Preface ........................................................ xv
1 Introduction ............................................... l
1.1 Scientific method .......................................... 1
1.1.1 Pattern description ................................. 2
1.1.2 Models .............................................. 2
1.1.3 Hypotheses and tests ................................ 3
1.1.4 Alternatives to falsification ....................... 4
1.1.5 Role of statistical analysis ........................ 5
1.2 Experiments and other tests ................................ 5
1.3 Data, observations and variables ........................... 7
1.4 Probability ................................................ 7
1.5 Probability distributions .................................. 9
1.5.1 Distributions for variables ........................ 10
1.5.2 Distributions for statistics ....................... 12
2 Estimation ................................................ 14
2.1 Samples and populations ................................... 14
2.2 Common parameters and statistics .......................... 15
2.2.1 Center (location) of distribution .................. 15
2.2.2 Spread or variability .............................. 16
2.3 Standard errors and confidence intervals for the mean ..... 17
2.3.1 Normal distributions and the Central Limit
Theorem ............................................ 17
2.3.2 Standard error of the sample mean .................. 18
2.3.3 Confidence intervals for population mean ........... 19
2.3.4 Interpretation of confidence intervals for
population mean .................................... 20
2.3.5 Standard errors for other statistics ............... 20
2.4 Methods for estimating parameters ......................... 23
2.4.1 Maximum likelihood (ML) ............................ 23
2.4.2 Ordinary least squares (OLS) ....................... 24
2.4.3 ML vs OLS estimation ............................... 25
2.5 Resampling methods for estimation ......................... 25
2.5.1 Bootstrap .......................................... 25
2.5.2 Jackknife .......................................... 26
2.6 Bayesian inference - estimation ........................... 27
2.6.1 Bayesian estimation ................................ 27
2.6.2 Prior knowledge and probability .................... 28
2.6.3 Likelihood function ................................ 28
2.6.4 Posterior probability .............................. 28
2.6.5 Examples ........................................... 29
2.6.6 Other comments ..................................... 29
3 Hypothesis testing ........................................ 32
3.1 Statistical hypothesis testing ............................ 32
3.1.1 Classical statistical hypothesis testing ........... 32
3.1.2 Associated probability and Type I error ............ 34
3.1.3 Hypothesis tests for a single population ........... 35
3.1.4 One-and two-tailed tests ........................... 37
3.1.5 Hypotheses for two populations ..................... 37
3.1.6 Parametric tests and their assumptions ............. 39
3.2 Decision errors ........................................... 42
3.2.1 Type I and II errors ............................... 42
3.2.2 Asymmetry and scalable decision criteria ........... 44
3.3 Other testing methods ..................................... 45
3.3.1 Robust parametric tests ............................ 45
3.3.2 Randomization (permutation) tests .................. 45
3.3.3 Rank-based non-parametric tests .................... 46
3.4 Multiple testing .......................................... 48
3.4.1 The problem ........................................ 48
3.4.2 Adjusting significance levels and/or P values ...... 49
3.5 Combining results from statistical tests ............. 50
3.5.1 Combining P values ................................. 50
3.5.2 Meta-analysis ...................................... 50
3.6 Critique of statistical hypothesis testing ................ 51
3.6.1 Dependence on sample size and stopping rules ....... 51
3.6.2 Sample space - relevance of data not observed ...... 52
3.6.3 P values as measure of evidence .................... 53
3.6.4 Null hypothesis always false ....................... 53
3.6.5 Arbitrary significance levels ...................... 53
3.6.6 Alternatives to statistical hypothesis testing ..... 53
3.7 Bayesian hypothesis testing ............................... 54
4 Graphical exploration of data ............................. 58
4.1 Exploratory data analysis ................................. 58
4.1.1 Exploring samples ................................... 58
4.2 Analysis with graphs ...................................... 62
4.2.1 Assumptions of parametric linear models ............. 62
4.3 Transforming data ......................................... 64
4.3.1 Transformations and distributional assumptions ..... 65
4.3.2 Transformations and linearity ...................... 67
4.3.3 Transformations and additivity ..................... 67
4.4 Standardizations .......................................... 67
4.5 Outliers .................................................. 68
4.6 Censored and missing data ................................. 68
4.6.1 Missing data ....................................... 68
4.6.2 Censored (truncated) data .......................... 69
4.7 General issues and hints for analysis ..................... 71
4.7.1 General issues ...................................... 71
5 Correlation and regression ................................ 72
5.1 Correlation analysis ...................................... 72
5.1.1 Parametric correlation model ....................... 72
5.1.2 Robust correlation ................................. 76
5.1.3 Parametric and non-parametric confidence regions ... 76
5.2 Linear models ............................................. 77
5.3 Linear regression analysis ................................ 78
5.3.1 Simple (bivariate) linear regression ............... 78
5.3.2 Linear model for regression ........................ 80
5.3.3 Estimating model parameters ........................ 85
5.3.4 Analysis of variance ............................... 88
5.3.5 Null hypotheses in regression ...................... 89
5.3.6 Comparing regression models ........................ 90
5.3.7 Variance explained ................................. 91
5.3.8 Assumptions of regression analysis ................. 92
5.3.9 Regression diagnostics ............................. 94
5.3.10 Diagnostic graphics ................................ 96
5.3.11 Transformations .................................... 98
5.3.12 Regression through the origin ...................... 98
5.3.13 Weighted least squares ............................. 99
5.3.14 X random (Model II regression) .................... 100
5.3.15 Robust regression ................................. 104
5.4 Relationship between regression and correlation .......... 106
5.5 Smoothing ................................................ 107
5.5.1 Running means ..................................... 107
5.5.2 LO(W)ESS .......................................... 107
5.5.3 Splines ........................................... 108
5.5.4 Kernels ........................................... 108
5.5.5 Other issues ...................................... 109
5.6 Power of tests in correlation and regression ............. 109
5.7 General issues and hints for analysis .................... 110
5.7.1 General issues .................................... 110
5.7.2 Hints for analysis ................................ 110
6 Multiple and complex regression .......................... 111
6.1 Multiple linear regression analysis ...................... 111
6.1.1 Multiple linear regression model .................. 114
6.1.2 Estimating model parameters ....................... 119
6.1.3 Analysis of variance .............................. 119
6.1.4 Null hypotheses and model comparisons ............. 121
6.1.5 variance explained ................................ 122
6.1.6 Which predictors are important? ................... 122
6.1.7 Assumptions of multiple regression ................ 124
6.1.8 Regression diagnostics ............................ 125
6.1.9 Diagnostic graphics ............................... 125
6.1.10 Transformations ................................... 127
6.1.11 Collinearity ...................................... 127
6.1.12 Interactions in multiple regression ............... 130
6.1.13 Polynomial regression ............................. 133
6.1.14 Indicator (dummy) variables ....................... 135
6.1.15 Finding the "best" regression model ............... 137
6.1.16 Hierarchical partitioning ......................... 141
6.1.17 Other issues in multiple linear regression ........ 142
6.2 Regression trees ......................................... 143
6.3 Path analysis and structural equation modeling ........... 145
6.4 Nonlinear models ......................................... 150
6.5 Smoothing and response surfaces .......................... 152
6.6 General issues and hints for analysis .................... 153
6.6.1 General issues .................................... 153
6.6.2 Hints for analysis ................................ 154
7 Design and power analysis ................................ 155
7.1 Sampling ................................................. 155
7.1.1 Sampling designs .................................. 155
7.1.2 Size of sample .................................... 157
7.2 Experimental design ...................................... 157
7.2.1 Replication ....................................... 158
7.2.2 Controls .......................................... 160
7.2.3 Randomization ..................................... 161
7.2.4 Independence ...................................... 163
7.2.5 Reducing unexplained variance ..................... 164
7.3 Power analysis ........................................... 164
7.3.1 Using power to plan experiments (a priori power
analysis) ......................................... 166
7.3.2 Post hoc power calculation ........................ 168
7.3.3 The effect size ................................... 168
7.3.4 Using power analyses .............................. 170
7.4 General issues and hints for analysis .................... 171
7.4.1 General issues .................................... 171
7.4.2 Hints for analysis ................................ 172
8 Comparing groups or treatments - analysis of variance .... 173
8.1 Single factor (one way) designs .......................... 173
8.1.1 Types of predictor variables (factors) ............ 176
8.1.2 Linear model for single factor analyses ........... 178
8.1.3 Analysis of variance .............................. 184
8.1.4 Null hypotheses ................................... 186
8.1.5 Comparing ANOVA models ............................ 187
8.1.6 Unequal sample sizes (unbalanced designs) ......... 187
8.2 Factor effects ........................................... 188
8.2.1 Random effects: variance components ............... 188
8.2.2 Fixed effects ..................................... 190
8.3 Assumptions .............................................. 191
8.3.1 Normality ......................................... 192
8.3.2 Variance homogeneity .............................. 193
8.3.3 Independence ...................................... 193
8.3.1 ANOVA diagnostics ................................. 194
8.5 Robust ANOVA ............................................. 195
8.5.1 Tests with heterogeneous variances ................ 195
8.5.2 Rank-based ("non-parametric") tests ............... 195
8.5.3 Randomization tests ............................... 196
8.6 Specific comparisons of means ............................ 196
8.6.1 Planned comparisons or contrasts ................... 197
8.6.2 Unplanned pairwise comparisons ..................... 199
8.6.3 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
8.10.1 General issues .................................... 206
8.10.2 Hints for analysis ................................ 206
9 Multifactor analysis of variance ......................... 208
9.1 Nested (hierarchical) designs ............................ 208
9.1.1 Linear models for nested analyses ................. 210
9.1.2 Analysis of variance .............................. 214
9.1.3 Null hypotheses ................................... 215
9.1.4 Unequal sample sizes (unbalanced designs) ......... 216
9.1.5 Comparing ANOVA models ............................ 216
9.1.6 Factor effects in nested models ................... 216
9.1.7 Assumptions for nested models ..................... 218
9.1.8 Specific comparisons for nested designs ........... 219
9.1.9 More complex designs .............................. 219
9.1.10 Design and power .................................. 219
9.2 Factorial designs ........................................ 221
9.2.1 Linear models for factorial designs ............... 225
9.2.2 Analysis of variance .............................. 230
9.2.3 Null hypotheses ................................... 232
9.2.4 What are main effects and interactions really
measuring? ........................................ 237
9.2.5 Comparing ANOVA models ............................ 241
9.2.6 Unbalanced designs ................................ 241
9.2.7 Factor effects .................................... 247
9.2.8 Assumptions ....................................... 249
9.2.9 Robust factorial ANOVAs ........................... 250
9.2.10 Specific comparisons on main effects .............. 250
9.2.11 Interpreting interactions ......................... 251
9.2.12 More complex designs .............................. 255
9.2.13 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
9.5.1 General issues .................................... 261
9.5.2 Hints for analysis ................................ 261
10 Randomized blocks and simple repeated measures:
unreplicated two factor designs .......................... 262
10.1 Unreplicated two factor experimental designs ............. 262
10.1.1 Randomized complete block (RCB) designs ........... 262
10.1.2 Repeated measures (RM) designs .................... 265
10.2 Analyzing RCB and RM designs ............................. 268
10.2.1 Linear models for RCB and RM analyses ............. 268
10.2.2 Analysis of variance .............................. 272
10.2.3 Null hypotheses ................................... 273
10.2.4 Comparing ANOVA models ............................ 274
10.3 Interactions in RCB and RM models ........................ 274
10.3.1 Importance of treatment by block interactions ..... 274
10.3.2 Checks for interaction in unreplicated designs .... 277
10.4 Assumptions .............................................. 280
10.4.1 Normality, independence of errors ................. 280
10.4.2 Variances and covariances - sphericity ............ 280
10.4.3 Recommended strategy .............................. 284
10.5 Robust RCB and RM analyses ............................... 284
10.6 Specific comparisons ..................................... 285
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
10.11.1 Factorial randomized block designs ............... 290
10.11.2 Incomplete block designs ......................... 292
10.11.3 Latin square designs ............................. 292
10.11.4 Crossover designs ................................ 296
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
10.14.1 General issues ................................... 299
10.14.2 Hints for analysis ............................... 300
11 Split-plot and repeated measures designs: partly
nested analyses of variance .............................. 301
11.1 Partly nested designs .................................... 301
11.1.1 Split-plot designs ................................ 301
11.1.2 Repeated measures designs ......................... 305
11.1.3 Reasons for using these designs ................... 309
11.2 Analyzing partly nested designs .......................... 309
11.2.1 Linear models for partly nested analyses .......... 310
11.2.2 Analysis of variance .............................. 313
11.2.3 Null hypotheses ................................... 315
11.2.4 Comparing ANOVA models ............................ 318
11.3 Assumptions .............................................. 318
11.3.1 Between plots/subjects ............................ 318
11.3.2 Within plots/subjects and multisample sphericity .. 318
11.4 Robust partly nested analyses ............................ 320
11.5 Specific comparisons ..................................... 320
11.5.1 Main effects ...................................... 320
11.5.2 Interactions ...................................... 321
11.5.3 Profile (i.e. trend) analysis ..................... 321
11.6 Analysis of unbalanced partly nested designs ............. 322
11.7 Power for partly nested designs .......................... 323
11.8 More complex designs ..................................... 323
11.8.1 Additional between-plots/subjects factors ......... 324
11.8.2 Additional within-plots/subjects factors .......... 329
11.8.3 Additional between-plots/subjects and within-
plots/subjects factors ............................ 332
11.8.4 General comments about complex designs ............ 335
11.9 Partly nested designs and statistical software ........... 335
11.10 General issues and hints for analysis ................... 337
11.10.1 General issues ................................... 337
11.10.2 Hints for individual analyses .................... 337
12 Analyses of covariance ................................... 339
12.1 Single factor analysis of covariance (ANCOVA) ............ 339
12.1.1 linear models for analysis of covariance .......... 342
12.1.2 Analysis of (co)variance .......................... 347
12.1.3 Null hypotheses ................................... 347
12.1.4 Comparing ANCOVA models ........................... 348
12.2 Assumptions of ANCOVA .................................... 348
12.2.1 Linearity ......................................... 348
12.2.2 Covariate values similar across groups ............ 349
12.2.3 Fixed covariate (X) ............................... 349
12.3 Homogeneous slopes ....................................... 349
12.3.1 Testing for homogeneous within-group regression
slopes 349
12.3.2 Dealing with heterogeneous within-group
regression slopes ................................. 350
12.3.3 Comparing regression lines ........................ 352
12.4 Robust ANCOVA ............................................ 352
12.5 Unequal sample sizes (unbalanced designs) ................ 353
12.6 Specific comparisons of adjusted means ................... 353
12.6.1 Planned contrasts ................................. 353
12.6.2 Unplanned comparisons ............................. 353
12.7 More complex designs ..................................... 353
12.7.1 Designs with two or more covariates ............... 353
12.7.2 Factorial designs ................................. 354
12.7.3 Nested designs with one covariate ................. 355
12.7.4 Partly nested models with one covariate ........... 356
12.8 General issues and hints for analysis .................... 357
12.8.1 General issues .................................... 357
12.8.2 Hints for analysis ................................ 358
13 Generalized linear models and logistic regression ........ 359
13.1 Generalized linear models ................................ 359
13.2 Logistic regression ...................................... 360
13.2.1 Simple logistic regression ........................ 360
13.2.2 Multiple logistic regression ...................... 365
13.2.3 Categorical predictors ............................ 368
13.2.4 Assumptions of logistic regression ................ 368
13.2.5 Goodness-of-fit and residuals ..................... 368
13.2.6 Model diagnostics ................................. 370
13.2.7 Model selection ................................... 370
13.2.8 Software for logistic regression .................. 371
13.3 Poisson regression ....................................... 371
13.4 Generalized additive models .............................. 372
13.5 Models for correlated data ............................... 375
13.5.1 Multi-level (random effects) models ............... 376
13.5.2 Generalized estimating equations .................. 377
13.6 General issues and hints for analysis .................... 378
13.6.1 General issues .................................... 378
13.6.2 Hints for analysis ................................ 379
14 Analyzing frequencies .................................... 380
14.1 Single variable goodness-of-fit tests .................... 381
14.2 Contingency tables ....................................... 381
14.2.1 Two way tables .................................... 381
14.2.2 Three way tables .................................. 388
14.3 Log-linear models ........................................ 393
14.3.1 Two way tables .................................... 394
14.3.2 Log-linear models for three way tables ............ 395
14.3.3 More complex tables ............................... 400
14.4 General issues and hints for analysis .................... 400
14.4.1 General issues .................................... 400
14.4.2 Hints for analysis ................................ 400
15 Introduction to multivariate analyses .................... 401
15.1 Multivariate data ........................................ 401
15.2 Distributions and associations ........................... 402
15.3 Linear combinations, eigenvectors and eigenvalues ........ 405
15.3.1 Linear combinations of variables .................. 405
15.3.2 Eigenvalues ....................................... 405
15.3.3 Eigenvectors ...................................... 406
15.3.4 Derivation of components .......................... 409
15.4 Multivariate distance and dissimilarity measures ......... 409
15.4.1 Dissimilarity measures for continuous variables ... 412
15.4.2 Dissimilarity measures for dichotomous (binary)
variables ......................................... 413
15.4.3 General dissimilarity measures for mixed
variables ......................................... 413
15.4.4 Comparison of dissimilarity measures .............. 414
15.5 Comparing distance and/or dissimilarity matrices ......... 414
15.6 Data standardization ..................................... 415
15.7 Standardization, association and dissimilarity ........... 417
15.8 Multivariate graphics .................................... 417
15.9 Screening multivariate data sets ......................... 418
15.9.1 Multivariate outliers ............................. 419
15.9.2 Missing observations .............................. 419
15.10 General issues and hints for analysis ................... 423
15.10.1 General issues ................................... 423
15.10.2 Hints for analysis ............................... 424
16 Multivariate analysis of variance and discriminant
analysis ................................................. 425
16.1 Multivariate analysis of variance (MANOVA) ............... 425
16.1.1 Single factor MANOVA .............................. 426
16.1.2 Specific comparisons .............................. 432
16.1.3 Relative importance of each response variable ..... 432
16.1.4 Assumptions of MANOVA ............................. 433
16.1.5 Robust MANOVA ..................................... 434
16.1.6 More complex designs .............................. 434
16.2 Discriminant function analysis ........................... 435
16.2.1 Description and hypothesis testing ................ 437
16.22 Classification and prediction ...................... 439
16.2.3 Assumptions of discriminant function analysis ..... 441
16.2.4 More complex designs .............................. 441
16.3 MANOVA vs discriminant function analysis ................. 441
16.4 General issues and hints for analysis .................... 441
16.4.1 General issues .................................... 441
16.4.2 Hints for analysis ................................ 441
17 Principal components and correspondence analysis ......... 443
17.1 Principal components analysis ............................ 443
17.1.1 Deriving components ............................... 447
17.1.2 Which association matrix to use? .................. 450
17.1.3 Interpreting the components ....................... 451
17.1.4 Rotation of components ............................ 451
17.1.5 How many components to retain? .................... 452
17.1.6 Assumptions ....................................... 453
17.1.7 Robust PCA ........................................ 454
17.1.8 Graphical representations ......................... 454
17.1.9 Other uses of components .......................... 456
17.2 Factor analysis .......................................... 458
17.3 Correspondence analysis .................................. 459
17.3.1 Mechanics ......................................... 459
17.3.2 Scaling and joint plots ........................... 461
17.3.3 Reciprocal averaging .............................. 462
17.3.4 Use of CA with ecological data .................... 462
17.3.5 Detrending ........................................ 463
17.4 Canonical correlation analysis ........................... 463
17.5 Redundancy analysis ...................................... 466
17.6 Canonical correspondence analysis ........................ 467
17.7 Constrained and partial "ordination" ..................... 468
17.8 General issues and hints for analysis .................... 471
17.8.1 General issues .................................... 471
17.8.2 Hints for analysis ................................ 471
18 Multidimensional scaling and cluster analysis ............ 473
18.1 Multidimensional scaling ................................. 473
18.1.1 Classical scaling - principal coordinates
analysis (PCoA) ................................... 474
18.1.2 Enhanced multidimensional scaling ................. 476
18.1.3 Dissimilarities and testing hypotheses about
groups of objects ................................. 482
18.1.4 Relating MDS to original variables ................ 487
18.1.5 Relating MDS to covariates ........................ 487
18.2 Classification ........................................... 488
18.2.1 Cluster analysis .................................. 488
18.3 Scaling (ordination) and clustering for biological data .. 491
18.4 General issues and hints for analysis .................... 493
18.4.1 General issues .................................... 493
18.4.2 Hints for analysis ................................ 493
19 Presentation of results .................................. 494
19.1 Presentation of analyses ................................. 494
19.1.1 Linear models ..................................... 494
19.1.2 Other analyses .................................... 497
19.2 Layout of tables .................................... 497
19.3 Displaying summaries of the data ......................... 498
19.3.1 Bar graph ......................................... 500
19.3.2 Line graph (category plot) ........................ 502
19.3.3 Scatterplots ...................................... 502
19.3.4 Pie charts ........................................ 503
19.4 Error bars ............................................... 504
19.4.1 Alternative approaches ............................ 506
19.5 Oral presentations ....................................... 507
19.5.1 Slides, computers, or overheads? .................. 507
19.5.2 Graphics packages ................................. 508
19.5.3 Working with color ................................ 508
19.5.4 Scanned images .................................... 509
19.5.5 Information content ............................... 509
19.6 General issues and hints ............................ 510
References .................................................... 511
Index ......................................................... 527
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