This page intentionally left blank Experimental Design and Data Analysis for Biologists An essential textbook for any student or researcher in Gerry Q u i n n is in the School of Biological biology needing to design experiments, sampling Sciences at Monash University, with research inter- programs or analyze the resulting data. The text ests in marine and freshwater ecology, especially begins with a revision of estimation and hypothesis river floodplains and their associated wetlands. testing methods, covering both classical and Bayesian philosophies, before advancing to the analysis of M i c h a e l Keough is in the Department of Zoology linear and generalized linear models. Topics covered at the University of Melbourne, with research inter- include linear and logistic regression, simple and ests in marine ecology, environmental science and complex ANOVA models (for factorial, nested, block, conservation biology.
split-plot and repeated measures and covariance designs), and log-linear models. Multivariate tech- Both authors have extensive experience teaching niques, including classification and ordination, are experimental design and analysis courses and have then introduced. Special emphasis is placed on provided advice on the design and analysis of sam- checking assumptions, exploratory data analysis and pling and experimental programs in ecology and presentation of results. The main analyses are illus- environmental monitoring to a wide range of envi- trated with many examples from published papers ronmental consultants, university and government and there is an extensive reference list to both the scientists.
statistical and biological literature. The book is sup- ported by a website that provides all data sets, ques- tions for each chapter and links to software. Experimental Design and Data Analysis for Biologists Gerry P. Quinn Monash University Michael J.
Keough University of Melbourne Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. Keough 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents Preface page xv 1 Introduction 1 1.3 Hypotheses and tests 3 1.4 Alternatives to falsification 4 1.5 Role of statistical analysis 5 1.2 Experiments and other tests 5 1.3 Data, observations and variables 7 1.1 Distributions for variables 10 1.2 Distributions for statistics 12 2 Estimation 14 2.1 Samples and populations 14 2.2 Common parameters and statistics 15 2.1 Center (location) of distribution 15 2.2 Spread or variability 16 2.3 Standard errors and confidence intervals for the mean 17 2.1 Normal distributions and the Central Limit Theorem 17 2.2 Standard error of the sample mean 18 2.3 Confidence intervals for population mean 19 2.4 Interpretation of confidence intervals for population mean 20 2.5 Standard errors for other statistics 20 2.4 Methods for estimating parameters 23 2.2 Ordinary least squares (OLS) 24 2.3 ML vs OLS estimation 25 2.5 Resampling methods for estimation 25 2.6 Bayesian inference – estimation 27 2.2 Prior knowledge and probability 28 2.6 Other comments 29 vi CONTENTS 3 Hypothesis testing 32 3.1 Statistical hypothesis testing 32 3.1 Classical statistical hypothesis testing 32 3.2 Associated probability and Type I error 34 3.3 Hypothesis tests for a single population 35 3.4 One- and two-tailed tests 37 3.5 Hypotheses for two populations 37 3.6 Parametric tests and their assumptions 39 3.1 Type I and II errors 42 3.2 Asymmetry and scalable decision criteria 44 3.3 Other testing methods 45 3.1 Robust parametric tests 45 3.3 Rank-based non-parametric tests 46 3.2 Adjusting significance levels and/or P values 49 3.5 Combining results from statistical tests 50 3.6 Critique of statistical hypothesis testing 51 3.1 Dependence on sample size and stopping rules 51 3.2 Sample space – relevance of data not observed 52 3.3 P values as measure of evidence 53 3.4 Null hypothesis always false 53 3.5 Arbitrary significance levels 53 3.6 Alternatives to statistical hypothesis testing 53 3.7 Bayesian hypothesis testing 54 4 Graphical exploration of data 58 4.1 Exploratory data analysis 58 4.2 Analysis with graphs 62 4.1 Assumptions of parametric linear models 62 4.1 Transformations and distributional assumptions 65 4.2 Transformations and linearity 67 4.3 Transformations and additivity 67 4.6 Censored and missing data 68 4.7 General issues and hints for analysis 71 4.1 General issues 71 CONTENTS vii 5 Correlation and regression 72 5.1 Parametric correlation model 72 5.3 Parametric and non-parametric confidence regions 76 5.3 Linear regression analysis 78 5.1 Simple (bivariate) linear regression 78 5.2 Linear model for regression 80 5.3 Estimating model parameters 85 5.4 Analysis of variance 88 5.5 Null hypotheses in regression 89 5.6 Comparing regression models 90 5.8 Assumptions of regression analysis 92 5.12 Regression through the origin 98 5.13 Weighted least squares 99 5.4 Relationship between regression and correlation 106 5.6 Power of tests in correlation and regression 109 5.7 General issues and hints for analysis 110 5.2 Hints for analysis 110 6 Multiple and complex regression 111 6.1 Multiple linear regression analysis 111 6.1 Multiple linear regression model 114 6.2 Estimating model parameters 119 6.3 Analysis of variance 119 6.4 Null hypotheses and model comparisons 121 6.6 Which predictors are important? 122 6.7 Assumptions of multiple regression 124 6.11 Collinearity 127 viii CONTENTS 6.12 Interactions in multiple regression 130 6.15 Finding the “best” regression model 137 6.17 Other issues in multiple linear regression 142 6.3 Path analysis and structural equation modeling 145 6.5 Smoothing and response surfaces 152 6.6 General issues and hints for analysis 153 6.2 Hints for analysis 154 7 Design and power analysis 155 7.2 Size of sample 157 7.5 Reducing unexplained variance 164 7.1 Using power to plan experiments (a priori power analysis) 166 7.2 Post hoc power calculation 168 7.3 The effect size 168 7.4 Using power analyses 170 7.4 General issues and hints for analysis 171 7.2 Hints for analysis 172 8 Comparing groups or treatments – analysis of variance 173 8.1 Single factor (one way) designs 173 8.1 Types of predictor variables (factors) 176 8.2 Linear model for single factor analyses 178 8.3 Analysis of variance 184 8.5 Comparing ANOVA models 187 8.6 Unequal sample sizes (unbalanced designs) 187 8.1 Random effects: variance components 188 8.3 Independence 193 CONTENTS ix 8.1 Tests with heterogeneous variances 195 8.2 Rank-based (“non-parametric”) tests 195 8.6 Specific comparisons of means 196 8.1 Planned comparisons or contrasts 197 8.2 Unplanned pairwise comparisons 199 8.3 Specific contrasts versus unplanned pairwise comparisons 201 8.7 Tests for trends 202 8.8 Testing equality of group variances 203 8.9 Power of single factor ANOVA 204 8.10 General issues and hints for analysis 206 8.2 Hints for analysis 206 9 Multifactor analysis of variance 208 9.1 Linear models for nested analyses 210 9.2 Analysis of variance 214 9.4 Unequal sample sizes (unbalanced designs) 216 9.5 Comparing ANOVA models 216 9.6 Factor effects in nested models 216 9.7 Assumptions for nested models 218 9.8 Specific comparisons for nested designs 219 9.9 More complex designs 219 9.10 Design and power 219 9.1 Linear models for factorial designs 225 9.2 Analysis of variance 230 9.4 What are main effects and interactions really measuring? 237 9.5 Comparing ANOVA models 241 9.9 Robust factorial ANOVAs 250 9.10 Specific comparisons on main effects 250 9.12 More complex designs 255 9.13 Power and design in factorial ANOVA 259 9.3 Pooling in multifactor designs 260 9.4 Relationship between factorial and nested designs 261 9.5 General issues and hints for analysis 261 9.2 Hints for analysis 261 x CONTENTS 10 Randomized blocks and simple repeated measures: unreplicated two factor designs 262 10.1 Unreplicated two factor experimental designs 262 10.1 Randomized complete block (RCB) designs 262 10.2 Repeated measures (RM) designs 265 10.2 Analyzing RCB and RM designs 268 10.1 Linear models for RCB and RM analyses 268 10.2 Analysis of variance 272 10.4 Comparing ANOVA models 274 10.3 Interactions in RCB and RM models 274 10.1 Importance of treatment by block interactions 274 10.2 Checks for interaction in unreplicated designs 277 10.1 Normality, independence of errors 280 10.2 Variances and covariances – sphericity 280 10.5 Robust RCB and RM analyses 284 10.7 Efficiency of blocking (to block or not to block?) 285 10.8 Time as a blocking factor 287 10.9 Analysis of unbalanced RCB designs 287 10.10 Power of RCB or simple RM designs 289 10.11 More complex block designs 290 10.1 Factorial randomized block designs 290 10.2 Incomplete block designs 292 10.3 Latin square designs 292 10.12 Generalized randomized block designs 298 10.13 RCB and RM designs and statistical software 298 10.14 General issues and hints for analysis 299 10.2 Hints for analysis 300 11 Split-plot and repeated measures designs: partly nested analyses of variance 301 11.1 Partly nested designs 301 11.1 Split-plot designs 301 11.2 Repeated measures designs 305 11.3 Reasons for using these designs 309 11.2 Analyzing partly nested designs 309 11.1 Linear models for partly nested analyses 310 11.2 Analysis of variance 313 11.4 Comparing ANOVA models 318 11.1 Between plots/subjects 318 11.2 Within plots/subjects and multisample sphericity 318 CONTENTS xi 11.4 Robust partly nested analyses 320 11.6 Analysis of unbalanced partly nested designs 322 11.7 Power for partly nested designs 323 11.8 More complex designs 323 11.1 Additional between-plots/subjects factors 324 11.2 Additional within-plots/subjects factors 329 11.3 Additional between-plots/subjects and within-plots/ subjects factors 332 11.4 General comments about complex designs 335 11.9 Partly nested designs and statistical software 335 11.10 General issues and hints for analysis 337 11.2 Hints for individual analyses 337 12 Analyses of covariance 339 12.1 Single factor analysis of covariance (ANCOVA) 339 12.1 Linear models for analysis of covariance 342 12.2 Analysis of (co)variance 347 12.4 Comparing ANCOVA models 348 12.2 Assumptions of ANCOVA 348 12.2 Covariate values similar across groups 349 12.1 Testing for homogeneous within-group regression slopes 349 12.2 Dealing with heterogeneous within-group regression slopes 350 12.3 Comparing regression lines 352 12.5 Unequal sample sizes (unbalanced designs) 353 12.6 Specific comparisons of adjusted means 353 12.7 More complex designs 353 12.1 Designs with two or more covariates 353 12.3 Nested designs with one covariate 355 12.4 Partly nested models with one covariate 356 12.8 General issues and hints for analysis 357 12.2 Hints for analysis 358 xii CONTENTS 13 Generalized linear models and logistic regression 359 13.1 Generalized linear models 359 13.1 Simple logistic regression 360 13.2 Multiple logistic regression 365 13.4 Assumptions of logistic regression 368 13.5 Goodness-of-fit and residuals 368 13.8 Software for logistic regression 371 13.4 Generalized additive models 372 13.5 Models for correlated data 375 13.1 Multi-level (random effects) models 376 13.2 Generalized estimating equations 377 13.6 General issues and hints for analysis 378 13.2 Hints for analysis 379 14 Analyzing frequencies 380 14.1 Single variable goodness-of-fit tests 381 14.1 Two way tables 381 14.2 Three way tables 388 14.3 Log-linear models 393 14.1 Two way tables 394 14.2 Log-linear models for three way tables 395 14.3 More complex tables 400 14.4 General issues and hints for analysis 400 14.2 Hints for analysis 400 15 Introduction to multivariate analyses 401 15.2 Distributions and associations 402 15.3 Linear combinations, eigenvectors and eigenvalues 405 15.1 Linear combinations of variables 405 15.4 Derivation of components 409 15.4 Multivariate distance and dissimilarity measures 409 15.1 Dissimilarity measures for continuous variables 412 15.2 Dissimilarity measures for dichotomous (binary) variables 413 15.3 General dissimilarity measures for mixed variables 413 15.4 Comparison of dissimilarity measures 414 15.5 Comparing distance and/or dissimilarity matrices 414 CONTENTS xiii 15.7 Standardization, association and dissimilarity 417 15.9 Screening multivariate data sets 418 15.