Experimental Design and Data Analysis for Biologists

Transcription

1 Experimental Design and Data Analysis for Biologists Gerry P. Quinn Monash University Michael J. Keough University of Melbourne CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface page xv I I Introduction Scientific method l Pattern description Models Hypotheses and tests Alternatives to falsification Role of statistical analysis Experiments and other tests Data, observations and variables Probability Probability distributions Distributions for variables Distributions for statistics 12 2 I Estimation Samples and populations Common parameters and statistics Center (location) of distribution Spread or variability Standard errors and confidence intervals for the mean Normal distributions and the Central limit Theorem Standard error of the sample mean Confidence intervals for population mean Interpretation of confidence intervals for population mean Standard errors for other statistics Methods for estimating parameters Maximum likelihood (ML) Ordinary least squares (OLS) ML vs OLS estimation Resampling methods for estimation Bootstrap Jackknife Bayesian inference - estimation Bayesian estimation Prior knowledge and probability Likelihood function Posterior probability Examples Other comments 29

3 3 I Hypothesis testing Statistical hypothesis testing Classical statistical hypothesis testing Associated probability and Type I error Hypothesis tests for a single population One-and two-tailed tests Hypotheses for two populations Parametric tests and their assumptions Decision errors Type I and II errors Asymmetry and scalable decision criteria Other testing methods Robust parametric tests Randomization (permutation) tests Rank-based non-parametric tests Multiple testing The problem Adjusting significance levels and/or P values Combining results from statistical tests Combining P values Meta-analysis Critique of statistical hypothesis testing Dependence on sample size and stopping rules Sample space-relevance of data not observed P values as measure of evidence Null hypothesis always false Arbitrary significance levels Alternatives to statistical hypothesis testing Bayesian hypothesis testing 54 4 I Graphical exploration of data Exploratory data analysis Exploring samples Analysis with graphs Assumptions of parametric linear models Transforming data Transformations and distributional assumptions Transformations and linearity Transformations and additivity Standardizations Outliers Censored and missing data Missing data Censored (truncated) data General issues and hints for analysis General issues 71

4 5 Correlation and regression Correlation analysis Parametric correlation model Robust correlation Parametric and non-parametric confidence regions Linear models Linear regression analysis Simple (bivariate) linear regression Linear model for regression Estimating model parameters Analysis of variance Null hypotheses in regression Comparing regression models Variance explained Assumptions of regression analysis Regression diagnostics Diagnostic graphics Transformations Regression through the origin Weighted least squares X random (Model II regression) Robust regression Relationship between regression and correlation Smoothing Running means LO(W)ESS Splines Kernels Other issues Power of tests in correlation and regression General issues and hints for analysis no General issues Hints for analysis I Multiple and complex regression in 6.1 Multiple linear regression analysis ill Multiple linear regression model Estimating model parameters Analysis of variance Null hypotheses and model comparisons Variance explained Which predictors are important? Assumptions of multiple regression Regression diagnostics Diagnostic graphics Transformations Collinearity 127

5 Interactions in multiple regression Polynomial regression Indicator (dummy) variables Finding the "best" regression model Hierarchical partitioning Other issues in multiple linear regression Regression trees Path analysis and structural equation modeling Nonlinear models Smoothing and response surfaces General issues and hints for analysis General issues Hints for analysis I Design and power analysis Sampling Sampling designs Size of sample Experimental design Replication Controls Randomization Independence Reducing unexplained variance Power analysis Using power to plan experiments (a priori power analysis) Post hoc power calculation The effect size Using power analyses General issues and hints for analysis General issues Hints for analysis I Comparing groups or treatments - analysis of variance Single factor (one way) designs Types of predictor variables (factors) Linear model for single factor analyses Analysis of variance Null hypotheses Comparing ANOVA models Unequal sample sizes (unbalanced designs) Factor effects Random effects: variance components Fixed effects Assumptions Normality Variance homogeneity Independence 193

6 8.4 ANOVA diagnostics Robust ANOVA Tests with heterogeneous variances Rank-based ("non-parametric") tests Randomization tests Specific comparisons of means Planned comparisons or contrasts Unplanned pairwise comparisons Specific contrasts versus unplanned pairwise comparisons Tests for trends Testing equality of group variances Power of single factor ANOVA General issues and hints for analysis General issues Hints for analysis I Multifactor analysis of variance Nested (hierarchical) designs Linear models for nested analyses Analysis of variance Null hypotheses Unequal sample sizes (unbalanced designs) Comparing ANOVA models Factor effects in nested models Assumptions for nested models Specific comparisons for nested designs More complex designs Design and power Factorial designs Linear models for factorial designs Analysis of variance Null hypotheses What are main effects and interactions really measuring? Comparing ANOVA models Unbalanced designs Factor effects Assumptions Robust factorial ANOVAs Specific comparisons on main effects Interpreting interactions More complex designs Power and design in factorial ANOVA Pooling in multifactor designs Relationship between factorial and nested designs General issues and hints for analysis General issues Hints for analysis 261

7 10 Randomized blocks and simple repeated measures: unreplicated two factor designs Unreplicated two factor experimental designs Randomized complete block (RCB) designs Repeated measures (RM) designs Analyzing RCB and RM designs Linear models for RCB and RM analyses Analysis of variance Null hypotheses Comparing ANOVA models Interactions in RCB and RM models Importance of treatment by block interactions Checks for interaction in unreplicated designs Assumptions Normality, independence of errors Variances and covariances - sphericity Recommended strategy Robust RCB and RM analyses Specific comparisons Efficiency ofblocking (to block or not to block?) Time as a blocking factor Analysis of unbalanced RCB designs Power of RCB or simple RM designs More complex block designs Factorial randomized block designs Incomplete block designs Latin square designs Crossover designs Generalized randomized block designs RCB and RM designs and statistical software General issues and hints for analysis General issues Hints for analysis 300 I I Split-plot and repeated measures designs: partly nested analyses of variance 30i 11.1 Partly nested designs 30i Split-plot designs 30i Repeated measures designs Reasons for using these designs Analyzing partly nested designs Linear models for partly nested analyses Analysis of variance Null hypotheses Comparing ANOVA models Assumptions 3i Between plots/subjects Within plots/subjects and multisample sphericity 318

8 11.4 Robust partly nested analyses Specific comparisons Main effects Interactions Profile (i.e. trend) analysis Analysis of unbalanced partly nested designs Power for partly nested designs More complex designs Additional between-plots/subjects factors Additional within-plots/subjects factors Additional between-plots/subjects and within-plots/ subjects factors General comments about complex designs Partly nested designs and statistical software General issues and hints for analysis General issues Hints for individual analyses I Analyses of covariance Single factor analysis of covariance (ANCOVA) Linear models for analysis of covariance Analysis of (co)variance Null hypotheses Comparing ANCOVA models Assumptions of ANCOVA Linearity Covariate values similar across groups Fixed covariate (X) Homogeneous slopes Testing for homogeneous within-group regression slopes Dealing with heterogeneous within-group regression slopes Comparing regression lines Robust ANCOVA Unequal sample sizes (unbalanced designs) Specific comparisons of adjusted means Planned contrasts Unplanned comparisons More complex designs Designs with two or more covariates Factorial designs Nested designs with one covariate Partly nested models with one covariate General issues and hints for analysis General issues Hints for analysis 358

9 13 I Generalized linear models and logistic regression Generalized linear models 13.2 Logistic regression Simple logistic regression Multiple logistic regression Categorical predictors Assumptions of logistic regression Goodness-of-fit and residuals Model diagnostics Model selection Software for logistic regression Poisson regression Generalized additive models Models for correlated data Multi-level (random effects) models Generalized estimating equations General issues and hints for analysis General issues Hints for analysis I Analyzing frequencies Single variable goodness-of-fit tests Contingency tables Two way tables Three way tables Log-linear models Two way tables Log-linear models for three way tables More complex tables General issues and hints for analysis General issues Hints for analysis I Introduction to multivariate analyses Multivariate data Distributions and associations Linear combinations, eigenvectors and eigenvalues Linear combinations of variables Eigenvalues Eigenvectors Derivation of components Multivariate distance and dissimilarity measures Dissimilarity measures for continuous variables Dissimilarity measures for dichotomous (binary) variables General dissimilarity measures for mixed variables Comparison of dissimilarity measures Comparing distance and/or dissimilarity matrices 414

10 15.6 Data standardization Standardization, association and dissimilarity Multivariate graphics Screening multivariate data sets Multivariate outliers Missing observations General issues and hints for analysis General issues Hints for analysis I Multivariate analysis of variance and discriminant analysis Multivariate analysis of variance (MANOVA) Single factor MANOVA Specific comparisons Relative importance of each response variable Assumptions of MANOVA Robust MANOVA More complex designs Discriminant function analysis Description and hypothesis testing Classification and prediction Assumptions of discriminant function analysis More complex designs MANOVA vs discriminant function analysis General issues and hints for analysis General issues Hints for analysis I Principal components and correspondence analysis Principal components analysis Deriving components Which association matrix to use? Interpreting the components Rotation of components How many components to retain? Assumptions Robust PCA Graphical representations Other uses of components Factor analysis Correspondence analysis Mechanics Scaling and joint plots Reciprocal averaging Use of CA with ecological data Detrending Canonical correlation analysis 463

11 17.5 Redundancy analysis Canonical correspondence analysis Constrained and partial "ordination" General issues and hints for analysis General issues Hints for analysis I Multidimensional scaling and cluster analysis Multidimensional scaling Classical scaling - principal coordinates analysis (PCoA) Enhanced multidimensional scaling Dissimilarities and testing hypotheses about groups of objects Relating MDS to original variables Relating MDS to covariates Classification Cluster analysis Scaling (ordination) and clustering for biological data General issues and hints for analysis Generalissues Hints for analysis Presentation of results Presentation of analyses Linear models Other analyses Layout of tables Displaying summaries of the data Bar graph Line graph (category plot) Scatterplots Pie charts Error bars Alternative approaches Oral presentations Slides, computers, or overheads? Graphics packages 5Og Working with color Scanned images 5Og Information content General issues and hints 5]0 References 511 Index 527