Statistics Explained: An Introductory Guide for Life Scientists

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
1.1 Why do life scientists need to know about experimental design and statistics?......Page 15
1.2 What is this book designed to do?......Page 19
2.2 Basic scientific method......Page 21
2.3 Making a decision about an hypothesis......Page 24
2.5 \'Negative\' outcomes......Page 25
2.6 Null and alternate hypotheses......Page 26
2.7 Conclusion......Page 27
3.2 Variables, experimental units, and types of data......Page 28
3.3.1 Histograms......Page 30
3.3.2 Frequency polygons or line graphs......Page 31
3.3.3 Cumulative graphs......Page 32
3.4 Displaying ordinal or nominal scale data......Page 34
3.5 Bivariate data......Page 37
3.6 Multivariate data......Page 39
3.7 Summary and conclusion......Page 40
4.1 Introduction......Page 41
4.2.1 Confusing a correlation with causality......Page 42
4.2.2 The inadvertent inclusion of a third variable: sampling confounded in time......Page 43
4.2.3 The need for independent samples in mensurative experiments......Page 44
4.3.1 Independent replicates......Page 46
4.3.2 Control treatments......Page 47
4.3.3 Other common types of manipulative experiments where treatments are confounded with time......Page 49
4.3.4 Pseudoreplication......Page 50
4.4 Sometimes you can only do an unreplicated experiment......Page 53
4.5 Realism......Page 54
4.7 Designing a \'good\' experiment......Page 55
4.8 Conclusion......Page 56
5.1 Introduction......Page 58
5.2 Statistical tests and significance levels......Page 59
5.4 Making the wrong decision......Page 63
5.5 Other probability levels......Page 64
5.6 How are probability values reported?......Page 65
5.8 A very simple example – the chi-square test for goodness of fit......Page 66
5.11 Summary and conclusion......Page 69
6.3 The normal distribution......Page 71
6.3.2 The variance of a population......Page 73
6.3.4 The Z statistic......Page 75
6.4.2 The sample variance......Page 77
6.5 Your sample mean may not be an accurate estimate of the population mean......Page 79
6.6 What do you do when you only have data from one sample?......Page 81
6.7 Why are the statistics that describe the normal distribution so important?......Page 85
6.8 Distributions that are not normal......Page 86
6.9 Other distributions......Page 87
6.10.1 The median......Page 88
6.11 Conclusion......Page 89
7.2 The 95% confidence interval and 95% confidence limits......Page 91
7.3 Using the Z statistic to compare a sample mean and population mean when population statistics are known......Page 92
7.4 Comparing a sample mean with an expected value......Page 95
7.4.1 Degrees of freedom and looking up the appropriate critical value of t......Page 96
7.4.2 One-tailed and two-tailed tests......Page 97
7.4.3 The application of a single sample t test......Page 100
7.5 Comparing the means of two related samples......Page 102
7.6 Comparing the means of two independent samples......Page 104
7.7 Are your data appropriate for a t test?......Page 106
7.7.2 Have the sample(s) been taken at random?......Page 107
7.8 Distinguishing between data that should be analysed by a paired sample test or a test for two independent samples......Page 108
7.9 Conclusion......Page 109
8.2 Type 1 error......Page 110
8.3.1 A worked example showing Type 2 error......Page 111
8.4 The power of a test......Page 114
8.4.1 What determines the power of a test?......Page 115
8.5 What sample size do you need to ensure the risk of Type 2 error is not too high?......Page 116
8.7 Conclusion......Page 118
9.1 Introduction......Page 119
9.2 Single factor analysis of variance......Page 120
9.3.1 Preliminary steps......Page 126
9.3.2 Calculation of within group variation (error)......Page 128
9.3.3 Calculation of among group variation (treatment)......Page 129
9.3.4 Calculation of the total variation......Page 130
9.5 An ANOVA does not tell you which particular treatments appear to be from different populations......Page 131
9.6 Fixed or random effects......Page 132
10.2 Multiple comparison tests after a Model I ANOVA......Page 133
10.3.1 The effects of dietary supplements on pig growth......Page 136
10.4 Other a-posteriori multiple comparison tests......Page 137
10.5 Planned comparisons......Page 138
11.1.1 Why do an experiment with more than one factor?......Page 141
11.2 What does a two factor ANOVA do?......Page 143
11.3 How does a two factor ANOVA analyse these data?......Page 145
11.4 How does a two factor ANOVA separate out the effects of each factor and interaction?......Page 150
11.5 An example of a two factor analysis of variance......Page 153
11.6.1 A-posteriori testing is still needed when there is a significant effect of a fixed factor......Page 154
11.6.2 An interaction can obscure a main effect......Page 155
11.6.3 Fixed and random factors......Page 159
Factor A fixed, Factor B random, and an interaction......Page 161
11.8 More complex designs......Page 163
12.2 Homogeneity of variances......Page 165
12.3.1 Skew and outliers......Page 166
12.3.2 A worked example of a box and whiskers plot......Page 168
12.4 Independence......Page 169
12.5.1 The square root transformation......Page 170
12.6 Are transformations legitimate?......Page 172
12.7 Tests for heteroscedasticity......Page 173
13.2 Two factor ANOVA without replication......Page 176
13.3 A-posteriori comparison of means after a two factor ANOVA without replication......Page 180
13.4 Randomised blocks......Page 181
13.5 Nested ANOVA as a special case of a one factor ANOVA......Page 182
13.6 A pictorial explanation of a nested ANOVA......Page 184
13.7 A final comment on ANOVA – this book is only an introduction......Page 189
14.1 Introduction......Page 190
14.3 Linear correlation......Page 191
14.4 Calculation of the Pearson r statistic......Page 192
14.7 Summary and conclusion......Page 198
15.2 Linear regression......Page 200
15.3 Calculation of the slope of the regression line......Page 202
15.4 Calculation of the intercept with the Y axis......Page 206
15.5.1 Testing the hypothesis that the slope is significantly different to zero......Page 207
15.5.3 The coefficient of determination r2......Page 212
15.6 An example – mites that live in your hair follicles......Page 213
15.8 Predicting a value of X from a value of Y......Page 215
15.10 Assumptions of linear regression analysis......Page 216
15.11 Further topics in regression......Page 218
16.2 The danger of assuming normality when a population is grossly non-normal......Page 219
16.3 The value of making a preliminary inspection of the data......Page 221
17.1 Introduction......Page 222
17.2 Comparing observed and expected frequencies – the chi-square test for goodness of fit......Page 223
17.2.1 Small sample sizes......Page 225
17.3 Comparing proportions among two or more independent samples......Page 226
17.3.1 The chi-square test for heterogeneity......Page 227
17.3.2 The G test or log-likelihood ratio......Page 228
17.4 Bias when there is one degree of freedom......Page 229
17.4.1 The Fisher Exact Test for 2x2 tables......Page 230
17.5 Three-dimensional contingency tables......Page 233
17.6 Inappropriate use of tests for goodness of fit and heterogeneity......Page 234
17.7 Recommended tests for categorical data......Page 235
17.8 Comparing proportions among two or more related samples of nominal scale data......Page 236
18.1 Introduction......Page 238
18.2 A non-parametric comparison between one sample and an expected distribution......Page 239
18.3.1 The Mann–Whitney test......Page 241
18.3.2 Randomisation tests for two independent samples......Page 242
18.3.3 Exact tests for two independent samples......Page 243
18.4.1 The Kruskal–Wallis test......Page 246
18.4.5 Recommended non-parametric tests for three or more independent samples......Page 249
18.5.1 The Wilcoxon paired-sample test......Page 250
18.6 Non-parametric comparisons among three or more related samples......Page 252
18.6.1 The Friedman test......Page 253
18.7 Analysing ratio, interval, or ordinal data that show gross differences in variance among treatments and cannot be satisfactorily transformed......Page 255
18.8.1 Spearman\'s rank correlation......Page 257
18.9 Other non-parametric tests......Page 259
19.1 Introduction......Page 260
20.2.1 Plagiarism......Page 269
20.2.4 Acknowledging the input of others......Page 270
20.3.2 Ethics......Page 271
20.4 Evaluating and reporting results......Page 272
20.4.1 Pressure from peers or superiors......Page 273
20.5 Quality control in science......Page 274
References......Page 275
Index......Page 277