Statistics

Michael J. Crawley, FRS, Department of Biological Sciences, Imperial College of Science, Technology and Medicine. Author of three bestselling Wiley statistics titles and five life science books.

• Preface xiChapter 1 Fundamentals 1Everything Varies 2Significance 3Good and Bad Hypotheses 3Null Hypotheses 3p Values 3Interpretation 4Model Choice 4Statistical Modelling 5Maximum Likelihood 6Experimental Design 7The Principle of Parsimony (Occam’s Razor) 8Observation, Theory and Experiment 8Controls 8Replication: It’s the ns that Justify the Means 8How Many Replicates? 9Power 9Randomization 10Strong Inference 14Weak Inference 14How Long to Go On? 14Pseudoreplication 15Initial Conditions 16Orthogonal Designs and Non-Orthogonal Observational Data 16Aliasing 16Multiple Comparisons 17Summary of Statistical Models in R 18Organizing Your Work 19Housekeeping within R 20References 22Further Reading 22Chapter 2 Dataframes 23Selecting Parts of a Dataframe: Subscripts 26Sorting 27Summarizing the Content of Dataframes 29Summarizing by Explanatory Variables 30First Things First: Get to Know Your Data 31Relationships 34Looking for Interactions between Continuous Variables 36Graphics to Help with Multiple Regression 39Interactions Involving Categorical Variables 39Further Reading 41Chapter 3 Central Tendency 42Further Reading 49Chapter 4 Variance 50Degrees of Freedom 53Variance 53Variance: A Worked Example 55Variance and Sample Size 58Using Variance 59A Measure of Unreliability 60Confidence Intervals 61Bootstrap 62Non-constant Variance: Heteroscedasticity 65Further Reading 65Chapter 5 Single Samples 66Data Summary in the One-Sample Case 66The Normal Distribution 70Calculations Using z of the Normal Distribution 76Plots for Testing Normality of Single Samples 79Inference in the One-Sample Case 81Bootstrap in Hypothesis Testing with Single Samples 81Student’s t Distribution 82Higher-Order Moments of a Distribution 83Skew 84Kurtosis 86Reference 87Further Reading 87Chapter 6 Two Samples 88Comparing Two Variances 88Comparing Two Means 90Student’s t Test 91Wilcoxon Rank-Sum Test 95Tests on Paired Samples 97The Binomial Test 98Binomial Tests to Compare Two Proportions 100Chi-Squared Contingency Tables 100Fisher’s Exact Test 105Correlation and Covariance 108Correlation and the Variance of Differences between Variables 110Scale-Dependent Correlations 112Reference 113Further Reading 113Chapter 7 Regression 114Linear Regression 116Linear Regression in R 117Calculations Involved in Linear Regression 122Partitioning Sums of Squares in Regression: SSY = SSR + SSE 125Measuring the Degree of Fit, r 2 133Model Checking 134Transformation 135Polynomial Regression 140Non-Linear Regression 142Generalized Additive Models 146Influence 148Further Reading 149Chapter 8 Analysis of Variance 150One-Way ANOVA 150Shortcut Formulas 157Effect Sizes 159Plots for Interpreting One-Way ANOVA 162Factorial Experiments 168Pseudoreplication: Nested Designs and Split Plots 173Split-Plot Experiments 174Random Effects and Nested Designs 176Fixed or Random Effects? 177Removing the Pseudoreplication 178Analysis of Longitudinal Data 178Derived Variable Analysis 179Dealing with Pseudoreplication 179Variance Components Analysis (VCA) 183References 184Further Reading 184Chapter 9 Analysis of Covariance 185Further Reading 192Chapter 10 Multiple Regression 193The Steps Involved in Model Simplification 195Caveats 196Order of Deletion 196Carrying Out a Multiple Regression 197A Trickier Example 203Further Reading 211Chapter 11 Contrasts 212Contrast Coefficients 213An Example of Contrasts in R 214A Priori Contrasts 215Treatment Contrasts 216Model Simplification by Stepwise Deletion 218Contrast Sums of Squares by Hand 222The Three Kinds of Contrasts Compared 224Reference 225Further Reading 225Chapter 12 Other Response Variables 226Introduction to Generalized Linear Models 228The Error Structure 229The Linear Predictor 229Fitted Values 230A General Measure of Variability 230The Link Function 231Canonical Link Functions 232Akaike’s Information Criterion (AIC) as a Measure of the Fit of a Model 233Further Reading 233Chapter 13 Count Data 234A Regression with Poisson Errors 234Analysis of Deviance with Count Data 237The Danger of Contingency Tables 244Analysis of Covariance with Count Data 247Frequency Distributions 250Further Reading 255Chapter 14 Proportion Data 256Analyses of Data on One and Two Proportions 257Averages of Proportions 257Count Data on Proportions 257Odds 259Overdispersion and Hypothesis Testing 260Applications 261Logistic Regression with Binomial Errors 261Proportion Data with Categorical Explanatory Variables 264Analysis of Covariance with Binomial Data 269Further Reading 272Chapter 15 Binary Response Variable 273Incidence Functions 275ANCOVA with a Binary Response Variable 279Further Reading 284Chapter 16 Death and Failure Data 285Survival Analysis with Censoring 287Further Reading 290Appendix Essentials of the R Language 291R as a Calculator 291Built-in Functions 292Numbers with Exponents 294Modulo and Integer Quotients 294Assignment 295Rounding 295Infinity and Things that Are Not a Number (NaN) 296Missing Values (NA) 297Operators 298Creating a Vector 298Named Elements within Vectors 299Vector Functions 299Summary Information from Vectors by Groups 300Subscripts and Indices 301Working with Vectors and Logical Subscripts 301Addresses within Vectors 304Trimming Vectors Using Negative Subscripts 304Logical Arithmetic 305Repeats 305Generate Factor Levels 306Generating Regular Sequences of Numbers 306Matrices 307Character Strings 309Writing Functions in R 310Arithmetic Mean of a Single Sample 310Median of a Single Sample 310Loops and Repeats 311The ifelse Function 312Evaluating Functions with apply 312Testing for Equality 313Testing and Coercing in R 314Dates and Times in R 315Calculations with Dates and Times 319Understanding the Structure of an R Object Using str 320Reference 322Further Reading 322Index 323