÂ” will be made with respect to the median confidence intervals. Finally, comparisons of the mean confidence interval obtained using the nave method, the Cox method and a propos ed method are also made. 1.5 Focus of Chapter Three A focus of this chapter is to investigate the appl icability of the normal probability distribution assumption made on the pseudovalues ob tained from the jackknifing
29 procedure for the Shannon-Wiener diversity index us ed in the works presented by Yanev et al. (2003a, b, 2004), and Tsokos et al. (2005, 2 006a) for the CREMP. The normality assumption was made when we constructed the 95% con fidence interval for the true Shannon-Wiener diversity indices for the entire san ctuary for years from 1996 to 2005. The validity of normality with respect to the 12 st ations in the Dry Tortugas area of the CREMP is also investigated. The accuracy of the co nfidence interval is investigated using the bootstrapping resampling procedure. Testi ng for normality of the pseudovalues is done using the Shapiro-Wilks normality test. We also propose in this chapter that the underlyin g probability structure of the species abundance be found if possible so that it g ives a better approximation of the true diversity index. Once the probability distribution of the species abundance is fitted then we can use the probability distribution instead of the species abundance data in obtaining as estimate of the true Shannon-Wiener diversity in dex or true SimpsonÂ’s diversity index for the entire sanctuary from 1996 to 2006. The Sha nnon-Wiener diversity index for the two-parameter lognormal probability distribution is obtained from literature under Shannon entropy. We solved the SimpsonÂ’s diversity index for the two-parameter lognormal probability distribution in this study. 1.6 Focus of Chapter Four The aim of this chapter is to develop the best pos sible estimate of the probability distribution of the jackknifing pseudovalues for th e sanctuary using the nonparametric kernel density method. Once the best kernel density estimate is obtained, a comparison with the parametric approach under the normal assum ption will be made. We compare the mean, standard deviation, standard error and th e 95% confidence interval of the true mean.
30 1.7 Focus of Chapter Five The aim of this chapter is to develop a predictive statistical model of stony coral cover proportions. Such a model would identify the attributable variables that influence the stony coral cover data of the lower Florida Key s. Having the attributable variables for different coral reef areas, we would be able to pre dict an estimate of the stony coral cover of that area.
31 Chapter 2 Parametric Analysis of Stony Coral Cover From the F lorida Keys 2.1 Introduction Coral reef habitats have been, and continue to be, the subject of many research studies. The main goals of these studies are to in vestigate and document the health status of these habitats. Unfortunately many of the publi shed studies are short in duration and narrow in scope. Such limitations have posed many difficulties in analyzing existing data for temporal or spatial differences and trends. Th e main difficulty is a limited sample size. Small sample size limits the investigation o f the underlying probability distribution of such data. Such information is invaluable in pa rametric analysis. If the probability distribution function is not easily identified, the tendency is to use non-parametric analysis to analyze the data. Lirman et al. (2007) used Kruskal-Wallis test to compare the percent cover of eight main benthic categories among his different survey methods. Yanev et al. (2003a, b, 2004) and Tsokos et al. (2 005, 2006a) used the nonparametric Friedman Repeated-Measure Two-way Analysis of Varia nce by Ranks to detect the differences in percent stony coral cover over the y ears. Non-parametric tests are best used when the distribution of the data is unknown o r it cannot be safely detected; however, non-parametric tests are less powerful tha n parametric tests. On the other hand, if parametric analyses are prematurely used, they h ave less power than the nonparametric analyses. While scientists have used dat a transformations to employ parametric analysis, the most widely used parametri c analysis seems to be ANOVA, which assumes normality, homogeneity of variance, a nd random independent samples. Many follow suggested transformations given by vari ous authors (Sokal and Rohlf 1995,
32 Zar 1996, Hayek and Buzas 1997, Krebs 1999) to achi eve the assumptions required for parametric analysis. Rogers, Gilnack and Fitz (198 3), Carleton and Done (1995), Murdoch and Aronson (1999), and Wielgus et al. (200 4) used the arcsine transformation to be able to use ANOVA. Brown et al. (2004) used the arcsine-square root transformation to be able to use the paired t-test. Pante et al. (2007), on the other hand, used the log transformation to apply the t-test to test for change in percent cover of stony coral between 1991 and 2004. It is imperative to k now the probability distribution of percent coral cover. This would ensure that the pr oper statistical test is employed and ecosystem managers could make more meaningful infer ences and better managerial decisions. With the development of convenient reliable sampli ng techniques, such as videography, monitoring programs such as CREMP, Cor al Reef Assessment and Monitoring Program (CRAMP) in HawaiÂ’I, and the Aust ralian Institute of Marine Science (AIMS) monitoring program are sampling larg er areas of reefs over longer periods of time. Monitoring of coral reefs to est imate coral cover through camera and video technology (Pavlov et al. 2004, Wielgus et al 2004) at small scale is becoming more common. These studies use the point count met hod on the photos or video-frames to obtain coral cover values. Many studies have do cumented the effectiveness and disadvantages of videography as a coral sampling me thod. Carleton and Done (1995) and Lirman et al. (2007) concluded that proper vide o sampling procedure provides a quantitative measure of spatial variability and tem poral change in benthic communities on coral reefs. This provides a cost saving method for sampling larger areas and produces a permanent record of the sampling sites as shown i n Figure 1.5 (see Chapter 1). It can be seen that Acropora palmata was thriving in this transect from a CREMP station in 1996 but was nonexistent in 1999. The mosaic shows the return of this very important stony coral species in 2004. One limitation of vid eography is the taxonomic resolution. As a result, it does not give a good estimate of th e presence of rare species. Lirman et al. (2007) also concluded that the video-mosaic did hav e problems in revealing juvenile corals. Nevertheless, for a monitoring project whos e focus is to document temporal change over a large area, videography is a time sav ing method that is economically
33 feasible. It is still important to identify the pr obabilistic structure of percent stony coral cover. Larger samples will facilitate the investig ation of the probability distribution function of percent stony coral cover. The data used in this study comes from the Florida Keys Coral Reef Evaluation and Monitoring Project (CREMP). To accomplish the objectives of the CREMP a thorough and complete analysis of the statistical p roperties of the stony coral cover proportions is vital. We address these objectives in this study. We shall analyze the stony coral cover proportions data of all the stati ons within the Sanctuary Region from 1996 to 2006. The main purpose is to identify the probability density function (pdf) of stony coral cover proportions in the Florida Keys a nd to investigate if the reported mean values of coral cover in the Technical Reports (Yan ev et al. 2003a, b, 2004, Tsokos et al. 2005, 2006a) are good estimates of the true mean s tony coral cover of the sanctuary region of the CREMP over the years of the project. In addition, the 90% and 95% confidence intervals for the true median and the tr ue mean of the stony coral cover are given. A comparison of the Â“proposed methodÂ” and t he Â“nave methodÂ” will be made with respect to the median confidence intervals. F inally, comparisons of the mean confidence interval obtained using the nave method the Cox method and a proposed method are also made. 2.2 Descriptive Statistic: Proportion of Stony Co ral Cover The stony coral cover proportions are obtained fro m the point count method used by the CREMP. The mosaics used in the point count method come from 37 sites over the Florida Keys. These include transects from 105 sta tions for the years 1996 to 2005 and 103 stations in 2006. The data demonstrate right s kewness and display leptokurtic behavior as shown by the histogram given in Figure 2.1. The stony coral cover proportion data for the other years from 1996 to 20 05 display similar histograms as 2006, (Fig. 2.1).
34 Figure 2.1 Histogram for the Stony Coral Cover Pro portions for 2006. 0 10 20 30 40 50 0.000.050.100.22.214.171.124 Percent 0 10 20 30 40 50 0.000.050.100.126.96.36.199 Percent Table 2.1 gives the descriptive statistics of the stony coral cover proportions from 1996 to 2006. A skewness measure that is greater t hat zero means the data are skewed to the right and a value less than zero means it is sk ewed to the left. Thus, Table 2.1 presents further evidence that the stony coral cove r proportions data from 1996 to 2006 displays right skewness as the skewness values rang e from 1.17 to 1.81. A kurtosis measure greater than zero means the distribution di splays leptokurtic behavior and a value less than zero means the distribution display s a platykurtic behavior. A kurtosis value of zero means the distribution displays mesok urtic behavior. Leptokurtic behaviors in the stony coral proportions data are also eviden t from Table 2.1, since the kurtosis values are all greater than zero.
35 Table 2.1 Descriptive Statistics for Proportion St ony Coral Cover. The boxplots, shown below by Figure 2.2, display t he spread of the stony coral cover proportions data from 1996 to 2006. The boxp lots show that many of the data values are found between 0 and 0.1. The whiskers o f the boxplots are longer towards the larger values, further evidence that the distributi on of the stony coral cover proportions data are skewed to the right. Thus, any attempt at identifying a probability density function for these data sets must be one that takes into consideration right skewness. 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2 006 Mean 0.119 0.114 0.096 0.074 0.075 0.074 0.073 0.072 0.066 0.067 0.061 Median 0.091 0.089 0.057 0.047 0.047 0.041 0.045 0.042 0.035 0.039 0.030 Std Error 0.010 0.009 0.009 0.007 0.008 0.007 0.008 0.008 0.007 0.008 0.008 Std Dev 0.101 0.094 0.092 0.076 0.078 0.076 0.080 0.081 0.076 0.079 0.079 Kurtosis 0.988 1.084 1.734 1.353 1.485 1.924 2.211 2.361 2.871 2.019 2.088 Skewness 1.171 1.236 1.473 1.487 1.509 1.600 1.707 1.767 1.866 1.746 1.771
36 Figure 2.2 Boxplots for Stony Coral Cover Proport ion Data from 1996 to 2006. 19961997199819992000200120022003200420052006 0 0.1 0.2 0.3 0.4 0.5Year Proprotion 19961997199819992000200120022003200420052006 0 0.1 0.2 0.3 0.4 0.5Year Proprotion 2.3 Procedure in Fitting a Three Parameter Lognor mal Probability Density Function In this section we shall introduce the theory used in estimating the parameters of the 3-parameter lognormal probability density funct ion and evaluating its fit to the subject data. A random variable has the three parameter lognormal probability dist ribution function if its probability density function is giv en by: 2 21/211ln() (2)()exp () 2 0 x xifx fx ifxqm psqq s q ----> = n Â£ (2.1) is the threshold parameter, is the scale parameter and is the shape parameter.
37 2.3.1 Maximum Likelihood Estimation Procedure The method of maximum likelihood estimator (MLE) i s considered to be more robust than other methods, such as methods based on moments or quantiles. In general MLE possesses many desirable statistical properties such as consistency, invariant, asymptotical normality, efficiency, and sufficiency The MLE method was the method of choice to approximate the parameters of the three p arameter lognormal probability density function (equation (2.1)) considered here. Even though the MLE for the three parameter lognor mal pdf seems not to have the usual asymptotic efficiency, Calitz (1973) conc luded that the variance is much smaller than that of the estimators based on moment s and quantiles, and is close to the theoretical asymptotic variance with large samples. Through a Monte Carlo study Calitz (1973) compared the estimates using the methods of MLE as described by Cohen (1951) method of generalized moments and the method of qua ntiles for 50 samples of size 100 with varying parameters. Calitz (1973) concluded t hat the method of MLE is much better than the other methods and that it should be consid ered as the prime method for estimating the parameters of the three parameter lo gnormal distribution. He strongly suggested using the procedure as shown by Cohen (19 51) since other studies did not have success with alternative procedures. Cohen (1951) w as one of the first to suggest the use of a local MLE now known as the local maximum likel ihood estimator (LMLE). The MLE of the three unknown parameters q m and 2 s maximizes the likelihood function of a random sample. The likelih ood function, 2 1 (,...,;,,) nLxx qms of the random sample consisting of n independent observations 1 (,...,) n xxx = from n independent random variables 1 (,...,) n XXX =each with the three parameter lognormal probability density function ( () fx ) is as follows:
38 2 1 (,...,;,,) nLxxqms = 2 2/21 1 1ln() 1 (2)()exp 2 0 n n n i i i i i ix x ifallx ifsomexqm psq q s q -= = ----> n Â£ (2.2) Without loss of generality, assume that the i x s are in increasing order, such that 1 x is the smallest data value. Hill (1963) has shown that arb itrarily large likelihoods can be achieved by allowing Âˆ q to converge on 1 x Thus, the true maximum likelihood estimates should be 1 Âˆ x q = Âˆ m =- and 2Âˆs = regardless of the sample. This leads to inadmissible estimates for the parameters. However, Cohen (1951) had already shown that a localized estimate of the threshold paramete r q was sufficient in the identification of the three parameter lognormal. Calitz (1973), Co hen and Whitten (1980), and Chen (2006) and others have since investigated the valid ity of the estimates from such procedure. CohenÂ’s procedure (Cohen 1951) identifi es the LMLE for the threshold parameter q and then finds the MLE for the parameters m and s To find these estimates we take the logarithm of the likelihood f unction 2 1 (,...,;,,) nLxx qms equation (2.2), and then set the partial derivatives with re spect to q m and s equal to zero. The log-likelihood function is given by: 2 1 ln(,...,;,,) nLxxqms = 2 2 111 ln(2)lnln()(ln()). 22nn ii iin nxx psqqm s==------(2.3) The partial derivatives give: 2 1ln1 (ln())0 n i iL xqm ms= =--= , (2.4) 2 3 1ln1 (ln())0 n i iLn xqm sss= =-+--= (2.5) and
39 2 11ln() ln11 0 nn i ii iix L xxqm qsqq==- =+= --. (2.6) Solving the first two equations (2.4) and (2.5), gi ves the estimates for m and 2 s respectively: 11 Âˆ Âˆ ln() n i ix n mq = =(2.7) and 2 22 1111 ÂˆÂˆ Âˆ (ln())ln() nn ii iixx nn sqq == =--. (2.8) Substituting equations (2.7) and (2.8) in equation (2.6), we have 2 11 2 1111 Âˆ ln(ÂˆÂˆÂˆ ()ln()(ln())n i innn iii iiix nxxxqqqq==== rr +---1 Âˆ ln() 0. Âˆn i i ix n xq q=-= (2.9) Equation (2.9) is solved iteratively for the local maximum likelihood estimate (LMLE), Âˆ q for q Only admissible roots for which Âˆ q is less than 1 x are considered. According to Cohen and Whitten (1980), usually only one such roo t is found. In cases where there are more than one admissible root, the root that gives the closest agreement between the expected value of the three parameter lognormal dis tribution and the mean of the data, is taken as the estimate. Once the estimate Âˆ q is found then the estimates of m and 2 s are found using equations (2.7) and (2.8). Cohen and Wh itten (1980) concluded that the LMLE is suitable whenever the skewed value of the d ata is greater than one and suggested other modified maximum likelihood estimat ors when the data has a skewed value less than one.
40 2.3.2 Goodness-of-fit Procedure Having calculated the LMLE and MLE parameter estim ates for the threeparameter lognormal distribution, one may ask how w ell this models the stony coral cover proportion data. It is important to use the g oodness of fit technique that examines how well the sample agrees with a given probability distribution of its population. For a continuous probability distribution, the appropriat e goodness of fit technique is the empirical distribution function (edf). Figure 2.3 s hows the edf used in the statistics that will be described below and the assumed probability cumulative curve for the 2007 stony coral cover proportion data. Figure 2.3 Cumulative Distribution Function Plot f or 1997 Stony Coral Cover Proportion Data. Lognormal Curve:Thresh=-0.02 Shape=0.7268 Scale=-2. 26 0 20 40 60 80 100 0.00.10.20.30.4 Cumulative Percent cdf edfLognormal Curve:Thresh=-0.02 Shape=0.7268 Scale=-2. 26 0 20 40 60 80 100 0.00.10.20.30.4 Cumulative PercentLognormal Curve:Thresh=-0.02 Shape=0.7268 Scale=-2. 26 0 20 40 60 80 100 0.00.10.20.30.4 Cumulative Percent cdf edf
41 The edf is a step function that takes a step of hei ght 1 n at each observation of the sample. The edf statistic measures the differences between the edf and the given cumulative distribution function (cdf) values whose parameters are either known or unknown (which are estimated from the sample). The edf statistic i s based on the differences between the edf from the proportion stony coral cover and the t hree-parameter lognormal distribution, equation 2.1, with estimates for the threshold, sca le and shape parameters from the sample. The edf is defined for a set of n independent observations 1 ,..., n XX with a common cumulative distribution function () Fx such as the one from the three-parameter lognormal distribution. Suppose that the sample of size n is arranged in increasing order (1)(2)() ... n XXX <<<, then the empirical distribution function () n Fx is defined as follows: (1) ()0,n FxxX =< ()(1) (),1,...,1 niii FxXxXin n + =Â£Â£=() ()1,. nn FxXx =Â£ At any value x () n Fx is the proportion of observations that are less th an or equal to x () Fx is the theoretical probability of an observation th at is less than or equal to x It is expected that () n Fx estimates () Fx which is a consistent estimator of () Fx as n meaning that ()()0 P nFxFx - In order to compute the edf statistics, the proba bility integral transformation () UFX = is used. If () FX is the cdf of X the random variable U is uniformly distributed between 0 and 1. There are several goodness of fit tests that are b ased on edf. They are based on the vertical differences between () n Fx and () Fx These tests are usually divided into two different classes: the supremum and the quadratic. We use the well known KolmogorovSmirnov statistic D which is in the supremum class and from the quadr atic class, we use both the Cramer-von Mises statistic 2 W and the Anderson-Darling statistic 2 A
42 The Kolmogorov-Smirnov statistic D is defined as: sup()()max(,). xn DFxFxDD +=-= This statistic is calculated by taking the maximum value of D + and D D + is the largest vertical difference between the edf and the cdf whe never the edf is greater than the cdf. D is the largest vertical difference between the edf and the cdf whenever the edf is less than the cdf. () maxiii DU n+ =() 1 maxiii DU n =In the case of the quadratic class the general sta tistic is defined as: 2 (()())()() n QnFxFxxdFx y -=where () x y is a suitable function which gives weights to the s quared difference 2 (()()) n FxFx For the Cramer-von Mises statistic 2 W () x y is equal to 1. The Cramer-von Mises statistic 2 W is therefore: 22 (()())()(). n WnFxFxxdFx y -=It is computed as: 2 2 () 1 211 212 n i ii WU nn ==-+ For the Anderson-Darling statistic 2 A () x y is equal to () 1 ()1() FxFx r The Anderson-Darling statistics 2 A is: ()1 22 (()())()1()(). n AnFxFxFxFxdFx -r =-It is computed using: 2 ()11 (21)(lnln(1)). iniAniUU n=r =---+
43 It is very important to note that if the parameter s are estimated from the sample, the edf techniques give different p-values than if the parameters are known. When the true parameters are known, the () i U follow the uniform distribution but that is not the case when the parameters are estimated from the sample. In such a case, the formulae are the same but the distribution of the edf statistics wil l depend on the distribution being tested, the parameters estimated, the method of estimation and the sample size. To obtain the critical values, simulation is used. Many tables of critical value have been created, e.g. Stephens (1986) and Chen (2006). The null hypothesi s for general goodness-of-fit tests based on edf is that the random sample of n X -values comes from () Fx The alternative hypothesis is that the random sample of n X -values do not come from () Fx Chen (2006) showed that the modified Kolmogorov-Smirnov statistic nD the Cramer-von Mises statistic 2 W and the Anderson-Darling statistic 2 A are very accurate goodness-offit tests for the three-parameter lognormal when th e estimates are found using the LMLE and the MLE estimators. He concluded that Anderson -Darling statistic 2 A was the most powerful edf test followed by Cramer-von Mises stat istic 2 W and then the modified Kolmogorov-Smirnov statistic nD for the three parameter lognormal distribution. Eve n though the Anderson-Darling statistic was the most powerful of the three, the other two functioned very well. The null hypothesis of these tests is that the edf and cdf are the same. Thus, low values of the Anderson-Darling sta tistic 2 A Cramer-von Mises statistic 2 W and Kolmogorov-Smirnov statistic nD give evidence that the distribution being tested fits the subject data properly. 2.4 Results in Fitting a Three Parameter Lognormal Probability Density Function In this section we shall show that the three param eter lognormal probability density function will fit the stony coral cover pro portion data quite well. Other probability density functions, such as Weibull, bet a and gamma, were considered but did not give a good fit to the stony coral cover propor tion data. The logarithm transformation
44 has been used to achieve normality in many geologic al data sets. Pante et al. (2007) used it to transform the percent coral cover in his stud y to obtain normality. In what follows, we will present the results obtained in estimating the parameters, using the MLE procedure, and the evaluation of the goodness-of fi t of the 3-parameter lognormal probability density function to the stony coral cov er proportion data in the Florida Keys from CREMP. Using the procedure discussed above, the three par ameters of the lognormal probability distribution were estimated using the s tony coral cover proportions data from 1996 to 2006. The estimates for the parameters are given in Table 2.2 for all the years from 1996 to 2006. Table 2.2 Parameter Estimates for the Three-Parame ter Lognormal Probability Distribution. Year Threshold Scale Shape 1996 -0.014 -2.33 0.835 1997 -0.020 -2.26 0.727 1998 -0.008 -2.64 0.913 1999 -0.004 -2.99 0.983 2000 -0.003 -3.05 1.060 2001 -0.004 -3.01 0.987 2002 -0.003 -3.09 1.080 2003 -0.003 -3.12 1.060 2004 -0.002 -3.23 1.100 2005 -0.001 -3.26 1.120 2006 0.000 -3.57 1.320 We used the goodness of fit tests, that are based on edf: the Anderson-Darling statistic 2 A the Cramer-von Mises statistic 2 W and the Kolmogorov-Smirnov statistic D to evaluate how well the three parameter lognorma l distribution using the parameter estimates in Table 2.2 fit the stony cora l cover proportions data for all years from 1996 to 2006. Table 2.3 gives the KolmogorovSmirnov statistic D Cramer-von Mises statistic 2 W and Anderson-Darling statistic 2 A and their corresponding p-values
45 for the three-parameter lognormal distribution usin g the LMLE and MLE method of estimation. Figure 2.3 shows the close fit of the cdf from the three-parameter lognormal probability distribution to the edf of the 1997 sto ny coral cover proportions data. Table 2.3 Goodness-of-fit Statistics for the Three -Parameter Lognormal Probability Distribution Fit. Year Statistic p-value D 2 W 2 A D 2 W 2 A 1996 0.076 0.103 0.669 0.09 0.06 0.04 1997 0.045 0.032 0.242 > 0.50 > 0.50 > 0.50 1998 0.052 0.046 0.324 > 0.50 > 0.25 > 0.25 1999 0.070 0.050 0.452 0.18 > 0.25 0.18 2000 0.052 0.041 0.351 > 0.50 > 0.50 > 0.25 2001 0.046 0.049 0.364 > 0.50 > 0.25 > 0.25 2002 0.063 0.081 0.525 > 0.25 0.14 0.11 2003 0.073 0.109 0.677 0.13 0.05 0.04 2004 0.074 0.096 0.627 0.13 0.09 0.06 2005 0.079 0.099 0.659 0.07 0.08 0.05 2006 0.063 0.070 0.564 > 0.25 0.22 0.10 There is no evidence to reject the null hypothesis that the proportion stony coral cover has a three parameter lognormal distribution using the Kolmogorov-Smirnov statistic D and its p-values in Table 2.3 at 0.05 a = for all the years from 1996 to 2006. The same can be seen with the Cramer-von Mises statistic 2 W for all the years. These two goodness-of-fit tests are in agreement that the sto ny coral cover proportions data from 1996 to 2006 follow the three parameter lognormal p robability distribution. In the case of the Anderson-Darling statistic 2 A the null hypothesis cannot be rejected at 0.05 a = for the years 1997 to 2002 and 2004 to 2006, whereas th e null hypothesis is rejected for years 1996 and 2003 at 0.05 a = To further check if the distribution fit is adeq uate, especially for the years where some doubt is presen t, the data were transformed to obtain a normally distributed sample using the estimates o f the parameters. The transformed values were then tested for normality using the Sha piro-Wilks normality test (see Table
46 2.4). It shows that all the samples, for all the y ears after transforming the data, are normally distributed. Table 2.4 Shapiro-WilkÂ’s Normality Test of Transfo rmed Data. Year Statistic p-value 1996 0.9783 0.08 1997 0.9861 0.34 1998 0.9852 0.29 1999 0.9815 0.15 2000 0.9829 0.20 2001 0.9856 0.32 2002 0.9820 0.17 2003 0.9797 0.11 2004 0.9785 0.09 2005 0.9778 0.08 2006 0.9785 0.09 This test provides further evidence, as was shown w ith the Kolmogorov-Smirnov statistic D and Cramer-von Mises statistic 2 W that fitting the three-parameter lognormal distribution to the proportion stony coral cover fo r 1996 and 2003 is justifiable. Thus the LMLE and MLE estimates for the three-parameter logn ormal distribution given in Table 2.2 provide an accurate fit to the proportion stony coral cover. The three-parameter lognormal probability distribution function for 199 6 is given in equation 2.10. Similar functions can be obtained for the years 1997 to 200 6 by substituting the threshold, scale and shape estimates from Table 2.2 in equation (2.1 ). Thus, for the 1996 data the three parameter lognormal pdf is given by: 2 11ln(0.014)2.33 0.4778(0.014)exp0.014 () 20.835 0 0.014 x xifx fx ifx++ +->= n Â£. (2.10) Researchers of the monitoring project will be able to use this new finding that the stony coral proportions data follow the three-parameter l ognormal probability distribution to
47 effectively and accurately analyze the stony coral cover proportions data for the Sanctuary region in the Florida Reef Tract. 2.5 Comparison of Descriptive Statistics vs. Parame tric Analysis The annual works given by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) reported the descriptive mean and used non-p arametric procedure to detect changes in the proportion stony coral cover over t he years of the monitoring project. It is now possible to get better results and to use param etric procedure which is more powerful that itsÂ’ non-parametric counterpart in analyzing t he data. Here, a comparison of the mean, median and standard deviation between the des criptive statistics and the probability distribution is made. The main advanta ge of knowing the probability distribution of a sample is that much information c an be deduced about the population. The mean, median, mode, variance and standard devia tion of the population for the threeparameter lognormal distribution are calculated fro m the equations given below: 22ExpectedValueem q s+ =+ Medianem q =+ 2 Modeems q =+ 2 2 2 1 Varianceee ms s + =, and ( ) 22 21 ee sms +=-. The estimates: Âˆ m (scale), Âˆ s (shape) and Âˆ q (threshold) for the parameters: m s and q are obtained from Table 2.2. Table 2.5 gives the r esult of using the three parameter lognormal probability distribution in obtaining the mean, median and standard deviation. The descriptive statistics results for the mean, me dian and standard deviation are given in Table 2.1.
48 Table 2.5 Probability Distribution Statistics for Stony Coral Cover Proportion Data. 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Mean 0.124 0.116 0.100 0.078 0.080 0.076 0.079 0.074 0.070 0.071 0.067 Median 0.083 0.084 0.063 0.046 0.044 0.045 0.043 0.041 0.038 0.037 0.028 Std Dev 0.138 0.113 0.124 0.104 0.120 0.103 0.121 0.112 0.111 0.114 0.146 In comparing the results for the mean, median and s tandard deviation using the threeparameter lognormal distribution for years 1996 to 2006 (Table 2.5) as opposed to the ones obtained from descriptive statistics (Table 2. 1) several differences are apparent. We will proceed to discuss these findings. The standard deviation obtained from the descripti ve statistics is much lower than that obtained from the lognormal probability distri bution for all years from 1996 to 2006. Figure 2.4 shows the standard deviation obtained by both the descriptive statistics and the parametric analysis.
49 Figure 2.4 Standard Deviation from the Probability Distribution and from Descriptive Statistics. StdDevPD = Standard Deviation from Prob ability Distribution, StdDevDS = Standard Deviation from Descriptive Statistic. 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 19961997199819992000200120022003200420052006Year StdDevPD StdDevDS This is expected, since the sites chosen in the stu dy represent the different types of coral reef habitat. On the other hand, the probability d istribution reflects the results of the population. Moreover, one can expect this because the lognormal probability distribution is a skewed distribution, where observ ations are realized quite far apart from the mean. The median is higher for the descriptive statistic than from the probability distributions for the years 1996, 1997, 1999, 2000, 2002, 2003, 2005 and 2006 as seen in Figure 2.5. The percent difference compared to the probability distribution for the mentioned years are 10, 6, 2, 6, 6, 2, 3 and 5, res pectively. For 1998, 2001 and 2004 the median obtained from the descriptive statistic is l ower than the value obtained from the probability distribution (Figure 2.5) by 9, 8 and 6 percent, respectively. Interestingly, the median obtained from descriptive statistic underest imates the value every third year
50 starting from 1996 the beginning of CREMP, Figure 2 .5. The confidence interval for the median will be discussed later. Figure 2.5 Mean and Median from the Probability Di stribution and from Descriptive Statistics. MeanPD = Mean from Probability Distribu tion, MeanDS = Mean from Descriptive Statistic, MedianPD = Median from Proba bility Distribution, and MedianDS = Median from Descriptive Statistic. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 19961997199819992000200120022003200420052006Year MeanPD MeanDS MedianPD MedianDS In the case of the mean, the descriptive statistic always underestimates the values from the probability distribution. This is clearly seen from Figure 2.5, where the curve from the descriptive statistic values is lower than the values from the lognormal probability distribution. The percentage difference s from years 1996 to 2006 are 4, 2, 4, 4, 6, 3, 6, 4, 6, 5 and 9, respectively. Thus the r eported mean values from the works of
51 (Yanev et al. 2003a, b, 2004, Tsokos et al. 2005, 2 006a) were always lower than the ones obtained by the fitted lognormal probability distri bution. Nevertheless, both curves show a decreasing trend for the mean stony coral cover f rom 1996 to 2006. They both show the rapid decrease that was reported for the first thre e years of the monitoring project and a slower decreasing rate for the subsequent years. Si nce many scientists prefer to report the mean for their data, the confidence interval for th e mean will be addressed later. The main concern with these differences should be the underestimation of the mean when using descriptive statistics. This is a n ew and significant finding that is part of the mission and objectives of the CREMP in monitori ng the stony coral cover. The values obtained from the three parameter lognormal probabi lity distribution are more accurate estimates of the true mean, median and standard dev iations than the ones obtained from descriptive statistics. 2.6 Confidence Interval for the Median Equation (2.11) gives the (1)100% a confidence interval of a sample having a two-parameter lognormal distribution as given by Mo hn (1979). Let the random variable X have a two-parameter lognormal pdf, then log YX = has a normal distribution with mean m and variance 2 s Then, the lower and upper confidence interval for the median, e m are given by: 1,21,2exp,expYY nnss YtYt nnaa-r -+ (2.11) where 1n i i Y Y n == 2 2 1() 1 n Y iYY s n == and 1,2 nt a -is the upper 2 a point of the studentt distribution with 1 n degrees of freedom. This confidence interval is mis used by many researchers. They use it as the confidence interval of the true mean instead of the confidence interval for the true median when the su bject data follows the two-parameter lognormal probability distribution (Land 1971, Zhou and Gao 1997). This procedure of obtaining the confidence limits is known as the na ve method. If the confidence interval
52 of the stony coral cover proportions for the true m edian is calculated using equation (2.11), the confidence interval suffers from a hori zontal shift factor due to the threshold parameter ( q ) of the three-parameter lognormal probability dist ribution. In this study, we propose that the estimates (Table 2.2) for the thre shold parameter obtained from the probability density function fit be used in the tra nsformation of the X random variable to obtain log() WX q =. This transformation is well known to have a norm al distribution with mean m and variance 2 s as required for the confidence interval in equatio n (2.11). The (1)100% a confidence interval for the three-parameter lognor mal probability distribution can be obtained from the two-parameter because theta causes only a horizontal shift to the probability distribution cu rve. The proposed (1)100% a confidence interval of the true median for the thre e-parameter lognormal probability distribution is given by: 1.21.2 ÂˆÂˆ exp,expW W n nss WtWt nna a qq r -+++ (2.12) The main challenge here is how to obtain the value of Âˆ q from the random sample. The estimators W and 2 W S are well known. For Âˆ q one can use the same procedure as proposed by Cohen (1951), among others. The LMLE e stimator for q can be found by solving equation (2.9). With the advances made in computer technology and sophisticated algorithms, such calculations are pos sible. In calculating the confidence interval using the n ave method from the stony coral cover proportion data, it was necessary to omit the lowest data values to carry out the log transformation of the data. The 90% and 95% confid ence intervals of both the nave method (NM) and the proposed method (PM1) (where th e inclusion of the LMLE estimator for theta is considered) are given in Tab le 2.6. There is a notable difference in values between the confidence intervals obtained fr om both methods at the 90% and 95% confidence level with the exception of 2006. Table 2.6 also gives the median estimate of the true median from the three-parameter lognormal probability distribution (PD) and the median estimate from the sample, assuming that the sample follows the three-parameter lognormal probability distribution. The estimates ( found under the column of the median
53 labeled PM1), obtained directly from the sample wit h the assumption that its underlying probability distribution is the three-parameter log normal, are close to the estimates (found under the column of the median labeled PD) obtained from the three-parameter lognormal probability distribution fitted to the da ta. Table 2.6 90% and 95% Confidence Interval for the True Median: Nave Method and Proposed Method. PD = Median from the Three-Paramet er Lognormal PDF PM1 = Median from Sample Assuming the Three-Parameter Log normal PDF Nave Method Proposed Method (PM1) Median 90% 95% 90% 95% Lower Upper Lower Upper Lower Upper Lower Upper PD PM1 19960.064 0.092 0.062 0.095 0.071 0.098 0.069 0.101 0.083 0.084 19970.064 0.091 0.062 0.094 0.073 0.098 0.071 0.100 0.084 0.085 19980.049 0.071 0.047 0.073 0.053 0.075 0.052 0.077 0.063 0.063 19990.039 0.055 0.038 0.057 0.039 0.055 0.038 0.057 0.046 0.046 20000.036 0.052 0.034 0.054 0.037 0.054 0.036 0.056 0.044 0.045 20010.037 0.053 0.036 0.055 0.039 0.055 0.037 0.056 0.045 0.046 20020.034 0.050 0.033 0.052 0.036 0.052 0.034 0.053 0.043 0.043 20030.033 0.048 0.031 0.050 0.034 0.050 0.033 0.051 0.041 0.041 20040.029 0.043 0.027 0.045 0.031 0.045 0.030 0.047 0.038 0.037 20050.029 0.043 0.028 0.045 0.030 0.044 0.029 0.046 0.037 0.036 20060.023 0.035 0.022 0.037 0.023 0.035 0.022 0.037 0.028 0.029 Figure 2.6 shows the lower and upper confidence lim its of the true median under the proposed method and the nave method. It also show s the median estimate of the true median using the three-parameter lognormal probabil ity distribution for all years from 1996 to 2006. It is evident that the confidence in terval from both the proposed method and the nave method include the median estimate. There is a notable difference in the confidence intervals for the years 1996 to 1998 and minor differences in the other years. The confidence interval band obtained from the prop osed method is shifted higher than the confidence interval band obtained from the nav e method.
54 Figure 2.6 90 % Confidence Interval for the Median : Proposed Method vs. Nave Method. PM1 = Proposed Method NM = Nave Method 0.000 0.020 0.040 0.060 0.080 0.100 0.120 19961997199819992000200120022003200420052006 Year Median PM1 NM To further compare these intervals, we calculate t he confidence range (upper confidence limit Â– lower confidence limit), Table 2 .7. Although the confidence ranges are closed for both methods, the confidence range f rom the proposed method is smaller or equal to the confidence range from the nave method The percentage difference between the naive method and the proposed method are given in Table 2.7 in brackets. The percentage difference is found as follows: 1 100 1 NMPM Percentdifferencex PM = Only in 1999 was the confidence range for the propo sed method a bit wider than confidence range for the nave method. This means that the proposed method is slightly
55 better than the nave method when constructing the (1)100% a confidence interval of the true median. Table 2.7 Confidence Range: Proposed Method vs. Na ve Method. Confidence Range Nave Method Proposed Method (PD1) Year 90% 95% 90% (% difference) 95%(% difference) 1996 0.028 0.033 0.026 (5) 0.032 (5) 1997 0.026 0.032 0.025 (7) 0.029 (7) 1998 0.022 0.027 0.021 (5) 0.025 (5) 1999 0.016 0.019 0.016 (-1) 0.019 (-1) 2000 0.017 0.020 0.016 (2) 0.020 (2) 2001 0.016 0.019 0.016 (0) 0.019 (1) 2002 0.016 0.019 0.016 (0) 0.019 (0) 2003 0.016 0.019 0.015 (2) 0.018 (2) 2004 0.014 0.017 0.014 (2) 0.017 (2) 2005 0.014 0.017 0.014 (1) 0.017 (2) 2006 0.012 0.015 0.012 (0) 0.015 (0) 2.7 Confidence Interval for the Mean The annual works given by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) on the stony coral proportions for the CREM P does not include the confidence intervals for the true mean. They present only a de scriptive statistic of the mean. Since we have identified the underlying probability distr ibution of the stony coral cover proportions of the sanctuary region in this study a s the three-parameter lognormal probability distribution, a good confidence interva l for the true mean can be formulated from the random samples. Many studies have been carried out to obtain an ef fective confidence interval procedure for the two-parameter lognormal probabili ty distribution, e.g. Land (1971, 1972, 1974), Mohn (1979), Angus (1994), Zhou and Ga o. (1997), Zhou and Tu (2000), and Tian and Wu (2006), among others. Procedures th at give optimum confidence interval for finite samples are difficult to comput e. Land (1971) found the uniformly
56 most accurate unbiased upper and lower confidence b ounds for the true mean. However, the procedure in finding such confidence bounds is tedious and requires many tables of critical values. Because this procedure is defined in terms of a conditional distribution of a test statistic given the value of another statist ic, these tables can never be large enough to cover every sample. The numerical algorithms fo r computing the optimum intervals are unstable for certain ranges of the estimates of 2 (,) x Xs For these reasons Land and others have searched for other reasonable confidenc e intervals. There are several approaches that have been investigated such as, usi ng various transformations, direct and bootstrap procedures (Land 1972, 1974, Mohn 1979, Z hou and Gao 1997, 2000). Land (1972, 1974), Mohn (1979), and Zhou and Gao (1997) have shown that among several procedures, the CoxÂ’s method in approximating the c onfidence interval of the true mean from a sample, that follows the two-parameter logno rmal probability distribution, is the best one. We will explain the theory that governs the CoxÂ’s method. Let the random variable X have a two-parameter lognormal probability distrib ution, then log YX = as previously stated, has a normal probability distrib ution with mean, m and variance, 2 s Since 1n i i Y Y n == and 2 2 1() 1 n Y iYY s n == are complete sufficient estimates for m and 2 s inferences on ()log() EYEX z == can be made based on Y and 2 Y s The minimum variance unbiased estimator (MVUE) of z is 2 1 Âˆ ÂˆÂˆ 2 zms =+ and the MVUE of the variance of Âˆ z is 24 2(1) YYss nn + + Assuming normality for Âˆ z the approximate confidence interval for 2()exp 2 EX s m =+ is obtained from: 224224 /2 /2 exp,exp. 22(1)22(1) YYYYYYssssss YzYz nnnna a +-++++ ++ (2.13) This procedure reduces to obtain the CoxÂ’s method o f confidence interval for the true mean of the two-parameter lognormal probability dis tribution.
57 Herein, three procedures are applied to the propor tion stony coral samples for years 1996 to 2006 in constructing the confidence i nterval for the true mean. The nave method, previously explained, will be presented her e since it is a popular method in finding the confidence interval of the true mean af ter log transformation of the data. We will only present it to show how inadequate it is a s a confidence interval for the true mean of a three-parameter lognormal probability dis tribution. It is known that the nave method gives the confidence interval for the true m edian and not the true mean. The second procedure, we use is the CoxÂ’s procedure. T he third procedure is one that we propose in this study which is a modified version o f the CoxÂ’s procedure. We proceed to develop the proposed method for conf idence interval of the true mean from a three-parameter lognormal probability d istribution. Let the random variable X have a three-parameter lognormal, then log() WX q =has a normal distribution with mean, m and variance, 2 s Since 1n i iW W n == and 2 2 1() 1 n W iWW s n == are complete sufficient estimates for m and 2 s inferences on ()log() EWEX zq ==can still be made based on W and 2 W s Having the same MVUE of z being 2 1 Âˆ ÂˆÂˆ 2 zms =+ the MVUE of the variance of Âˆ z being 24 2(1) WWss nn + + and assuming normality for Âˆ z the approximate confidence interval for 2 ()exp 2 EX s qm -=+ can be obtained from equation (2.13). Therefore, the confidence interva l for the true mean of the threeparameter lognormal probability distribution 2()exp 2 EXs mq =++ is obtained from: 224224 /2 /2 ÂˆÂˆ exp,exp 22(1)22(1)WWWWWWssssss WzWz nnnna a qq r +-++++++ ++ (2.14) For Âˆ q one can use the same procedure as proposed by Coh en (1951) and recommended by many others. The LMLE estimator for theta can be obtained by solving equation (2.9). This confidence interval would apply to a three-para meter lognormal distribution.
58 The results of the 90% and 95% confidence interval using the CoxÂ’s method and the proposed method (PM2) are given in Table 2.8. The confidence interval using the nave method is given in Table 2.6. There are obvi ous differences that can be seen between the confidence intervals obtained from all t hree methods considered here. Table 2.8 also gives the mean estimate of the true mean f rom the three-parameter lognormal probability distribution and the mean estimate from the sample assuming that the sample follows the three-parameter lognormal probability di stribution. The estimates (found under the column of the mean labeled PM2), obtained directly from the sample under the assumption that its underlying probability distribu tion is the three-parameter lognormal, are close to the estimates (found under the column of the mean labeled PD) obtained from the three-parameter lognormal probability dist ribution fitted to the data. Table 2.8 90% and 95% Confidence Interval for the True Mean: CoxÂ’s Method and Proposed Method. PD = Mean from the Three-Parameter Lognormal PDF PM2 = Mean from Sample Assuming the Three-Parameter Lognormal P DF CoxÂ’s Method Proposed Method (PM2) Mean 90% 95% 90% 95% Lower Upper Lower Upper Lower Upper Lower Upper PD PM2 1996 0.113 0.177 0.108 0.184 0.104 0.147 0.101 0.15 2 0.1240.124 1997 0.108 0.164 0.103 0.171 0.099 0.135 0.097 0.13 9 0.1160.116 1998 0.090 0.146 0.086 0.153 0.083 0.121 0.080 0.12 5 0.1000.100 1999 0.065 0.098 0.062 0.102 0.063 0.094 0.061 0.09 8 0.0780.077 2000 0.068 0.112 0.065 0.117 0.064 0.100 0.062 0.10 4 0.0800.080 2001 0.065 0.101 0.062 0.105 0.063 0.093 0.060 0.09 7 0.0760.076 2002 0.065 0.106 0.062 0.111 0.062 0.097 0.060 0.10 1 0.0790.078 2003 0.064 0.106 0.061 0.111 0.060 0.093 0.057 0.09 7 0.0740.075 2004 0.059 0.100 0.056 0.106 0.056 0.088 0.053 0.09 2 0.0700.070 2005 0.058 0.098 0.055 0.103 0.056 0.090 0.053 0.09 5 0.0710.071 2006 0.050 0.089 0.047 0.094 0.050 0.089 0.047 0.09 4 0.0670.067 To display the differences seen in Table 2.8, we p lot the graph in Figure 2.7. Figure 2.7 shows the mean obtained from the three-pa rameter lognormal probability distribution that were fitted to the stony coral cov er proportion data. Figure 2.7 also
59 shows the confidence interval bands obtained from th e nave method, CoxÂ’s method and the proposed method. The confidence interval band f rom the nave method are much lower than the confidence interval bands obtained fr om both the CoxÂ’s method and the proposed method, as shown in Figure 2.7. The confide nce interval band from the nave method does not include the estimated mean. This was expected as the nave method should be used to construct the confidence interval of the true median and not for the true mean. From here on, we will only compare the confidence in tervals obtained from the CoxÂ’s method and the proposed method as these are c onfidence intervals of the true mean. The lower bound of the confidence interval obt ained from the proposed method is lower than the lower bound of the confidence interval obtained from the CoxÂ’s method. The same can be observed for the upper bounds from both methods. Furthermore the confidence intervals from both methods include the estimated mean obtained from the three-parameter lognormal probability distribution. From Figure 2.7, the confidence range of the proposed method is shorter than the co nfidence interval band obtained from the CoxÂ’s method (CM). To investigate this we calcul ated the confidence range from both methods, Table 2.9, at confidence levels of 90% and 95%. In addition, Table 2.9 gives the percent difference in the confidence range valu es between both methods in brackets. We calculated the percent difference as follows: 2 100 2 CMPM Percentdifferencex PM = The improvements of the confidence range are very g ood when using the proposed method as compared to the CoxÂ’s method. For all yea rs, the confidence ranges are much smaller for the proposed method as opposed to the C oxÂ’s method. In the first three years the confidence range of the proposed method is near ly half the confidence range from the CoxÂ’s method. The proposed method is superior to th e CoxÂ’s method in the construction of the confidence interval for the true mean of a t hree-parameter lognormal probability distribution.
60 Figure 2.7: The 90 % Confidence Interval for the T rue Mean: Nave Method, CoxÂ’s Method and Proposed M ethod. NM = Nave Method PM 2= Proposed Method CU = CoxÂ’ s Method 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 19961997199819992000200120022003200420052006 Year NM CM PM2 Mean
61 Table 2.9 Confidence Range: Proposed Method vs. Co xÂ’s Method. Confidence Range Cox's Method Method: Adjustment for Theta Year 90% 95% 90% 95% 1996 0.064 0.076 0.043(49) 0.051(50) 1997 0.057 0.068 0.036(59) 0.042(59) 1998 0.056 0.067 0.038(47) 0.045(47) 1999 0.033 0.040 0.031(6) 0.037(6) 2000 0.044 0.052 0.035(23) 0.042(24) 2001 0.036 0.043 0.031(18) 0.037(18) 2002 0.041 0.049 0.035(18) 0.042(18) 2003 0.042 0.050 0.033(27) 0.040(27) 2004 0.042 0.050 0.032(28) 0.039(29) 2005 0.040 0.048 0.034(16) 0.041(17) 2006 0.039 0.046 0.039(0) 0.046(0) 2.8 Conclusion The three-parameter lognormal probability distribut ion has been shown to be a very good fit for the stony coral cover proportion data obtained from the CREMP of the sanctuary stations for years 1996 to 2006. This is a very important finding that allows us to proceed with parametric analysis of the subject d ata. Such analysis will significantly improve previous results that used non-parametric m ethods to test for spatial and temporal trends on the status of the Florida coral reefs. In addition it allows for better estimations of the true central measures and confid ence intervals of these measures from the given data sets. Furthermore, descriptive stat istics information when given was not consistent with those obtained by using the three-pa rameter lognormal probability distribution. It was shown that the mean was undere stimated using descriptive statistics. This implies that the reported values by Yanev et a l. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) underestimated the true mean stony co ral cover. Thus, the results of this study should be used to understand and monitor the stony coral cover proportions as required in the CREMP.
62 In the present study we have extensively reviewed all possible methods and introduced new methods for obtaining confidence lim its of the true mean and median of the stony coral cover proportions data that is bein g probabilistically characterized by the three-parameter lognormal probability distribution. Two methods were used by applying the subject data and obtained 90% and 95% confidenc e limits for each of the true mean and median. For each central measure, we proposed a method (PM1 and PM2) that gives better results than the other methods we utilized. The criteria of evaluation and comparison were the confidence ranges of the true me an and median.
63 Chapter 3 Statistical Modeling of the Health of the Reefs: Di versity Indices 3.1 Introduction The main purpose of collecting species data is to m onitor the status of the reef including the diversity of coral species that inhab it it. Many scientists believe diversity is an indicator of the stability of the ecosystem (Xu 1 996, Hayek and Buzas 1997, Datta and Jana 2002, Foggo, Rundle and Bilton 2003, Jrgensen Costanza and Xu 2005, Pinckney, Paerl and Valdes-Weaver 2005). Two commonly used dive rsity indices are the ShannonWiener diversity index and the SimpsonÂ’s index. The concept of the diversity index is the measurement of the number of species and the spread of these species in their habitat. We will focus mainly on the Shannon-Wiener diversity in dex, because that was the diversity index of choice in the works by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a). Shannon-Wiener diversity index is a measure of the uncertainty that an individual in an ecosystem belongs to a certain species. The S hannon-Wiener diversity index is defined as follows: lnln 1 ln 1 s nnff ii s i Hpp ii n i -==-= = (3.1) where / ii pfn = is the proportion of points where the ith species is identified, i f is the abundance of the ith species, s is the number of species present and n is the total abundance of all species in the sample. The maximu m value of the Shannon-Wiener diversity index for a fixed set of species is obtai ned whenever each species have the same
64 abundance. This creates the most uncertainty of kn owing which species an individual in the habitat belongs to. On the other hand if a habi tat is dominated by a few species, then the uncertainty of knowing which species an individua l belongs to becomes low. If there is only one species present, then the uncertainty o f which species an individual belongs to is zero. This means that the higher the number of species and the more evenly distributed their abundance, the higher the index will be. This suggests that the ecosystem should be more stable than one with fewer species and more unba lanced abundance among the species. SimpsonÂ’s diversity index is the probability that two individuals in an ecosystem belong to different species. The SimpsonÂ’s diversi ty index is defined as follows: 2 1 1s p i il=-= (3.2) where i p is the proportion of individual where the ith species is identified, and s is the number of species present in the sample. SimpsonÂ’s diversity index ranges between 0 and 1. The closer the SimpsonÂ’s diversity index is to 1 the more diverse the ecosystem is said to be. It is important to note that the Shannon-Weiner di versity index is one of the best indices available to scientists, but it also has it s disadvantages. One of the disadvantages of the Shannon-Wiener diversity index is that the r are species do not contribute much to its value. The Shannon-Wiener diversity index acco unts for the rare species more than the SimpsonÂ’s index. Another disadvantage that plag ues not only the Shannon-Wiener diversity index, but many other diversity indices, is that it is possible for two very different ecosystems to have the same Shannon-Wiene r diversity index. In estimating the Shannon-Wiener diversity in the works of Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a), the jackk nife procedure was used. This procedure was used for several reasons. First, it re duces the bias in the estimator to the Shannon-Wiener diversity index when applying it to t he various stations. Miller (1974), Zahl (1977), and Routledge 1980) showed that the jac kknife procedure reduces the bias that exists in sampling by quadrat, because, in pra ctice, the observed values are normally dependent. Another reason is that no probability d istribution structural assumption is
65 made on the original set of data in applying the ja ckknife procedure. The most important reason is that Tukey (1958) conjectured that the ps eudovalues obtained as a result of the jackknife procedure could be treated as independent ly identical normally distributed random variables. Miller (1974) showed that the Tuk ey conjecture is true. These properties of the pseudovalues allow for the formula tion of the confidence intervals and for carrying out hypothesis testing using parametri c analysis. From the pseudovalues one can obtain estimates for the variance and thus the standard errors of the diversity index which is not possible otherwise. The jackknife procedure is not without problems esp ecially in practical sampling. Heltshe and Forrester (1985) used simulated data to investigate the jackknife procedure in estimating the BrillouinÂ’s and SimpsonÂ’s indices of diversity when sampling is done by selecting quadrats. He found that smaller quadrats tend to give less biased and smaller variances for the BrillouinÂ’s index, whereas, for t he SimpsonÂ’s index the estimate was unbiased using the quadrat samples but the estimate of the standard error was overestimated for larger sample causing over covera ge for observed confidence interval. Baxter (2001) questioned the validity of normality assumptions for the Shannon-Wiener diversity index pseudovalues obtained by using the jackknife procedure. Baxter (2001) used archaeological data of 15 non-empty classes fo r an archeological site, plotting the histogram of these pseudovalues and concluding from the histogram that the pseudovalues are non-normal. Baxter (2001) constru cted the 95% confidence interval assuming the normal probability distribution obtain ing the following confidence interval [3.16, 4.24]. He also found the empirical confidenc e interval through bootstrapping the 15 pseudovalues and obtaining the confidence interv al [3.12, 4.14]. In comparing them, Baxter (2001) concluded that the normality assumpti on should be questioned and that the bootstrap procedure should be adopted in cases wher e the sample size is small. The main problem using this procedure is the conclusion that these pseudovalues are non-normal by just using the histogram and the confidence interva l of the bootstrap procedure. It would be best to test for normality. We use the CREMP dat a to further investigate this. Zahl (1977) used quadrats of trees to estimate the diver sity of two blocks of forest. He used the jackknife procedure in estimating the Shannon-W iener diversity index and the
66 SimpsonÂ’s diversity index. He estimated 9 different Shannon-Wiener diversity indices and SimpsonÂ’s diversity indices. Out of the 18 sets of pseudovalues, 15 sets showed that they were normally distributed. His conclusions were very important. He showed that random sampling is not critical an important find ing since in practice, it is generally impossible to execute such random sampling. Zahl ( 1977) also concluded that spacing out the quadrats gave a better approximation of the diversity index for the type of forest he dealt with. In addition there was no significant advantage of using either the ShannonWiener diversity index or the SimpsonÂ’s diversity i ndex. Thus the jackknifing procedure provides a tool for scientists to check statistical ly significant differences between different samples. A focus of this study is to investigate the applica bility of the normal probability distribution assumption made on the pseudovalues ob tained from the jackknifing procedure for the Shannon-Wiener diversity index us ed in the works presented by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 20 06a) for the CREMP. The normality assumption was made when we constructed the 95% confi dence interval for the true Shannon-Wiener diversity indices for the entire san ctuary for years from 1996 to 2005. The validity of normality with respect to the 12 sta tions in the Dry Tortugas area of the CREMP is also investigated. The accuracy of the co nfidence interval is investigated using the bootstrapping resampling procedure. Test ing for normality of the pseudovalues is done using the Shapiro-Wilks normality test. Another major problem that affects diversity indice s, including the ShannonWeiner diversity index, is that indices tend to und erestimate the population diversity. This occurs because adding species always increases the diversity index. Since it is impossible to sample all the species in a populatio n, the diversity index will always be underestimated. We propose in this study that the u nderlying probability structure of the species abundance be found if possible so that it g ives a better approximation of the true diversity index. In this manner all the possible s pecies abundances within the population, as reflected by the sample, are accounted. This he lps in two ways: first, it accounts for the abundances of the missing species that may exis t in the population and secondly, it gives larger weights to the abundances of rare spec ies, since there are more species with
67 small abundances than with larger abundances. Once the probability distribution of the species abundance is fitted then we can use the prob ability distribution instead of the species abundance data in obtaining as estimate of the true Shannon-Wiener diversity Index or true SimpsonÂ’s diversity index for the ent ire sanctuary from 1996 to 2006. The Shannon-Wiener diversity index for the two-parameter lognormal probability distribution is obtained from literature under Shannon entropy. We solved the SimpsonÂ’s diversity index for the two-parameter lognormal probability di stribution in this study. 3.2 Methodology of Statistical Analysis of ShannonWeiner Diversity Index Yanev et al. (2003a, b, 2004) and Tsokos et al. (20 05, 2006a) used the jackknife procedure as explained by Zahl (1977), who used it for quadrats, which can be considered similar to the stations of the CREMP. W e calculated these pseudovalues for the Shannon-Wiener Diversity Index for the different habitats and regions. From these pseudovalues we obtain the jackknifing estimate and the confidence interval for the Shannon-Wiener diversity index for each year at the regional and habitat levels. Stations in each of the region or habitat level are used to calculate the jackknifing estimate of the Shannon-Wiener diversity index. Let n be the number of stations in the region or habitat The jackknifing procedure is as follows: Calculate the Shannon-Wiene r diversity index including all the n stations by summing the total abundances for each s pecies from all the stations, and then applying equation (3.1). Let o H denote the overall Shannon-Wiener diversity index. The diversity index is then calculated Â“ n Â” times but with the removal of the abundance from each station one at a time, using equation (3.1). Let i H -be the Shannon-Wiener diversity index without including station i Then the Â“ n Â” pseudovalues are obtained using the formula below: n i H n nH Hi o i... 1 )1 ( = =-. (3.3) i H is the ith pseudovalue. The jackknifing estimate of the Shannon-Wiener diversity index, H Âˆ for the region or habitat is obtained by taking t he average of all the pseudovalues, i H
68 We obtain the confidence interval to the Jackknifi ng estimate H Âˆ : s z H Â¢ 2 / Âˆa (3.4) where /2 z ais the standard normal and s Â¢ is the standard error obtained from the Â“ n Â” pseudo values. In this study, we shall focus on the 95% confidence interval for both the sanctuary and Dry Tortugas for each year that data were collect ed, i.e., from 1996 to 2006 for the sanctuary and from 1999 to 2006 for the Dry Tortugas The confidence intervals are obtained by using equation (3.4). We obtained 105 pseudovalues for the years 1996 to 2005 and 103 for 2006 for the sanctuary using equat ion (3.3) and 12 pseudovalues for Dry Tortugas for the yearÂ’s 1999 to 2006 using equat ion (3.3). We will compare the confidence intervals obtained assuming that the pse udovalues are normally distributed with the empirical confidence intervals obtained fro m the bootstrap procedure. The bootstrap procedure is a simulation method that doe s not make any assumptions about the probability structure of the sample of the populati on. Given a sample, the procedure resamples from this sample, with replacement. The re samples have the same size as the original sample. We obtain three sets of bootstrap repetitions having 1500, 5000 and 10,000 re-samples for each set of pseudovalues for the sanctuary region per year. For the Dry Tortugas we obtain three sets of bootstrap repet itions having 100, 200 and 500 resamples for each set of pseudovalues per year. For each bootstrap repetition the empirical confidence interval for the jackknife est imate to the Shannon-Wiener diversity index will be recorded and then compared to the conf idence interval obtained by just using the jackknife procedure. In addition the pse udovalues will be tested for normality using the Shapiro-Wilks normality test. 3.3 Comparison of the Bootstrap and Normality Confi dence Intervals We will present the confidence intervals for the tru e Shannon-Wiener diversity indices obtained by assuming that the pseudovalues are normally distributed and the empirical confidence intervals obtained from the bo otstrap procedure for the sanctuary and Dry Tortugas regions. We compare them graphical ly and by their confidence ranges
69 of the true Shannon-Wiener diversity indices. Fina lly, we present the results of the Shapiro-Wilks normality test. The results of the 95% confidence interval are giv en in Table 3.1. The table shows the lower and upper boundaries of the 95% confi dence interval obtained by assuming normality of the pseudovalues and the empi rical 95% confidence intervals from the bootstrap repetitions of size 1500, 5000 and 10 000. The first set of lower and upper bounds found in Table 3.1 are formed using the norm ality assumption and the others correspond to the empirical confidence intervals fo r the bootstrapping procedure at 1500, 5000 and 10000 repetitions, respectively. The empi rical confidence intervals for all bootstrap sets are very close to each other. Incre asing the number of repetitions in the bootstrap procedure does not change the confidence interval. In addition the empirical confidence intervals for all bootstrap sets are ver y close to the confidence interval obtained using the normality assumption of the pseu dovalues. Table 3.1 95% Confidence Interval for the True Sha nnon-Wiener Diversity Index for Sanctuary Region Normality Bootstrap Assumption N=1500 N=5000 N=10000 Year Lower Upper Lower Upper Lower Upper Lower Uppe r 1996 2.12 2.51 2.12 2.50 2.11 2.50 2.11 2.50 1997 2.10 2.50 2.09 2.49 2.10 2.49 2.09 2.49 1998 1.98 2.40 1.99 2.40 1.97 2.39 1.97 2.39 1999 1.76 2.17 1.75 2.16 1.76 2.16 1.75 2.17 2000 1.88 2.25 1.87 2.25 1.87 2.25 1.87 2.25 2001 1.83 2.24 1.83 2.23 1.83 2.23 1.82 2.23 2002 1.97 2.35 1.96 2.34 1.96 2.34 1.96 2.34 2003 1.86 2.31 1.86 2.30 1.86 2.29 1.85 2.29 2004 1.87 2.32 1.88 2.32 1.86 2.32 1.86 2.31 2005 1.88 2.29 1.87 2.28 1.88 2.28 1.87 2.28 2006 1.79 2.20 1.79 2.19 1.79 2.20 1.79 2.19 Figure 3.1 shows the graphical representation of Ta ble 3.1. The confidence interval obtained using the normality assumption is given in solid lines in Figure 3.2. The upper boundary of the confidence interval obtai ned by assuming normality is a bit
70 higher than the ones obtained from bootstrap for al l years. For most of the lower boundary, the bootstrap values are lower than the no rmality values.
71 Figure 3.1 95% Confidence Interval from the Normal ity Assumption and Bootstrapping for Sanctuary. J95 = Confidence Interval with Normality Assumption F95, S95 and T95 are the Confi dence Interval for the Bootstrap with 1500, 5000 an d 10000 Repetitions, Respectively 1.5 1.7 1.9 2.1 2.3 2.5 2.7 19941996199820002002200420062008YearsShannon Index F95 S95 T95 J95 ShannonIndex
72 To further compare these confidence intervals of t he true Shannon-Wiener diversity indices, we calculate the confidence rang e (upper confidence limit Â– lower confidence limit), Table 3.2. The confidence range s are similar from both procedures. The percentage difference between the confidence int erval obtained using the normality assumption on the pseudovalues and from the bootstr ap procedure is given in Table 3.2 in brackets. The percentage difference is found as fo llows: 100% Bn nLL x L (3.5) where n L is the confidence range obtained with the normality assumption and B L is the confidence range for the bootstrap procedure. The closeness of the confidence ranges are evident as the percentage differences are less than 4 %. The negative sign associated with the percentage indicate the confidence ranges wh ereby the confidence range obtained under the normality assumption is wider tha n the confidence range obtained through bootstrapping. The confidence intervals fr om the bootstrap procedure are shorter in range that the confidence intervals from the nor mality assumption for the most part. Table 3.2 Confidence Range: Bootstrap Confidence I nterval vs. Normality Confidence Interval for the Sanctuary Region. Normality Bootstrap Procedure Year Assumption N=1500 N=5000 N=10000 1996 0.39 0.38 (-3.1) 0.39 (-0.8) 0.39 ( 0.0) 1997 0.40 0.41 ( 0.5) 0.39 (-3.9) 0.40 (-1.0) 1998 0.42 0.40 (-4.4) 0.42 (-0.6) 0.42 (-1.1) 1999 0.41 0.41 ( 0.5) 0.40 (-2.5) 0.41 ( 0.0) 2000 0.37 0.37 (-0.6) 0.38 ( 2.1) 0.38 ( 1.3) 2001 0.41 0.40 (-1.8) 0.40 (-3.0) 0.41 (-0.5) 2002 0.38 0.38 ( 0.0) 0.38 ( 0.0) 0.38 (-1.3) 2003 0.44 0.44 ( 0.2) 0.43 (-1.9) 0.44 (-0.1) 2004 0.45 0.44 (-1.6) 0.45 ( 0.4) 0.45 (-0.5) 2005 0.41 0.41 (-0.7) 0.40 (-2.1) 0.41 (-0.4) 2006 0.41 0.41 (-0.7) 0.41 (-0.2) 0.41 ( 0.3)
73 The results of the 95% confidence interval of the true Shannon-Wiener diversity index are given in Table 3.3. The table gives the lower and upper boundaries of the 95% confidence interval obtained by assuming normality of the pseudovalues and the 95% empirical confidence intervals from the bootstrap r epetitions of size 100, 200 and 500. The empirical confidence intervals for all bootstra p sets are slightly different from each other. The empirical confidence intervals are also different from the confidence interval obtained under the normality assumption on the pseu dovalues. Table 3.3 95% Confidence Interval for the True Sha nnon-Wiener Diversity Index for Dry Tortugas. Normality Bootstrap Assumption N=100 N=200 N=500 Year Lower Upper Lower Upper Lower Upper Lower Uppe r 1999 1.81 2.16 1.813 2.158 1.826 2.153 1.829 2.18 2000 1.70 2.06 1.727 2.05 1.724 2.033 1.695 2.036 2001 1.66 2.02 1.670 2.009 1.685 1.992 1.677 2.014 2002 1.71 2.07 1.737 2.085 1.736 2.033 1.737 2.068 2003 1.75 2.20 1.741 2.185 1.774 2.166 1.768 2.196 2004 1.66 1.89 1.686 1.872 1.656 1.873 1.669 1.881 2005 1.61 1.88 1.618 1.860 1.606 1.862 1.601 1.864 2006 1.54 1.81 1.520 1.794 1.541 1.782 1.529 1.806 Figure 3.2 shows the confidence interval bands obta ined using both procedures. The confidence interval band of the true Shannon-Wi ener diversity index obtained using the normality assumption encompasses all the other three confidence interval bands obtained using the bootstrap procedure.
74 Figure 3.2 95% Confidence Interval from the Normal ity Assumption and Bootstrapping for Dry Tortugas. J95 = Confidence Interval with Normality Assumption F95, S95 and T95 are the Confidence Interval for the Bootstrap with 100, 200 and 500 Repetitions, Respectively 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 1998199920002001200220032004200520062007YearShannon Index F95 S95 T95 J95 ShannonIndex
75 We calculate the confidence range (upper bound Â– l ower bound) and percentage difference using equation (3.5), (see Table 3.4). The confidence ranges from the bootstrap procedure are shorter than the confidence range under the normality assumption. There is a percentage difference in co nfidence range up to 17 %. Thus, the confidence intervals from bootstrap are significant ly shorter than those under the normality assumption. Table 3.4 Confidence Range: Bootstrap Confidence I nterval vs. Normality Confidence Interval for the Dry Tortugas. Normality Bootstrap Procedure Year Assumption N=100 N=200 N=500 1999 0.35 0.35 ( -1.5) 0.33 ( -6.6) 0.35 ( 0.2) 2000 0.36 0.32 (-10.9) 0.31 (-14.8) 0.34 ( -5.9) 2001 0.36 0.34 ( -6.2) 0.31 (-15.0) 0.34 ( -6.7) 2002 0.36 0.35 ( -3.0) 0.30 (-17.2) 0.33 ( -7.8) 2003 0.45 0.44 ( -2.1) 0.39 (-13.5) 0.43 ( -5.6) 2004 0.22 0.19 (-17.2) 0.22 ( -3.4) 0.21 ( -5.7) 2005 0.27 0.24 (-10.6) 0.26 ( -5.4) 0.26 ( -2.8) 2006 0.28 0.27 ( -0.4) 0.24 (-12.4) 0.28 ( 0.7) The results of the normality test using the Shapir o-Wilks normality test for the pseudovalues for the sanctuary region from 1996 to 2006 and for Dry Tortugas from 1999 to 2006 are given in Table 3.5. There is no e vidence that the underlying probability distribution of the pseudovalues from the sanctuary follow the normal probability distribution. Since the sample size is large, one may argue that the normal assumption is valid due to the central limit theorem. It would be best if we can locate the underlying probability distribution of the pseudovalues rather than just assuming normality. In the case of the pseudovalues from Dry Tortugas, there is evidence that the pseudovalues follow the normal probability distribution.
76 Table 3.5 Normality Test of Pseudovalues. W is the Shapiro-Wilks Statistic. Sanctuary Dry Tortugas W P-value W P-value 1996 0.713 0 1997 0.7027 0 1998 0.7161 0 1999 0.7702 0 0.9534 0.69 2000 0.7748 0 0.8978 0.15 2001 0.6741 0 0.8672 0.06 2002 0.751 0 0.9102 0.21 2003 0.6655 0 0.8561 0.04 2004 0.687 0 0.9375 0.47 2005 0.7163 0 0.8789 0.08 2006 0.7024 0 0.9534 0.69 3.4 Probability Distribution Fit of the Species Abun dance Here, we investigate the probability structure of th e species abundance data from the sanctuary region. The species abundance data s ets for the years from 1996 to 2006 have similar statistical properties to the proporti on stony coral cover. The species abundance data is highly skewed to the right since a ll skewness values are 4 (Table 3.6). It displays strong leptokurtic behavior as the kurt osis values are greater than 15. Table 3.6 Descriptive Statistics for Species Abundan ce. 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Mean 596 579 487 445 371 437 390 429 415 414 369 Median 61 50 72 54 47 85 81 65 78 43 67 Std Dev 1338 1331 1182 1127 993 1159 947 1116 1023 1026 915 Kurtosis 20 22 21 19 20 20 16 20 19 17 15 Skewness 4 4 4 4 4 4 4 4 4 4 4 The two-parameter lognormal is the best fit for the species abundance data from the sanctuary region. A random variable has the twoparameter lognormal distribution function if its probability density function is giv en by:
77 2 21/211ln (2)exp0 () 2 0 0 x xifx fx ifxm ps s--> = n Â£ (3.6) is the scale parameter and is the shape parameter. The method of maximum likelihood (MLE) is used in the estimations for the parameters. The reasons for choosing this method are explained in section 2.3.1. Sectio n 2.3.1 also gives the explanation of the MLE for the three-parameter lognormal distribution and it is the same for the twoparameter lognormal probability distribution. We show that the two-parameter lognormal probability density function will fit the species abundance data very well for all the yea rs 1996 to 2006. The MLE estimates for the parameters are given in Table 3.7 for the y ears from 1996 to 2006. Table 3.7 Parameter Estimates for the Two-Parameter Lognormal Distribution. Year Scale Shape 1996 4.319 2.419 1997 4.399 2.279 1998 4.057 2.344 1999 4.119 2.060 2000 3.656 2.294 2001 3.904 2.250 2002 3.976 2.196 2003 3.887 2.264 2004 3.998 2.220 2005 3.895 2.234 2006 3.710 2.320 We used the goodness of fit tests (previously disc ussed), that are based on edf: the Anderson-Darling statistic 2 A the Cramer-von Mises statistic 2 W and the KolmogorovSmirnov statistic D to evaluate how well the two-parameter lognormal dis tribution using the parameter estimates in Table 3.6 fit the specie s abundance data for all years from 1996 to 2006. Table 3.8 gives the Kolmogorov-Smirno v statistic D Cramer-von Mises statistic 2 W and Anderson-Darling statistic 2 A and their corresponding p-values for the
78 two-parameter lognormal distribution using the MLE m ethod of estimation. There is no evidence to reject the null hypothesis that the spe cies abundance has the two-parameter lognormal probability distribution at 0.05 a = for all the years from 1996 to 2006. All three goodness-of-fit tests are in agreement with th is result. Table 3.8 Goodness-of-fit Statistics for the Two-Par ameter Lognormal Probability Distribution fit. Year Statistic P-value D 2 W 2 A D 2 W 2 A 1996 0.094 0.060 0.400 >0.15 0.398 0.367 1997 0.106 0.061 0.361 >0.15 0.382 0.445 1998 0.103 0.034 0.283 >0.15 >0.5 >0.5 1999 0.122 0.082 0.544 >0.15 0.193 0.152 2000 0.101 0.051 0.329 >0.15 0.491 >0.5 2001 0.109 0.087 0.488 >0.15 0.164 0.218 2002 0.114 0.061 0.353 >0.15 0.378 0.460 2003 0.088 0.041 0.249 >0.15 >0.5 >0.5 2004 0.141 0.096 0.542 0.093 0.124 0.156 2005 0.132 0.077 0.496 0.135 0.223 0.208 2006 0.113 0.071 0.421 >0.15 0.273 0.323 The two-parameter lognormal probability distributio n function for 1996 is given by: 2 11ln()4.319 0.165()exp0 () 22.419 00 x xifx fx ifx-> = n Â£ (3.7) Similar probability functions can be written for th e species abundance for the other years using the estimates in Table 3.7 and the pdf, equat ion (3.6). This enables us to calculate the Shannon-Wiener diversity index and SimpsonÂ’s di versity index of the CREMP for the sanctuary region in the Florida Reef Tract using th e underlying probability structure.
79 3.5 Shannon-Wiener and SimpsonÂ’s Diversity Index: S pecies Abundance Probability Distribution We can use the two-parameter lognormal probability distribution to obtain a good estimate of the true diversity index. The ShannonÂ’ s index and the SimpsonÂ’s index for the two-parameter lognormal distribution are given i n this section. The ShannonÂ’s index for the two-parameter lognormal probability distribu tion was obtained from literature regarding Shannon entropy. We will proceed to solve the SimpsonÂ’s index for the twoparameter lognormal probability distribution. 3.5.1 Shannon-Wiener Diversity Index for the 2-Para meter Lognormal Probability Distribution Shannon (1948) published the Shannon entropy for m any probability density functions. The Shannon procedure used was to intro duce the probability distributions in the probability structure ( () px ) of what is used as the Shannon-Wiener diversity in dex, equation 3.1 but applied to categorical variable. I n the case of the CREMP, the categorical variable is the stony coral species. T he Shannon entropy for continuous probability distribution takes the form: ()()log() Hxpxpxdx =. (3.8) When this is applied to the 2-pararmeter lognormal probability distribution (equation (3.6)), the Shannon entropy result is given by: 2 11 ()ln(2) 22 Hxpsm =++ (3.9) where m is the scale and s is the shape parameter. 3.5.2 SimpsonÂ’s Diversity Index for the 2-Paramete r Lognormal Probability Distribution We applied the same approach as Shannon (1948), us ing a probability distribution function to the probability structure of the Simpso nÂ’s index, as defined in equation (3.2)
80 (Simpson 1949). The SimpsonÂ’s index for continuous probability distribution takes the form: 2 ()1() xpxdx l=. (3.11) Using equation (3.6), we calculate the SimpsonÂ’s in dex for the 2-parameter lognormal probability distribution by replacing the () px in equation (3.11). Equation (3.12) gives the integral that must be solved to obtain the Simp sonÂ’s index for the two-parameter lognormal probability distribution. ()() 2 2 1 2 21ln() ()12exp 2 x xxdx qm lps s r r-=-. (3.12) The first step is to substitute ln zx = we have () 2 22exp() ()1exp 2 z z xdz m l pss - r =-. (3.13) Simplifying the exponential term from equation (3.1 3), we have: () 2 2z Zm s--= 2 224 224 24 z msmss s r ---+ (3.14) Inserting equation (3.14) in equation (3.13), we ha ve 2 2 2 222 2 14 ()1expexp 24 z x dz ms ms l pss -r r =--. (3.15) Simplifying equation (3.15), we have 2 2 2 2 2242 exp 42 1 ()1exp 2 2 2 2 2 z x dz smms l s ps s p - r -=-(3.16)
81 The integral part of the above expression is a nor mal probability density function with mean 2 2 2 ms and variance 2 2 s and it is equal to 1. The SimpsonÂ’s index for the 2-parameter lognormal probability distribution is: 2 2 14 ()1exp 4 2 x sm l ps =. (3.17) Equation (3.17) is also the SimpsonÂ’s index for the 3-parameter lognormal probability distribution. The Shannon-WienerÂ’s and SimpsonÂ’s di versity indices for the two parameter lognormal probability distribution are gi ven in Table 3.9. Table 3.9 Shannon-WienerÂ’s and SimpsonÂ’s Diversity Index for the Two-Parameter Lognormal Probability Distribution. Diversity Index Two-Parameter Lognormal Shannon-Wiener 2 11 ln(2) 22 psm ++ Simpson 2 2 14 1exp 4 2 sm ps 3.5.3 The Diversity Indices for the Probability Dis tribution of Species Abundance Knowing the underlying probability distribution of the species abundance data sets and the corresponding Shannon-Wiener and Simps onÂ’s diversity indices for these underlying distributions (Table 3.10), we can estim ate the indices from the sample of the population using the underlying probability distrib ution. The estimates for the true Shannon-WienerÂ’s and Si mpsonÂ’s diversity indices using the two-parameter lognormal probability distr ibution for the years from 1996 to 2006 are given in Table 3.9. This table also has t he Shannon-Wiener diversity index obtained in the reports by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) obtained using the jackknife procedure and the Shan non-Wiener diversity index without the jackknife procedure (Direct procedure). The Sh annon-Wiener diversity index
82 obtained from the distribution fit approach is much higher than both estimates obtained through the direct procedure and by using the jackk nife procedure. This means that the estimate under the two-parameter lognormal probabil ity distribution indicates that the ecosystem is more diverse than the estimates from t he other two procedures. We know that the two estimates obtained from the direct pro cedure and through the jackknifing procedure always under represent the true Shannon-W iener diversity index parameter. The estimate from the jackknifing procedure is slig htly greater than the estimate from the direct procedure, since it corrects the bias that e xists as a result of the sampling procedure. From the results of the three procedure s in obtaining an estimate for the true Shannon-Wiener, we can say that there are changes i n the species distribution over the years of the CREMP. In Table 3.10, there are two t ime periods: from 1996 to 1999 and from 2000 to 2006 that show different patterns of t he Shannon-Wiener diversity index estimates. The values from 1996 to 1999 are higher than the values from 2000 to 2006. This means that the ecosystem in the sanctuary had more uncertainty of knowing which species an individual belongs too prior to 1999 as compared to post 1999. Table 3.10 Shannon-WienerÂ’s and SimpsonÂ’s Diversit y Index for The Species Abundance Data. Shannon-Wiener Simpson Year Distribution Jackknife Original Distribution 1996 6.62 2.31 2.29 0.9933 1997 6.64 2.30 2.27 0.9944 1998 6.33 2.19 2.16 0.9918 1999 6.26 1.97 1.95 0.9936 2000 5.91 2.06 2.04 0.9882 2001 6.13 2.04 2.02 0.9910 2002 6.18 2.16 2.14 0.9920 2003 6.12 2.08 2.07 0.9908 2004 6.21 2.10 2.08 0.9920 2005 6.12 2.08 2.07 0.9911 2006 5.97 2.00 1.98 0.9886 The SimpsonÂ’s diversity index estimates from 1996 to 2006, Table 3.10, are close to each other. These values indicate that there is a high probability that two individuals
83 in the sanctuary region, through the sample, belong to different species. We can interpret this as meaning that the ecosystem is very diverse. The percentage improvement of the estimates of the true Shannon-Wiener diversity index using the underlying probability di stribution of the species abundance data as compared to the other two procedures are gi ven in Table 3.11. The percentage improvement is very encouraging that we can estimat e the true Shannon-Wiener diversity index from a sample of the population. Table 3.11 Percentage Differences of the Shannon-W iener Diversity Index: PDF vs. Jackknifing Procedure and Direct Procedure. Percentage Change Year Jackknife Original 1996 186190 1997 189192 1998 189192 1999 218221 2000 186190 2001 201204 2002 186189 2003 194196 2004 196199 2005 194196 2006 199202 Figure 3.3 shows the differences between the Shann on-Wiener diversity indices using the three approaches. The estimates from the two-parameter lognormal probability distribution are much larger than from the other tw o procedures. It can be seen that all three seem to follow the same pattern through the y ears. Thus, the results of this study should be used to understand and monitor the divers ity of the stony coral as required by CREMP.
84 Figure 3.3 Shannon-Wiener Diversity Indices for t he Sanctuary Region. DSV = Index from the PDF JSV = Index from the Jackknife Procedure OSV = Index directly from the Sample 0 1 2 3 4 5 6 7 1994199619982000200220042006Year Shannon Index DSV JSV OSV
85 3.6 Conclusion In investigating the normality assumption of the p seudovalues of the jackknife procedure for the Shannon-Wiener diversity index es timate, we found that the Dry Tortugas with only 12 pseudovalues tested positive for normality through the ShapiroWilks normality test. While this is true of the ps eudovalues for the Dry Tortugas for each year (1999 to 2006), the same cannot be said for th e 105 pseudovalues from the sanctuary region for each year from 1996 to 2006. The sanctu ary pseudovalues failed to support the null hypothesis that the data came from a norma l distribution under the Shapiro-Wilks normality test. Also the bootstrap empirical confi dence intervals for the pseudovalues from the jackknife procedure are shorter than the o nes obtained using the normality assumption. This was true for the sample sizes of 12 and 105 using the two sets of bootstrap repetitions 100, 200 and 500, and 1500, 5 000 and 10000. The difference in confidence range was greater for the 12 pseudovalue samples than for the 105 samples. In the case of the sample size of 12, the bootstrap em pirical confidence interval was always contained inside the confidence interval with the n ormality assumption. Although this was not the case with the sample size of 105, the b ootstrap empirical confidence intervals were lower than the confidence intervals under the normality assumption but they had very close values for both upper and lower confiden ce bounds. The jackknife procedure is still essential in the study of diversity indice s due to its biased reduction capability and the advantages of having the pseudovalues. The adva ntages of the jackknife procedure can be enhanced with the use of bootstrapping in cr eating the confidence interval of the Jackknife estimates. The results of the Shapiro-Wilks normality test sh ow that the normality assumption plays no role in the difference found be tween the confidence intervals for the jackknife estimate from the normality assumption on the pseudovalues and from the bootstrap procedure. This statement can be defended using two angles of analytical reasoning. First we looked at the approach of using the normality statistical testing of the samples. The pseudovalues from Dry Tortugas tested positive for normality and the pseudovalues from the sanctuary did not. Yet the co nfidence intervals from both
86 approaches for the non-normal distributed samples ( sanctuary) were much closer than for the normally distributed samples (Dry Tortugas). Th e second point of view is that one can assume normality for the large sample through t he central limit theorem. This again leads to the same conclusion that the difference in confidence interval is not due to the normality assumption. In this case this is true, si nce both samples would be normally distributed but still have differences in the confi dence intervals. It is quite possible that the difference is due to the sample size but this h as to be further investigated. With respect to investigating the diversity index through the probability distribution fit of the sanctuary species abundance data, the two-parameter lognormal was a very good fit to the species abundance for all th e years. We can therefore use parametric analysis to improve the results from the works of Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) for the CREMP From the probability distribution, we calculated the estimate of the true Shannon-Wien er diversity indices. These are much higher than the ones obtained from the jackknife pr ocedure or from the direct method. By finding the probability distribution of the species abundance, more weight was given to the rare or less abundant species, while still cons idering the contribution of the most abundance species to the habitat. The high values a re also a result of the attempt to account for species not represented or recorded due to sampling. The high values of the Shannon-Wiener indicate high diversity among the st ony coral species within the sanctuary. There is an indication that a decrease i n diversity occurred between 1996 and 2006 in the sanctuary region. We were able to solve the SimpsonÂ’s diversity inde x for the two-parameter lognormal probability. This is important as it prov ides scientists, who prefer to use the SimpsonÂ’s diversity index, another better approach of estimating the true SimpsonÂ’s diversity index. We can now use the results here to improve the pre viously obtained results from the works of Yanev et al. (2003a, b, 2004) and Tsok os et al. (2005, 2006a). More importantly the results can be used in general to e ffectively examine stony coral species data that contain the different types of species an d their abundances.
87 Chapter 4 Nonparametric Statistical Analysis of Diversity Ind ex 4.1 Introduction The focus of Chapter 3 was to investigate the norm al probability distribution assumption made on the pseudovalues obtained from t he jackknife procedure for the Shannon-Wiener diversity index used in the works of Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) for the CREMP. The Dry Tortugas pseudovalues showed statistical evidence that they are normally distrib uted but this was not so for the sanctuary pseudovalues. The normality assumption was used in the construction of the 95% confidence interval for the Shannon-Wiener diversit y indices for the entire sanctuary for years from 1996 to 2005. Since the underlying assu mption of normality was used to obtain the confidence intervals using the pseudoval ues, they cannot be used for decision making. Such a decision would be misleading. We tr ied to fit other continuous probability distributions so that we can proceed wi th parametric analysis without any success. Thus, we will proceed nonparametrically b y estimating the probability distribution function using the kernel density appr oach. This nonparametric procedure has been applied successfully (e.g. Baxter, Beardah and Wright 1997, Ker and Goodwin 2000, Wolf and Sumner 2001, Lotti and Santarelli 20 04, Buskirk and Lohr 2005) in estimating the probability distribution function wh en a classical well defined probability distribution function could not be found to fit the data. Hence, the aim of this chapter is to develop the best possible estimate of the probab ility distribution of the jackknifing pseudovalues for the sanctuary using the kernel den sity method. Once the best kernel density estimate is obtained, a comparison with the parametric approach under the normal
88 assumption will be made. We compare the mean, stand ard deviation, standard error and the 95% confidence interval of the true mean. 4.2 Kernel Probability Density Many scientists have studied the development of non parametric density estimation methods using kernels and bandwidths. Fo r example Tapia and Thompson (1978), Bean and Tsokos (1980), Silverman (1986), a nd Liu (1998) have found interesting analytical results and documented the i mportance of nonparametric density estimation methods to various applications. Silverm an (1986), Liu (1998), and Liu and Tsokos (2002), among others, have shown that once a good bandwidth estimate has been identified the kernel density estimation is very ac curate and effective. Liu (1998) and Liu and Tsokos. (2002) showed that the optimal bandwidt h is different for both the kernel probability distribution function and kernel cumula tive distribution function. Tapia and Thompson (1978), Silverman (1986), Liu (1998), and Liu and Tsokos (2002), all stress that the bandwidth is much more important than the choice of the kernel used. Silverman (1986) suggested that the choice of kernel be consi dered on the degree of differentiability required or the computation effort involved. Let 1 ,..., n XX be a sample of independent, identically distribute d observations having a common pdf, () fx then Âˆ () fx will denote the kernel density estimate to () fx Âˆ () fx is defined by 11 Âˆ ()n i i xX fxK nhh== (4.1) where K is the kernel function, n is the sample size and h is the bandwidth. From equation (4.1), both the kernel function and bandwi dth are needed to obtain the kernel density estimate of () fx It is also assumed that the underlying probabilit y density function () fx has continuous derivatives of all orders.
89 First, we shall summarize some of the basic proper ties of Âˆ () fx The kernel function K is usually restricted to be a symmetric probabilit y density function. It means that the kernel function satisfies the following co nditions: ()1 Ktdt - = ()0 tKtdt - = and 2 2 ()0 tKtdtk - = Properties of K define the properties of the resulting kernel esti mate Âˆ () fx Since K is a probability density function, then Âˆ () fx is also a probability density function. Table 4.1 gives a list of commonly used kernel functions. Fr om the commonly used kernel function, all except the Gaussian kernel function h ave a bounded sample space. Table 4.1 Some Kernels and Their Inefficiencies. Kernel () Kt Inefficiency Epanechnikov 2 31 1 5 45 t for 5 t < 1 Biweight () 2 215 1 16 t for 1 t < 1.0061 Triangular 1 t for 1 t < 1.0143 Gaussian 2 2 1 2 t e p 1.0513 Uniform 1 2 for 1 t < 1.0758 From the kernel density estimate, equation (4.1), the bandwidth ( h ) controls how many observations appear around x in the kernel density estimate. If h is too small, too few observations around x are considered in the stability of the kernel dens ity estimate. The kernel density estimate produces too many bumps On the other hand, if the bandwidth is too large, many observations around x are considered in the stability of the kernel density estimate, causing over smoothness of the kernel density estimate. The inefficiency of the kernel function gives an i ndication of its performance on the kernel density estimate based on the criteria o f the mean integrated square error (MISE). The Epanechnikov kernel function is the mos t efficient under the MISE criteria
90 but the others are very close. Thus the choice of t he bandwidth is more important than the choice of kernel function in trying to fit the best kernel density estimate. 4.2.1 Statistical Properties of the Kernel Density Estimator The kernel density estimate of the given data unde r the Gaussian kernel is defined by 2 1 2 111 Âˆ (). 2ixX n h ifxe nhp== (4.2) The expected value, () Ex is given by 2 21 2 1 1 2 111 () 2 11 2i ixX n h i xX n h i Exxedx n h xedx n hp p = - = -= = (4.3) Using the substitution i xX z h = in equation (4.3), we obtain 22 1 11 () 2 1 z n i i n i izhX Exedz n X np = - = + = = (4.4) The variance, () Varx is given by () ()22 2 1 2 2 2 1()()() 11 (). 2ixX n h iVarxExEx xedxEx n hp = -==(4.5)
91 We can show the results for 2 () Ex to be ()2 21 22 2 1 2 2 1 22 111 () 2 () 1 2 1 .ixX n h i z n i i n i i Exxedx n h zhX edz n hX np p- -= - = - == + = =+ (4.6) By combining equations (4.4) and (4.6), the varianc e is given as: () ()2 2 2 22 11 2 2 1()()() 111 .nn ii ii n i iVarxExEx hXX nn hXX n=====+=+(4.7) To obtain the interval that contains 95% of the tru e mean using the kernel density estimate, we will calculate the cumulative distribu tion function (cdf) of the kernel density estimate. The cdf is defined by 2 21 2 1 2 1 111 Âˆ () 2 11 2 1 .i itX x n h i xX z h n i n i i Fxedt n h edz n xX nhp p= - = - == = =F (4.8) From the cdf in equation (4.8), we can now formulat e the 95% confidence interval of the true mean of the unknown underlying probability dis tribution for the jackknifing
92 pseudovalues from the sanctuary. Table 4.2 gives a summary of the results for the expected value, variance and the cumulative density function. Table 4.2 Expected Value, Variance and Cumulative Distribution Function of the Kernel Density Estimate. Gaussian Kernel Density Estimate Expected Value 11n i i X n= Variance () 2 2 11n i ihXX n=+CDF 11n i i xX nh=F 4.2.2 Criteria for Quality of Fit Here we identify the criteria that we use to evalu ate the quality of Âˆ () fx We employ the mean integrated square error (MISE) whic h was developed from the mean square error (MSE) which is a measurement of error at single points. MSE is defined by ( ) 2 Âˆ ()() MSEEfxfx =. (4.9) Expanding equation (4.9) we have ( ) 2 ÂˆÂˆ ()()var() MSEEfxfxfx =-+ This expression shows that MSE has two components, the sum of squared bias and the variance. Rosenblatt (1956) introduced the concept of MISE which allows obtaining a better approximation of the MSE measurement. The MI SE is defined by ( ) ()2 2Âˆ ()() ÂˆÂˆ ()()var(). MISEEfxfxdx Efxfxdxfxdx ==-+ (4.10) The approximations for theses two integrals can be found in Silverman (1986).
93 It allows for the estimations of the bias portion o f the MISE to be obtained from () Âˆ ()() ()/ ()() biasEfxfx Kxyh fydyfx h ==(4.11) By using the substitution yxht =and a Taylor series expansion of equation (4.11), the following approximation to the bias portion of the MISE is obtained. 222 1 ()()() 2 biashfxtKtdtoh Â¢Â¢ =+. (4.12) The integral of the square of equation (4.12) is th en taken to complete the first part of equation (4.10). That is, 4 222 ()() 4 h biasdxtKtdtfxdx Â¢Â¢ . (4.13) The integral of the variance is found similarly giv ing the following approximation: 21 Âˆ var()() fxdxKtdt nh . (4.14) The importance of the results of equation (4.13) an d (4.14) is that MISE depends on the choice of h and K Since the efficiency of the K Â’s are very similar, the selection of the choice of h becomes very important. The value of h canÂ’t be too small or too large. A large value will increase the bias and a s mall value would increase the variance. We selected to work with the Gaussian kernel for t wo reasons: the Gaussian is differentiable everywhere and x can exist everywhere (not limited to a certain rang e). Even though we selected to work with the Gaussian k ernel, we tried the other commonly used kernel functiosn such as Epanechnikov, rectang ular, triangular, etc. and concluded that the Gaussian gave a better approximation. The major problem was obtaining the best bandwidth ( h ). Silverman (1986) suggested a few rules of thumb for bandwidths when using the Gaussian kernel. The rule of thumb bandwi dths are given below: 1/5 1 1/5 2 1/5 3Âˆ Âˆ Âˆ 1.06min(,) 1.34 Âˆ Âˆ 0.79 Âˆ Âˆ Âˆ 0.9min(,). 1.34 R hn hRn and R hns s-= = = (4.15)
94 where Âˆ s is the estimate of the standard deviation and Âˆ R is an estimate of the interquartile range of the sample. Silverman (1986) also investigated the results of these three bandwidth. He suggests using 2 Âˆ h and 3 Âˆ h whenever the data sets are skewed. But we encounter problems with 2 Âˆ h whenever a bimodal behavior appears in the data. In such case, he suggests to use either 1 Âˆ h or 3 Âˆ h A good approximation of MISE, is given below, when using the Gaussian kernel: 2222 2211111 2 22 2 MISE nh hh hp s ss s =-++ ++ + (4.16) Using this approximation of MISE we obtain good res ults. 4.3 Procedure for Developing the Kernel Density Es timate We will fit the kernel density estimate for all th e jackknife pseudovalues samples from 1996 to 2006. It is important that the observ ations in each sample are independent. Zahl (1977) showed that these values can be conside red to be independent. The histogram will be used in the fitting process, it is necessary to standardize the plotting procedure of the histogram. The bin size of all the histogram will be 1 and the first break occurred at smallest integer in the dat a set. The bandwidth will be chosen by using the rule of thumb formulas in equation (4.15) and the MISE, equation (4.16). We used the rule of thumb values as a starting point, increasing its value by 0.05 until a suitable fit is seen along with a small MISE value. This will be considered the best kernel estimate for the unknown underlying probabil ity distribution of the sample. 4.4 The Kernel Density Estimate We fitted the kernel density estimate using the Ga ussian kernel to the jackknifing pseudovalues for the stations in the sanctuary. We found the maximum likelihood
95 estimates for the parameters of the normal probabil ity distribution that is assumed under the jackknifing procedure. Table 4.3 has the bandwidth and MISE for each of t he kernel density estimates to the underlying probability density distribution for the years 1996 to 2006. The bandwidths for all the years were found to be aroun d 0.3 to 0.5. As the results show the MISE values are all very small, less than 0.006. Table 4.3 Parameter Estimates for the Kernel Densi ty Estimate and the Normal Probability Distribution. Kernel Density Normal Distribution Year Bandwidth MISE Mean Std Dev 1996 0.350 0.0058 2.31 1.03 1997 0.364 0.0056 2.30 1.06 1998 0.332 0.0061 2.19 1.10 1999 0.495 0.0049 1.97 1.07 2000 0.419 0.0054 2.06 0.98 2001 0.366 0.0056 2.04 1.07 2002 0.362 0.0057 2.16 1.00 2003 0.329 0.0062 2.08 1.15 2004 0.333 0.0061 2.10 1.18 2005 0.390 0.0053 2.08 1.08 2006 0.460 0.0050 2.00 1.06 One criterion in selecting the bandwidth is to use the bandwidth with the minimum MISE. In 1996, as shown in Figure 4.1, the re exists a bandwidth with a minimum MISE. This MISE occurs at a bandwidth of 0 .452 with a MISE value of 0.0051. The bandwidth chosen was 0.35 with a MISE value of 0.0058 which is close to the lowest MISE value possible. We chose to use th e bandwidth 0.350 rather than 0.452, since it gave a better fit to the histogram, (Figur e 4.2).
96 Figure 4.1 MISE vs. Bandwidth for 1996. 0.10.20.30.188.8.131.52 0.0050.0100.0150.0200.025 hMISE Figure 4.2 Kernel Density Estimate Fit: 1996. ST1996 -6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0
97 Table 4.1 also shows the MLE for both parameters o f the normal distribution fit for the same jackknifing pseudo values. Here we mu st reiterate, that the goodness-of-fit statistics used in Chapter two all showed that the normal probability distribution is not a good fit to the jackknifing pseudovalues. Figure 4 .3 show the normal probability distribution fit to the jackknifing pseudovalues to the data. We can see from Figures 4.2 and 4.3 that the Gaussian kernel density estimate f its the data very well and is better than the normality probability distribution fit. We tri ed to fit other continuous probability distribution functions to the data such as the logn ormal, Weibull, gamma and others with no success. Figure 4.3 Normal Distribution Fit: 1996. The main problem with the data values is the high peaks that occur, giving the existing probability distribution trouble to fit bu t does not pose a problem for the kernel density estimate. Similarly, close fit results as seen for 1996 is seen for the data of the other nine years. The kernel density estimate vers us the normality probability ST1996 -6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0
98 distribution fit for the years from 1997 to 2006 ca n be seen in Figure 4.4. We can see that the kernel density estimate has a superior fit to the data for all the years compared to the normality probability distribution fit. Figure 4.4 Kernel Density Estimate vs. Normal Prob ability Distribution: 1997 to 2006. ST1997Density -6-20246 0.00.20.40.60.81.0 ST1997Density -6-20246 0.00.20.40.60.81.0 ST1998Density -6-20246 0.00.20.40.60.81.0 ST1998Density -6-20246 0.00.20.40.60.81.0
99 Figure 4.4 (Continued) ST1999Density -6-20246 0.00.20.40.60.81.0 ST1999Density -6-20246 0.00.20.40.60.81.0 ST2000Density -6-20246 0.00.20.40.60.81.0 ST2000Density -6-20246 0.00.20.40.60.81.0
100 Figure 4.4 (Continued) ST2001Density -6-20246 0.00.20.40.60.81.0 ST2001Density -6-20246 0.00.20.40.60.81.0 ST2002Density -6-20246 0.00.20.40.60.81.0 ST2002Density -6-20246 0.00.20.40.60.81.0
101 Figure 4.4 (Continued) ST2003Density -6-20246 0.00.20.40.60.81.0 ST2003Density -6-20246 0.00.20.40.60.81.0 ST2004Density -6-20246 0.00.20.40.60.81.0 ST2004Density -6-20246 0.00.20.40.60.81.0
102 Figure 4.4 (Continued) ST2005Density -6-20246 0.00.20.40.60.81.0 ST2005Density -6-20246 0.00.20.40.60.81.0 ST2006Density -6-20246 0.00.20.40.60.81.0 ST2006Density -6-20246 0.00.20.40.60.81.0
103 The Gaussian kernel density estimate for 1996 pseu dovalues data is given by 2 1 105 20.35 111 Âˆ (). 36.75 2ixX ifxep== (4.17) Similar Gaussian kernel density estimate can be wri tten for the pseudovalues data for the other years using the estimates in Table 4.3 and eq uation (4.2). 4.5 Comparison of the Nonparametric and Parametric Although using the normal probability distribution is not valid and is not recommended to be used, we proceed to compare the r esults with the kernel density estimate of the pseudovalues data. Table 4.4-A and Table 4.4-B gives the mean, stand ard deviation, standard error, and the 95% confidence interval of the true mean fr om the Gaussian kernel density estimate and from the normal probability distributi on, respectively. Table 4.4-A Statistical Properties of the Gaussian Kernel Density Estimate. 95% Year Mean Std Dev Std Error Upper Lower 1996 2.31 1.08 0.105 3.858 -0.969 1997 2.30 1.11 0.109 3.908 -1.271 1998 2.19 1.15 0.112 3.730 -1.494 1999 1.97 1.18 0.115 3.814 -1.625 2000 2.06 1.06 0.103 3.651 -1.170 2001 2.04 1.13 0.110 3.448 -1.859 2002 2.16 1.06 0.103 3.966 -1.204 2003 2.08 1.19 0.117 3.694 -1.937 2004 2.10 1.22 0.119 3.681 -2.028 2005 2.08 1.14 0.111 3.795 -1.501 2006 2.00 1.15 0.113 3.675 -1.617
104 Table 4.4-B Statistical properties of the Normal P robability Distribution. 95% Year Mean Std Dev Std Error Upper Lower 1996 2.31 1.03 0.100 4.323 0.305 1997 2.30 1.06 0.103 4.368 0.230 1998 2.19 1.10 0.108 4.356 0.026 1999 1.97 1.07 0.105 4.074 -0.139 2000 2.06 0.98 0.095 3.974 0.149 2001 2.04 1.07 0.105 4.145 -0.069 2002 2.16 1.00 0.098 4.120 0.196 2003 2.08 1.15 0.113 4.346 -0.176 2004 2.10 1.18 0.115 4.410 -0.212 2005 2.08 1.08 0.105 4.193 -0.026 2006 2.00 1.06 0.104 4.069 -0.071 Figure 4.5 shows the cdf of the Gaussian kernel de nsity estimate for the pseudovalues from 1996. The red lines show the 95% confidence interval of the true mean. We used the cdf to find the 95% confidence in tervals of the true mean for the Gaussian kernel density estimate. Figure 4.5 Cumulative Distribution Function for th e Gaussian Kernel Density Estimate.
105 The estimates for the true expected mean are equal for both the Gaussian kernel density estimate and the normal probability distrib ution, Tables 4.4-A and 4.4-B. The standard error is a monotone function of the standard deviation, comparing one of these is sufficient in comparing the other. We compare the standard deviations. Figure 4.6 shows a graph of the standard deviation for both probability distribution functions. The standard deviation for the Gaussian kernel density estimate is larger than that for the normal distribution for every year. Figure 4.6 Standard Deviations: Gaussian Kernel De nsity Estimate (KDE) vs. Normal Probability Distribution (N). 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 19941996199820002002200420062008 KDE N The results from the 95% confidence intervals for the true mean is important as shown in the works by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) for the CREMP. In those works, the confidence intervals for the true mean of the pseudovalues data sets were constructed under the n ormality assumption. This mean value is the estimate for the true Shannon-Wiener d iversity index of the stony corals. The results of the confidence interval are given in Tab les 4.4-A and 4.4-B. In Figure 4.7, the
106 blue solid line is the interval for the normal dist ribution and the red dotted line represents the interval from the kernel density estimator. The 95% confidence intervals are completely different using the Gaussian kernel dens ity estimate and the normal probability distribution. Figure 4.7 95 % Confidence Interval: Gaussian Kern el Density Estimate (KDE) vs. Normal Probability Distribution (N). -3 -1 1 3 519941996199820002002200420062008 KDE N 4.6 Conclusion We have shown that the Gaussian kernel density est imate is clearly a much better fit to the jackknifing pseudovalues compared to the normal probability distribution and other continuous probability distributions. A good estimate of the true probability structure of the pseudovalues data is imperative fo r sound decision making. The estimates of the statistical properties of the data obtained from the Gaussian kernel density estimate are different than the ones from the normal probabi lity distribution.
107 The results presented in this chapter can help ana lyze and interpret the ShannonWeiner diversity index of the stony coral for the C REMP. A better understanding of the underlying probability distribution of the jackknif ing pseudovalues can enhance the advantages of using the jackknifing method in obtai ning a good estimate of the ShannonWiener diversity index and providing data that can be used in the statistical analysis and interpretation of the Shannon-Wiener diversity inde x. Furthermore, we have shown that one cannot just assume that the pseudovalues obtain ed from the jackknifing procedure are normally distributed as suggested by Zahl (1977). T his assumption can lead to obtaining false information.
108 Chapter 5 Statistical Modeling of Stony Coral Cover 5.1 Introduction The coral reef communities are very important ecos ystems in the world. They are home to at least 4,000 species and almost a third o f the worldÂ’s marine fish species (Paulay 1996). Coral reefs have ecological and econ omic significance throughout the world. Ecologically for Florida and elsewhere, they provide habitat for fish and macroinvertebrates. They also serve as protection a gainst wave action, especially during storm surges and they provide source of coral rubbl e and sand. Coral reefs can also be the base for island formation, as well as maintaining a nd replenishing beaches. The economic significance for FloridaÂ’s coral reefs is enormous, especially for the tourist industry and the generation of marine products for commercial ex port. Tourism generates over $50 billion a year; in 2003 over 74 million visitors en gaged in reef-based activities in Florida (Andrews et al. 2005). Coral reefs are very specialized communities that are highly sensitive to local, regional, and global environmental changes; therefo re it is very important to learn as much as we can about this system. As Hallock (1997) warned, the recovery of reef building communities from extinction events require s millions of years. With this in mind, we believe a statistical model to predict an estimate stony coral cover proportion would be very beneficial to monitor ecosystem healt h of coral reefs. This would give scientists another tool to understand what environm ental factors drive the coral reef ecosystem. Since reefs are complex systems, it is v irtually impossible to represent all interactions in one statistical model of the ecosys tem with a few parameters.
109 Nevertheless, it is imperative to try to model the information that is available to us. Having such a statistical model will assist the man agers in making decisions that would be meaningful for ecosystem conservation. The focus of this chapter is to develop a statisti cal predictive model of stony coral cover proportion. Such a model would identify the a ttributable variables that influence the stony coral cover. The quality of the model wil l be a function of the limited data that are available on the subject matter. 5.2 Response and Attributable Variables The response variable is the stony coral cover pro portion data obtained from CREMP, which is one of the components of the Water Quality Protection Program (WQPP) instituted for the Florida Keys National Mar ine Sanctuary (FKNMS). The attributable variables were obtained from the both the CREMP and the Water Quality Monitoring Project (WQMP). WQMP is the othe r component of the WQPP instituted for the Florida Keys National Marine San ctuary by EPA in 1995. Data were provided by the Southeast Environmental Research Ce nter-Florida International University, Water Quality Monitoring Network which is supported by SFWMD/SERC Cooperative Agreements #C-10244 and #C-13178 as wel l as EPA Agreement #X99462194-0. Sampling is done on a quarterly basis for mor e than 200 stations in the FKNMS and Shelf since March 1995 by SERC (Boyer and Jones 200 2). The attributable variables obtained from CREMP are station location: latitude and longitude, depth of the station: inshore and offsho re and type of stony coral reef: patch, shallow, hardbottom and deep. The attributable vari ables obtained from WQMP project are total nitrogen, total phosphorus, chlorophyll a total organic carbon, turbidity, salinity and temperature. Both surface and bottom measuremen ts were considered. These attributable variables were chosen from among other variables collected by WQMP project on the advice of Walter Jaap, a leading sci entist in the field of corals and who has worked many years with Florida Fish and Wildlife Re search Institute (FWRI and CREMP until he retired). He suggested these as stro ng contributing variables that will
110 influence stony coral cover. Table 5.1 gives the ma thematical notations of the variables used in the present study. Table 5.1 Notation of Variables. Notation Variable CC Coral Cover T Arcsine transformation of coral cover lat Latitude of the CREMP station lon Longitude of the CREMP station INSD Inshore Depth of the CREMP station OSD Offshore Depth of the CREMP station P1 Patch reef S1 Shallow reef D1 Deep reef TN Total nitrogen TP Total phosporus CHLA Chlorophyll a TOC Total organic carbon TURB Turbidity SAL Salinity TEMP Temperature 5.3 Data Manipulation The first major problem that we encountered was th at the sister projects (CREMP and WQMP) do not sample at the exact stations. The matching of the stations from both projects were obtained from Callahan (2005). He use d an Arcview query tool developed by FWRI to create such matching. Water-quality stat ions were chosen based on four main criteria: 1) proximity to CREMP sites, 2) depth sim ilarity, 3) relative distance to shore, and 4) similarity of benthic cover under the WQMN s tation (i.e., reef/ hardbottom/ seagrass). Due to the close proximity of some of th e CREMP, deep and shallow reef stations, both stations were paired with the same w ater-quality station. This allowed us to obtain data for only 27 stations for which data was collected within the sanctuary region of the CREMP. The matching stations are given in Ta ble 5.2.
111 Table 5.2 CREMP and WQMP Stations Pairing List. CREMP Stations WQMP Stations 9P1 Turtle Patch 212 Turtle Harbor 9S1 Carysfort Shallow 216 Carysfort Reef 9D1 Carysfort Deep 216 Carysfort Reef 9S2 Grecian Rocks 400 Grecian Rocks 9P3 Porter Patch 400 Grecian Rocks 9H2 El Radabob 220 Radabob Key 9S3 Molasses Shallow 225 Molasses Reef 9P4 Admiral Patch 224 Molasses Reef Channel 9S4 Conch Shallow 228 Conch Reef 9D4 Conch Deep 264 Aquarius 7S1 Alligator Shallow 401 Alligator Reef 7S2 Tennessee Shallow 243 Tennessee Reef 7D2 Tennessee Deep 243 Tennessee Reef 7H2 Long Key 242 Long Key Channel 7P1 West Turtle 248 Coffins Patch Channel 7P2 Dustan Rocks 248 Coffins Patch Channel 5S1 Sombrero Shallow 402 Sombrero Key 5S2 Looe Key Shallow 263 Looe Key 5D2 Looe Key Deep 263 Looe Key 5P4 Jaap Reef 268 Saddlebunch Keys 5P1 W. Washer Woman 269 W. Washerwoman 5S3 Eastern Sambo Shallow 273 Eastern Sambo Offshore 5D3 Eastern Sambo Deep 273 Eastern Sambo Offshore 5S4 Western Sambo Shallow 403 Western Sambo 5D4 Western Sambo Deep 403 Western Sambo 2D1 Sand Key Deep 281 Middle Ground 3H1 Content Keys 302 Content Passage The second problem was one of missing values in so me of the environmental variables: TN, TP, TURB and TOC for stations of the WQMP numbered: 216, 224, 268 and 302. The second quarter sampling data values fo r 2004 were used from the WQMP that took place between April and June. Contour ana lysis was used to approximate the missing values by using all the available data for that quarter. We proceeded with kriging with linear variogram for the contour analysis, sin ce we have only 84 data values for each of the variables to generate contour maps. Kriging allows us to extrapolate values beyond the data set. It is a method of interpolation, whic h predicts unknown values from data
112 observed at known locations. This method uses vario grams to express the spatial variation, and it minimizes the error of predicted values, which are estimated by spatial distribution of the predicted values. Figure 5.1 sh ows the contour map obtained for TN with the WQMP stations, which will be used in build ing the stony coral cover statistical model. Figure 5.1 Contour map for total nitrogen (TN) -82-81.8-81.6-81.4-81.2-81-80.8-80.6-80.4-80.2 24.6 24.8 25 25.2 212 216 216 220 224 225 228 242 243 243 248 248 263 263 264 268 269 273 273 281 302 400 400 401 402 403 403 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42Total N From this contour map we extrapolated the missing TN values for WQMP stations numbered: 216, 224, 268 and 302. Table 5.3 gives the results of the contour analysis for the environmental parameters: TN, TP, TURB and TOC.
113 Table 5.3 Contour Analysis Results. Station TN TP TURB TOC 216 15.32 0.1181 0.150 107.9 224 18.02 0.1190 0.317 117.9 268 16.70 0.1586 0.932 126.8 302 14.88 0.2260 1.156 119.0 5.4 Multivariate Statistical Model Using environmental data only for 2004, we investi gated the best statistical model fit for the response variable, stony coral cover, w ith respect to the environmental variables mentioned in the previous section. We acc omplish this by obtaining the best parsimonious statistical models according to Mallow Â’s C(p) statistic, 2 R and adjusted 2 R In addition we investigated the interactions and higher order of the variables in the selected list of statistical models. The selection of the best statistical model was based on the smallest residuals using similar data from the year 2005. 5.4.1 Transformations of the Response Variable. We studied various statistical models after consid ering several transformations of the response variable. This was necessary since in Chapter two we found that the stony coral cover followed a three-parameter lognormal pr obability distribution for the data collected in 2004 and the transformation of the res ponse gave better results. Also, the 27 values available for formulating the present statis tical model showed to be positively skewed. Because this data set is small, we found i t necessary to look for the most parsimonious statistical model possible. We invest igate statistical models with the original, ln(0.002) CC + ln(0.005) CC + Box-Cox and arcsine transformations of the response variable. These statistical models were co nstructed using the 14 variables, previously mentioned. A sequential approach was us ed, in which the first statistical models consisted of only one variable, the second s tatistical models consisted of
114 combinations of two variables, the third consisted of combinations of three variables, etc. Out of these statistical models the best statistica l models, that were further investigated, were chosen according to the MallowÂ’s C(p) statisti c. MallowÂ’s C(p) statistic is a predictive oriented c riterion. MallowÂ’s C(p) statistic measures the significance of including additional v ariables and is defined by 22 2Âˆ ()() Âˆp snp Cps s -=+ 2 Âˆ s is the residual mean square of the full model. 2 s is the estimate of the residual mean square of the model under consideration. The best m odel according to MallowÂ’s C(p) statistic is one that has a MallowÂ’s C(p) statistic equivalent to the number of attributable variables in the model plus one. The first log transformation, ln(0.002) CC + was used since that was the variable adjustment deemed necessary to transform the sanctu ary data of 2004 to a normal probability distribution. Two statistical models w ere considered using the log transformation, ln(0.002) CC + which had 6 attributable variables. The MallowÂ’s C(p) statistics were very high for the 1 to 5 attributab le variables. The only reasonable C(p)Â’s were obtained with 6 attributable variables or high er but considering the samples size, these would be quite unreasonable models. For the second one, we used the log transformation that had the lowest ShapiroWilks normality test statistic. The statistical mod els after the transformation of ln(0.005) CC + were much improved over the ln(0.002) CC + transformation. They had lower MallowÂ’s C(p) statistic. Three statistical mo dels were considered using the log transformation, ln(0.005) CC + which had 5 attributable variables. The MallowÂ’s C (p) statistic was very high for the 1 to 4 attributable variables. The only reasonable MallowÂ’s C(p) statistic was obtained with 5 attributable var iables or higher but considering the samples size, these would still be quite unreasonab le models. We proceeded to utilize the Box-Cox transformation which assumes that a transformation from the power family may be appropr iate. The estimate of the best power is located by using maximum likelihood. The best ob tained value was 0.15 l =
115 The Box Â–Cox transformation is defined by: 1 0 ln0 y if w yifll l l = n = Three models were considered using the Box-Cox tran sformation with 6 attributable variables. The MallowÂ’s C(p) statistic were very hi gh for the 1 to 5attributable variables. Six attributable variables would be too high when c onsidering the sample size. Finally, we used the arcsine transformation which is based on multiplying the response variable by 0.01, followed by the square r oot and then taking the arcsine of that result. The transformation is defined by arcsin0.01 TCC = The arcsine transformation has been used extensive ly in analyzing environmental data e.g., Rogers et al. (1983), Carleton and Done (1995 ), Murdoch and Aronson (1999), and Wielgus et al. (2004), among others. This transfor mation gave the lowest number of attributable variables that contribute to the stati stical model in terms of the MallowÂ’s C(p) statistic. There were one 3and seven 4-attributab le variable statistical models that were considered according to the MallowÂ’s C(p) statistic 5.4.2 Best Initial Statistical Model The correlation matrix of the response variable an d the 14 environmental variables considered for the statistical model are summarized in Table 5.4. From here on we will refer to the response variable as T, unless stated otherwise. From the correlation matrix you can see that P1 and Sal are strongly cor related with T, whereas the others have a weaker correlation with T. In addition some of the variables are correlated with each other such as TOC and TN or SAL and P1. Upon c hoosing the model variables it will be necessary to check the variance inflation f actor to test for multicollinearity if correlated attributable variables exist in the fina l statistical model.
116 Table 5.4 Correlation Matrix of Response and Attri butable Variables. D1 S1 P1 lat lon INSD OSD TN TP CHLA TOC TURB SAL TEMP D1 S1 -0.454 P1 -0.350 -0.454 lat -0.231 0.006 0.116 lon -0.221 0.107 0.075 0.939 INSD 0.861 -0.341 -0.366 -0.121 -0.050 OSD 0.883 -0.401 -0.270 -0.148 -0.099 0.973 TN 0.266 0.005 -0.169 -0.262 -0.291 0.072 0.142 TP -0.185 0.093 -0.078 0.003 -0.076 -0.102 -0.196 0 .066 CHLA 0.157 -0.242 -0.191 -0.154 -0.368 0.012 0.078 0.465 0.072 TOC -0.079 -0.226 0.003 0.179 0.033 -0.219 -0.177 0 .467 0.164 0.402 TURB 0.139 -0.334 0.028 -0.184 -0.373 -0.113 -0.020 0.663 0.089 0.806 0.680 SAL -0.330 -0.210 0.437 -0.003 -0.028 -0.265 -0.242 -0.364 -0.023 -0.379 0.093 -0.179 TEMP -0.397 -0.156 0.328 0.726 0.730 -0.260 -0.288 -0.564 -0.150 -0.424 0.158 -0.345 0.532 T -0.234 -0.208 0.716 -0.254 -0.229 -0.204 -0.122 0.084 -0.151 -0.294 -0.158 -0.057 0.557 0.122
117 After choosing the best statistical models using M allowÂ’s C(p) statistic, we evaluate their suitability as the best statistical model via their 2 R statistic and adjusted 2 R statistic. The coefficient of determination 2 R is a criterion used in statistical model selection. The coefficient of determination 2 R is defined by () () 2 2 1 2 111n ii i n i i yy SSE R SST yy = ==-=) SSE is the error sums of squares and SST is the tot al sums of squares. It measures the statistical modelÂ’s capability to fit the stony cor al cover. This value is a measure of the variation about the mean explained by the regressio n model. The addition of a new variable will always increase 2 R Thus, the inclusion of a new variable must be che cked to see that it is of statistical significance to th e response. The adjusted 2 R is defined by 2 2 1 11(1) 1adjSSE n np RR SST np n =-=-. It is a modification of the 2 R statistic. This is done by replacing SSE and SST by their corresponding mean squares which is the division SS E and SST by their corresponding degrees of freedom, np and 1 n respectively. The adjusted 2 R is not automatically increased by introducing more variables to the stat istical model. If additional variables are added that do not contribute significantly to t he statistical model, this criterion penalizes the statistical model by lowering the val ue of the statistic. Thus adjusted 2 R helps in avoiding over fitting the model and must b e very close to the 2 R statistic. The best initial statistical model was the four va riable model with variables: patch reef, latitude of the station, total organic carbon and salinity. We identified and ranked the statistically significant attributable variable s in Table 5.5.
118 Table 5.5 Statistical Ranking of the Attributing V ariables to Stony Coral Cover. Rank Variable 2 R 1 Patch reef 0.513 2 Latitude of the CREMP station 0.628 3 Salinity 0.691 4 Total organic carbon 0.708 The model with these four attributing variables ex plains 70.81% of the variation in the response variable and has an adjusted 2 R value of 0.66. Since SAL and P1 had a correlation with each other, we investigated the po ssibility of multicollinearity using the variance inflation factor (VIF). Multicollinearity is present if the VIF values are greater than 10. The VIF of the four attributable variable s in our statistical model are close to 1; therefore, we should not be concerned with the corr elation that exists between P1 and SAL. 5.4.3 Interaction and Higher Order While the attributable variables in Table 5.5 expl ain 70.81% of the variation in the response, to improve the quality of the model w e tested for possible contribution to the stony coral cover by various interactions of th e attributable variables. We found that there are significant interactions between patch re ef and latitude, patch reef and total organic carbon, latitude and total organic carbon, and total organic carbon and salinity. We ranked the statistically significant attributabl e interaction variables in Table 5.6.
119 Table 5.6 Statistical Ranking of the Attributing V ariables with Interactions. Rank Variable 2 R 1 P1_TOC 0.527 2 latDD 0.652 3 SAL 0.704 4 P1 0.745 5 TOC 0.781 6 TOC_Sal 0.805 7 Lat_TOC 0.825 8 P1_Lat 0.839 The statistical model results in a 2 R statistic value of 0.839 and adjusted 2 R statistic value of 0.767. This is a significant improvement over the model without interactions. We found no statistically significant higher order variables. The theoretical statistical model that best descri bes the statistical behavior of the response variable, stony coral cover, according to the attributable variables (lat, P1, SAL and TOC) and their interactions is given by 012345 67811_ 1___ PlatSALTOCPlat Z PTOClatTOCTOCSALbbbbbb bbbe +++++=n ++++ (5.1) Z is the response variable (arcsine transformed of t he stony coral cover) given the attributable variables found in the model. i b s are the weights of the individual attributable variables and their interactions that drive the estimate of the attributable variables and e is the random error. The estimation of the i b s using the stony coral cover and the attributable variables and their interactions gives the resultan t statistical model 1.20.6210.0440.0640.00980.0261_ Âˆ 0.00051_0.0004_0.0005_ PlatSALTOCPlat Z PTOClatTOCTOCSAL-+-++-=n++. (5.2)
120 where Âˆ Z is the estimate of the arcsine transformed of the s tony coral cover. To obtain the stony coral cover, we untransformed the Âˆ Z by 2 Âˆ sin() Âˆ 0.01 Z Y=. 5.4.4 Model Predictive Capability Of the statistical models investigated, the best i s given in equation (5.2). This model contains the exploratory variables: patch ree f, total organic carbon, latitude and salinity and the interactions between patch reef an d latitude, patch reef and total organic carbon, latitude and total organic carbon, and tota l organic carbon and salinity. To investigate how well this model acts as a predictor model, the predictability of the stony coral cover for 22 stations sampled in 2005 is inve stigated. The attributable variables, TOC and salinity, are obtained from the WQMP sample d in the second quarter of 2005. The observed stony coral cover, P1 and lat are obta ined from the CREMP. The results are given in Table 5.7 as percentage cover.
121 Table 5.7 Prediction Results. CREMP Type Num WQMP Observed (%) Predicted (%) Turtle P 1 212 6.07 0.45 El Radabob HB 2 220 0.14 2.44 Molasses S 3 225 4.61 1.01 Conch S 4 228 2.28 3.02 Long Key HB 2 242 5.92 18.04 Tennessee D 2 243 5.19 4.72 Tennessee S 2 243 2.55 4.79 W. Turtle Shoal P 1 248 10.74 10.56 Dustan P 2 248 13.20 10.77 Looe Key D 2 263 4.87 5.84 Looe Key S 2 263 4.90 5.81 Conch D 4 264 2.90 1.86 W. Washer Women P 1 269 21.79 14.13 Eastern Sambo D 3 273 4.28 4.60 Eastern Sambo S 3 273 2.26 4.60 Sand Key D 1 281 2.60 8.89 Grecian Rocks S 2 400 2.18 2.05 Porter Patch P 3 400 1.49 1.73 Alligator S 1 401 0.57 5.54 Sombrero S 1 402 4.49 5.52 Western Sambo D 4 403 1.69 4.28 Western Sambo S 4 403 6.58 4.28 The modelÂ’s prediction had a 1 to 3 % difference fo r about 17 of the 22 stations. Good predictions were observed for various type of reef and high and low stony coral cover. The worst prediction occurred at Long Key (hardbott om) where the observed value was about 6 % but the predicted value was 18%. The stat ion with the highest stony coral cover in the data set was also not predicted proper ly: observed value of 22 % and the predicted value was 14%. Most of the medium stony c oral covered (2 to 7 % cover) stations were predicted closely.
122 5.4.5 Final Model We checked the model, equation (5.2), to see how w ell it predicted the 2005 values. Since we are satisfied with the model consi dering the amount of data we had, we refitted equation (5.2) with the initial 27 values plus the 22 data values. The resulting model is given by 0.5900.44010.00380.01940.0062 Âˆ 0.01841_0.000311_0.000022_0.00015_ PlatSALTOC YPlatPTOClatTOC TOCSAL -+-++=-+-nThis model can now be used to predict the stony co ral cover of other areas that have stony coral cover. To predict stony coral cover, al l that is needed for this model is the latitude (location), type of reef, salinity and tot al organic carbon. 5.5 Conclusion Despite the problems we encounter in this study an d the limited samples size, this predictive statistical model for the stony coral co ver has shown that it is possible to formulate a very good predictive model for the ston y coral cover. The statistically significant attributing variable s to the predictive statistical model are latitude (location), patch reef, salinity and t otal organic carbon. Patch reef contributes positively to arcsine transformed stony coral cover This is not surprising as the works by Tsokos et al. (2006a) showed that since 1996 up to 2005, the patch reefs had the highest coverage by reef type. Higher latitude shows to hav e a negative impact on the arcsine transformed stony coral cover. This again is substa ntiated by the fact that the high stony coral coverage are found in the lower keys and Dry Tortugas in the works of Tsokos et al. (2006a). Salinity has a positive effect to the prop ortion stony coral cover. Keeping all other variables in the model constant the proportio n stony coral cover increases by 0.0194 for every 1 unit increase in salinity not including its effect in the interaction with total
123 organic carbon. Total organic carbon has a positive contribution of 0.0062 for every unit increase of TOC while all other variables are kept constant and not including its contribution due to its interactions with patch ree f, latitude and with salinity. This positive contribution concurs with the chemical pro perties of TOC, of which colored dissolved organic matter (CDOM) and particulate org anic matter (POM) are two components. Screening from UV-radiation through li ght attenuation by suspended POM (Goreau, McClanahan, Hayes and Strong 1998), or thr ough light absorption by CDOM (Otis, Carder, English, Ivey and Warrior 2004), can also protect corals from solarradiation that sontributes to bleaching events. The interactions between patch and latitude, latitude and total organic, and salinity and total organic carbon all have a negative contribution to the proportion stony coral cover. The interaction between patch and total organic carbon has a positive contributio n to the proportion stony coral cover. A model such as the one formulated here for the FK NMS can be effectively used in many ways by local managers of reserve sites thr oughout the world. They can use it to ensure that influences from developing areas are no t affecting the coral reef by measuring the relevant environment variables. It can also be used in conjunction with GIS mapping of the habitat areas such as the one that has been done in the Florida Keys. This model can be used to increase the accuracy of such mappin g endeavors. This would also help governments in locating the best areas to create re levant reserves to ensure the existence of coral reefs.
124 Chapter 6 Future Research 6.1 Introduction From the results of the present study, we have iden tified several interesting and important extensions for this research. 6.2 Non-Parametric Kernel Density We will proceed to study the behavior of the bandw idth to obtain better nonparametric probabilistic estimates of the behavior of the pseudovalues. We will also study if any other kernel function will give better resul ts of such data. We will seek a combination of an estimate of the optimal bandwidth and the appropriate kernel function that will minimize the mean integrated square error We also propose to investigate the behavior of the kernel density estimate as a functi on of sample size. 6.3 Improving the Proposed Statistical Model We will continue to improve the proposed statistic al model by identifying and testing the significant contributions of additional attributable variables and their interactions with the ones we have already identifi ed. Attributable variables such as surface temperature, humidity, rainfall, wind speed current among others will be investigated. We believe that these additional attr ibutable variables will significantly improve the quality of the statistical model.
125 6.4 Surface Response Analysis We propose to apply surface analysis methodology t o the developed model and any improvements of the subject model. We would lik e to identify the behavior of the attributable variables so that we will maximize the response, percent stony coral cover, with a specified degree of accuracy. 6.5 Stony Coral Cover Parametric Analysis We will proceed to further investigate the works d one by Yanev et al. (2003a, b, 2004) and Tsokos et al. (2005, 2006a) as it pertai ns to hypothesis testing of the stony coral cover from over the years for the Coral Reef Evaluation and Monitoring Project. Since we have identified the probability distributi on of the stony coral cover, we can now proceed to test using parametric analysis as compar ed to the non-parametric analysis that was done by Yanev et al. (2003a, b, 2004) and Tsok os et al. (2005, 2006a).
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About The Author Armando J. Hoare was born in Corozal Town, Belize, Central America, the ninth child of 11 siblings. In 1992 he graduated magna cum laude with his Bachelor of Science degree in Mathematics and Chemistry from Regis University, Colorado. He taught mathematics for 10 years at St. JohnÂ’s Junior College, served a s department head for two years and was awarded Teacher of the Year in 1998. In 2002, Armando received his Master of Education in Educational Leadership from the University of North Florida with a distinc tion in mathematics. He achieved a Master of Arts in Mathematics at the University of South Florida. He then pursued his doctoral degree focusing on Statistics. During his time as a graduate student, Armando received a certificate of recognition for outstandi ng performance as a graduate teaching assistant, and was nominated for the ProvostÂ’s Awar d. Armando is married to Ana and they have a son, Armando Fulgencio.