Choosing and Using Statistics

Calvin Dytham has a wide range of research interests in ecology and evolutionary biology and is especially interested in the impacts of dispersal and the arrangement of individuals in space on ecological and evolutionary processes.  He is a Reader in the Department of Biology at the University of York, UK, and has been teaching statistics to undergraduate and postgraduate students since 1994.

Preface xiii The third edition xiv

How to use this book xiv

Packages used xv

Example data xv

Acknowledgements for the first edition xv

Acknowledgements for the second edition xv

Acknowledgements for the third edition xvi

1 Eight steps to successful data analysis 1

2 The basics 2

Observations 2

Hypothesis testing 2

P-values 3

Sampling 3

Experiments 4

Statistics 4

Descriptive statistics 5

Tests of difference 5

Tests of relationships 5

Tests for data investigation 6

3 Choosing a test: a key 7

Remember: eight steps to successful data analysis 7

The art of choosing a test 7

A key to assist in your choice of statistical test 8

4 Hypothesis testing, sampling and experimental design 23

Hypothesis testing 23

Acceptable errors 23

P-values 24

Sampling 25

Choice of sample unit 25

Number of sample units 26

Positioning of sample units to achieve a random sample 26

Timing of sampling 27

Experimental design 27

Control 28

Procedural controls 28

Temporal control 28

Experimental control 29

Statistical control 29

Some standard experimental designs 29

5 Statistics, variables and distributions 32

What are statistics? 32

Types of statistics 33

Descriptive statistics 33

Parametric statistics 33

Non-parametric statistics 33

What is a variable? 33

Types of variables or scales of measurement 34

Measurement variables 34

Continuous variables 34

Discrete variables 35

How accurate do I need to be? 35

Ranked variables 35

Attributes 35

Derived variables 36

Types of distribution 36

Discrete distributions 36

The Poisson distribution 36

The binomial distribution 37

The negative binomial distribution 39

The hypergeometric distribution 39

Continuous distributions 40

The rectangular distribution 40

The normal distribution 40

The standardized normal distribution 40

Convergence of a Poisson distribution to a normal distribution 41

Sampling distributions and the 'central limit theorem' 41

Describing the normal distribution further 41

Skewness 41

Kurtosis 43

Is a distribution normal? 43

Transformations 43

An example 44

The angular transformation 44

The logit transformation 45

The t-distribution 46

Confidence intervals 47

The chi-square distribution 47

The exponential distribution 47

Non-parametric 'distributions' 48

Ranking, quartiles and the interquartile range 48

Box and whisker plots 48

6 Descriptive and presentational techniques 49

General advice 49

Displaying data: summarizing a single variable 49

Box and whisker plot (box plot) 49

Displaying data: showing the distribution of a single variable 50

Bar chart: for discrete data 50

Histogram: for continuous data 51

Pie chart: for categorical data or attribute data 52

Descriptive statistics 52

Statistics of location or position 52

Arithmetic mean 53

Geometric mean 53

Harmonic mean 53

Median 53

Mode 53

Statistics of distribution, dispersion or spread 55

Range 55

Interquartile range 55

Variance 55

Standard deviation (SD) 55

Standard error (SE) 56

Confidence intervals (CI) or confidence limits 56

Coefficient of variation 56

Other summary statistics 56

Skewness 57

Kurtosis 57

Using the computer packages 57

General 57

Displaying data: summarizing two or more variables 62

Box and whisker plots (box plots) 62

Error bars and confidence intervals 63

Displaying data: comparing two variables 63

Associations 63

Scatterplots 64

Multiple scatterplots 64

Trends, predictions and time series 65

Lines 65

Fitted lines 67

Confidence intervals 67

Displaying data: comparing more than two variables 68

Associations 68

Three-dimensional scatterplots 68

Multiple trends, time series and predictions 69

Multiple fitted lines 69

Surfaces 70

7 The tests 1: tests to look at differences 72

Do frequency distributions differ? 72

Questions 72

G-test 72

An example 73

Chi-square test 75

An example 76

Kolmogorov–Smirnov test 86

An example 87

Anderson–Darling test 89

Shapiro–Wilk test 90

Graphical tests for normality 90

Do the observations from two groups differ? 92

Paired data 92

Paired t-test 92

Wilcoxon signed ranks test 96

Sign test 99

Unpaired data 103

t-test 103

One-way ANOVA 111

Mann–Whitney U 119

Do the observations from more than two groups differ? 123

Repeated measures 123

Friedman test (for repeated measures) 123

Repeated-measures ANOVA 127

Independent samples 128

One-way ANOVA 129

Post hoc testing: after one-way ANOVA 138

Kruskal–Wallis test 142

Post hoc testing: after the Kruskal–Wallis test 145

There are two independent ways of classifying the data 145

One observation for each factor combination (no replication) 146

Friedman test 146

Two-way ANOVA (without replication) 152

More than one observation for each factor combination (with

replication) 160

Interaction 160

Two-way ANOVA (with replication) 163

An example 164

Scheirer–Ray–Hare test 175

An example 175

There are more than two independent ways to classify the data 182

Multifactorial testing 182

Three-way ANOVA (without replication) 183

Three-way ANOVA (with replication) 184

An example 184

Multiway ANOVA 191

Not all classifications are independent 192

Non-independent factors 192

Nested factors 192

Random or fixed factors 193

Nested or hierarchical designs 193

Two-level nested-design ANOVA 193

An example 193

8 The tests 2: tests to look at relationships 199

Is there a correlation or association between two variables? 199

Observations assigned to categories 199

Chi-square test of association 199

An example 200

Cramér coefficient of association 208

Phi coefficient of association 209

Observations assigned a value 209

'Standard' correlation (Pearson's product-moment correlation) 210

An example 210

Spearman's rank-order correlation 214

An example 215

Kendall rank-order correlation 218

An example 218

Regression 219

An example 220

Is there a cause-and-effect relationship between two variables? 220

Questions 220

'Standard' linear regression 221

Prediction 221

Interpreting r2 222

Comparison of regression and correlation 222

Residuals 222

Confidence intervals 222

Prediction interval 223

An example 223

Kendall robust line-fit method 230

Logistic regression 230

An example 231

Model II regression 235

Polynomial, cubic and quadratic regression 235

Tests for more than two variables 236

Tests of association 236

Questions 236

Correlation 236

Partial correlation 237

Kendall partial rank-order correlation 237

Cause(s) and effect(s) 237

Questions 237

Regression 237

Analysis of covariance (ANCOVA) 238

Multiple regression 242

Stepwise regression 242

Path analysis 243

9 The tests 3: tests for data exploration 244

Types of data 244

Observation, inspection and plotting 244

Principal component analysis (PCA) and factor analysis 244

An example 245

Canonical variate analysis 251

Discriminant function analysis 251

An example 251

Multivariate analysis of variance (MANOVA) 256

An example 256

Multivariate analysis of covariance (MANCOVA) 259

Cluster analysis 259

DECORANA and TWINSPAN 263

Symbols and letters used in statistics 264

Greek letters 264

Symbols 264

Upper-case letters 265

Lower-case letters 266

Glossary 267

Assumptions of the tests 282

What if the assumptions are violated? 284

Hints and tips 285

Using a computer 285

Sampling 286

Statistics 286

Displaying the data 287

A table of statistical tests 289

Index 291