Preface ................................................... xiii
1 Introduction ................................................. 1
1.1 Why do life scientists need to know about experimental
design and statistics? .................................. 1
1.2 What is this book designed to do? ....................... 5
2 Doing science: hypotheses, experiments and disproof .......... 7
2.1 Introduction ............................................ 7
2.2 Basic scientific method ................................. 7
2.3 Making a decision about an hypothesis .................. 11
2.4 Why can't an hypothesis or theory ever be proven? ...... 11
2.5 'Negative' outcomes .................................... 12
2.6 Null and alternate hypotheses .......................... 12
2.7 Conclusion ............................................. 14
2.8 Questions .............................................. 14
3 Collecting and displaying data .............................. 15
3.1 Introduction ........................................... 15
3.2 Variables, experimental units and types of data ........ 15
3.3 Displaying data ........................................ 17
3.4 Displaying ordinal or nominal scale data ............... 23
3.5 Bivariate data ......................................... 25
3.6 Multivariate data ...................................... 26
3.7 Summary and conclusion ................................. 28
4 Introductory concepts of experimental design ................ 29
4.1 Introduction ........................................... 29
4.2 Sampling - mensurative experiments ..................... 30
4.3 Manipulative experiments ............................... 34
4.4 Sometimes you can only do an unreplicated experiment ... 41
4.1 Realism ................................................ 42
4.6 A bit of common sense .................................. 43
4.7 Designing a 'good' experiment .......................... 44
4.8 Reporting your results ................................. 45
4.9 Summary and conclusion ................................. 46
4.10 Questions .............................................. 46
5 Doing science responsibly and ethically ..................... 48
5.1 Introduction ........................................... 48
5.2 Dealing fairly with other people's work ................ 48
5.3 Doing the experiment ................................... 50
5.4 Evaluating and reporting results ....................... 52
5.5 Quality control in science ............................. 53
5.6 Questions .............................................. 54
6 Probability helps you make a decision about your results .... 56
6.1 Introduction ........................................... 56
6.2 Statistical tests and significance levels .............. 57
6.3 What has this got to do with making a decision about
your results? .......................................... 60
6.4 Making the wrong decision .............................. 60
6.5 Other probability levels ............................... 61
6.6 How are probability values reported? ................... 62
6.7 All statistical tests do the same basic thing .......... 63
6.8 A very simple example - the chi-square test for
goodness of fit ........................................ 64
6.9 What if you get a statistic with a probability
of exactly 0.05? ....................................... 66
6.10 Statistical significance and biological significance ... 67
6.11 Summary and conclusion ................................. 69
6.12 Questions .............................................. 70
7 Probability explained ....................................... 71
7.1 Introduction ........................................... 71
7.2 Probability ............................................ 71
7.3 The addition rule ...................................... 71
7.4 The multiplication rule for independent events ......... 72
7.5 Conditional probability ................................ 75
7.6 Applications of conditional probability ................ 77
8 Using the normal distribution to make statistical
decisions ................................................... 87
8.1 Introduction ........................................... 87
8.2 The normal curve ....................................... 87
8.3 Two statistics describe a normal distribution .......... 89
8.4 Samples and populations ................................ 93
8.5 The distribution of sample means is also normal ........ 95
8.6 What do you do when you only have data from one
sample? ................................................ 99
8.7 Use of the 95% confidence interval in significance
testing ............................................... 102
8.8 Distributions that are not normal ..................... 102
8.9 Other distributions ................................... 103
8.10 Other statistics that describe a distribution ......... 105
8.11 Summary and conclusion ................................ 106
8.12 Questions ............................................. 106
9 Comparing the means of one and two samples of normally
distributed data ........................................... 108
9.1 Introduction .......................................... 108
9.2 The 95% confidence interval and 95% confidence
limits ................................................ 108
9.3 Using the Z statistic to compare a sample mean and
population mean when population statistics are known .. 108
9.4 Comparing a sample mean to an expected value when
population statistics are not known ................... 112
9.5 Comparing the means of two related samples ............ 116
9.6 Comparing the means of two independent samples ........ 118
9.7 One-tailed and two-tailed tests ....................... 121
9.8 Are your data appropriate for a t test? ............... 124
9.9 Distinguishing between data that should be analysed
by a paired sample test and a test for two
independent samples ................................... 125
9.10 Reporting the results of t tests ...................... 126
9.11 Conclusion ............................................ 127
9.12 Questions ............................................. 128
10 Type 1 error and Type 2 error, power and sample size ....... 130
10.1 Introduction .......................................... 130
10.2 Type 1 error .......................................... 130
10.3 Type 2 error .......................................... 131
10.4 The power of a test ................................... 135
10.5 What sample size do you need to ensure the risk of
Type 2 error is not too high? ......................... 135
10.6 Type 1 error, Type 2 error and the concept of
biological risk ....................................... 136
10.7 Conclusion ............................................ 138
10.8 Questions ............................................. 139
11 Single-factor analysis of variance ......................... 140
11.1 Introduction .......................................... 140
11.2 The concept behind analysis of variance ............... 141
11.3 More detail and an arithmetic example ................. 147
11.4 Unequal sample sizes (unbalanced designs) ............. 152
11.5 An ANOVA does not tell you which particular
treatments appear to be from different populations .... 153
11.6 Fixed or random effects ............................... 153
11.7 Reporting the results of a single-factor ANOVA ........ 154
11.8 Summary ............................................... 154
11.9 Questions ............................................. 155
12 Multiple comparisons after ANOVA ........................... 157
12.1 Introduction .......................................... 157
12.2 Multiple comparison tests after a Model I ANOVA ....... 157
12.3 An a posteriori Tukey comparison following
a significant result for a single-factor Model I
ANOVA ................................................. 160
12.4 Other a posteriori multiple comparison tests .......... 162
12.5 Planned comparisons ................................... 162
12.6 Reporting the results of a posteriori comparisons ..... 164
12.7 Questions ............................................. 166
13 Two-factor analysis of variance ............................ 168
13.1 Introduction .......................................... 168
13.2 What does a two-factor ANOVA do? ...................... 170
13.3 A pictorial example ................................... 174
13.4 How does a two-factor ANOVA separate out the effects
of each factor and interaction? ....................... 176
13.5 An example of a two-factor analysis of variance ....... 180
13.6 Some essential cautions and important complications ... 181
13.7 Unbalanced designs .................................... 192
13.8 More complex designs .................................. 192
13.9 Reporting the results of a two-factor ANOVA ........... 193
13.10 Questions ............................................ 194
14 Important assumptions of analysis of variance,
transformations, and a test for equality of variances ...... 196
14.1 Introduction .......................................... 196
14.2 Homogeneity of variances .............................. 196
14.3 Normally distributed data ............................. 197
14.4 Independence .......................................... 201
14.5 Transformations ....................................... 201
14.6 Are transformations legitimate? ....................... 203
14.7 Tests for heteroscedasticity .......................... 204
14.8 Reporting the results of transformations and the
Levene test ........................................... 205
14.9 Questions ............................................. 207
15 More complex ANOVA ......................................... 209
15.1 Introduction .......................................... 209
15.2 Two-factor ANOVA without replication .................. 209
15.3 A posteriori comparison of means after a two-factor
ANOVA without replication ............................. 214
15.4 Randomised blocks ..................................... 214
15.5 Repeated-measures ANOVA ............................... 216
15.6 Nested ANOVA as a special case of a single-factor
ANOVA ................................................. 222
15.7 A final comment on ANOVA - this book is only an
introduction .......................................... 229
15.8 Reporting the results of two-factor ANOVA without
replication, randomised blocks design,
repeated-measures ANOVA and nested ANOVA .............. 229
15.9 Questions ............................................. 230
16 Relationships between variables: correlation and
regression ................................................. 233
16.1 Introduction .......................................... 233
16.2 Correlation contrasted with regression ................ 234
16.3 Linear correlation .................................... 234
16.4 Calculation of the Pearson r statistic ................ 235
16.5 Is the value of r statistically significant? .......... 241
16.6 Assumptions of linear correlation ..................... 241
16.7 Summary and conclusion ................................ 242
16.8 Questions ............................................. 242
17 Regression ................................................. 244
17.1 Introduction .......................................... 244
17.2 Simple linear regression .............................. 244
17.3 Calculation of the slope of the regression line ....... 246
17.4 Calculation of the intercept with the Taxis ........... 249
17.5 Testing the significance of the slope and the
intercept ............................................. 250
17.6 An example - mites that live in the hair follicles .... 258
17.7 Predicting a value of Y from a value of X ............. 260
17.8 Predicting a value of X from a value of Y ............. 260
17.9 The danger of extrapolation ........................... 262
17.10 Assumptions of linear regression analysis ............ 263
17.11 Curvilinear regression ............................... 266
17.12 Multiple linear regression ........................... 273
17.13 Questions ............................................ 281
18 Analysis of covariance ..................................... 284
18.1 Introduction .......................................... 284
18.2 Adjusting data to remove the effect of a confounding
factor ................................................ 285
18.3 An arithmetic example ................................. 288
18.4 Assumptions of ANCOVA and an extremely important
caution about parallelism ............................. 289
18.5 Reporting the results of ANCOVA ....................... 295
18.6 More complex models ................................... 296
18.7 Questions ............................................. 296
19 Non-parametric statistics .................................. 298
19.1 Introduction .......................................... 298
19.2 The danger of assuming normality when a population is
grossly non-normal .................................... 298
19.3 The advantage of making a preliminary inspection
of the data ........................................... 300
20 Non-parametric tests for nominal scale data ................ 301
20.1 Introduction .......................................... 301
20.2 Comparing observed and expected frequencies: the
chi-square test for goodness of fit ................... 302
20.3 Comparing proportions among two or more independent
samples ............................................... 305
20.4 Bias when there is one degree of freedom .............. 308
20.5 Three-dimensional contingency tables .................. 312
20.6 Inappropriate use of tests for goodness of fit and
heterogeneity ......................................... 312
20.7 Comparing proportions among two or more related
samples of nominal scale data ......................... 314
20.8 Recommended tests for categorical data ................ 316
20.9 Reporting the results of tests for categorical data ... 316
20.10 Questions ............................................ 318
21 Non-parametric tests for ratio, interval or ordinal scale
data ....................................................... 319
21.1 Introduction .......................................... 319
21.2 A non-parametric comparison between one sample and
an expected distribution .............................. 320
21.3 Non-parametric comparisons between two independent
samples ............................................... 325
21.4 Non-parametric comparisons among three or more
independent samples ................................... 331
21.5 Non-parametric comparisons of two related samples ..... 335
21.6 Non-parametric comparisons among three or more
related samples ....................................... 338
21.7 Analysing ratio, interval or ordinal data that show
gross differences in variance among treatments and
cannot be satisfactorily transformed .................. 341
21.8 Non-parametric correlation analysis ................... 342
21.9 Other non-parametric tests ............................ 344
21.10 Questions ............................................ 344
22 Introductory concepts of multivariate analysis ............. 346
22.1 Introduction .......................................... 346
22.2 Simplifying and summarising multivariate data ......... 347
22.3 An R-mode analysis: principal components analysis ..... 348
22.4 Q-mode analyses: multidimensional scaling ............. 361
22.5 Q-mode analyses: cluster analysis ..................... 368
22.6 Which multivariate analysis should you use? ........... 372
22.7 Questions ............................................. 374
23 Choosing a test ............................................ 375
23.1 Introduction .......................................... 375
Appendix: Critical values of chi-square, t and F ........... 388
References ................................................. 394
Index ...................................................... 396
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