Essential Statistics for the Pharmaceutical Sciences
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Content
Preface xiii
Statistical packages xix
About the Website xxi
PART 1: PRESENTING DATA 1
1 Data types 3
1.1 Does it really matter? 3
1.2 Interval scale data 4
1.3 Ordinal scale data 4
1.4 Nominal scale data 5
1.5 Structure of this book 6
1.6 Chapter summary 6
2 Data presentation 7
2.1 Numerical tables 8
2.2 Bar charts and histograms 9
2.3 Pie charts 14
2.4 Scatter plots 16
2.5 Pictorial symbols 21
2.6 Chapter summary 22
PART 2: INTERVAL-SCALE DATA 23
3 Descriptive statistics for interval scale data 25
3.1 Summarising data sets 25
3.2 Indicators of central tendency: Mean, median and mode 26
3.3 Describing variability - Standard deviation and coefficient of variation 33
3.4 Quartiles - Another way to describe data 36
3.5 Describing ordinal data 40
3.6 Using computer packages to generate descriptive statistics 43
3.7 Chapter summary 45
4 The normal distribution 47
4.1 What is a normal distribution? 47
4.2 Identifying data that are not normally distributed 48
4.3 Proportions of individuals within 1SD or 2SD of the mean 52
4.4 Skewness and kurtosis 54
4.5 Chapter summary 57
4.6 Appendix: Power, sample size and the problem of attempting to test for a normal distribution 58
5 Sampling from populations. The standard error of the mean 63
5.1 Samples and populations 63
5.2 From sample to population 65
5.3 Types of sampling error 65
5.4 What factors control the extent of random sampling error when estimating a population mean? 68
5.5 Estimating likely sampling error - The SEM 70
5.6 Offsetting sample size against SD 74
5.7 Chapter summary 75
6 95% Confidence Interval for the Mean and Data Transformation 77
6.1 What is a confidence interval? 78
6.2 How wide should the interval be? 78
6.3 What do we mean by '95%' confidence? 79
6.4 Calculating the interval width 80
6.5 A long series of samples and 95% C.I.s 81
6.6 How sensitive is the width of the C.I. to changes in the SD, the sample size or the required level of confidence? 82
6.7 Two statements 85
6.8 One-sided 95% C.I.s 85
6.9 The 95% C.I. for the difference between two treatments 88
6.10 The need for data to follow a normal distribution and data transformation 90
6.11 Chapter summary 94
7 The two-sample t-test (1): Introducing hypothesis tests 95
7.1 The two-sample t-test - an example of an hypothesis test 96
7.2 Significance 103
7.3 The risk of a false positive finding 104
7.4 What aspects of the data will influence whether or not we obtain a significant outcome? 106
7.5 Requirements for applying a two-sample t-test 108
7.6 Performing and reporting the test 109
7.7 Chapter summary 110
8 The two?]sample t-test (2): The dreaded P value 111
8.1 Measuring how significant a result is 111
8.2 P values 112
8.3 Two ways to define significance? 113
8.4 Obtaining the P value 113
8.5 P values or 95% confidence intervals? 114
8.6 Chapter summary 115
9 The two-sample t-test (3): False negatives, power and necessary sample sizes 117
9.1 What else could possibly go wrong? 118
9.2 Power 119
9.3 Calculating necessary sample size 122
9.4 Chapter summary 130
10 The two-sample t-test (4): Statistical significance, practical significance and equivalence 131
10.1 Practical significance - Is the difference big enough to matter? 131
10.2 Equivalence testing 135
10.3 Non-inferiority testing 139
10.4 P values are less informative and can be positively misleading 141
10.5 Setting equivalence limits prior to experimentation 143
10.6 Chapter summary 144
11 The two-sample t-test (5): One-sided testing 145
11.1 Looking for a change in a specified direction 146
11.2 Protection against false positives 148
11.3 Temptation! 149
11.4 Using a computer package to carry out a one-sided test 153
11.5 Chapter summary 153
12 What does a statistically significant result really tell us? 155
12.1 Interpreting statistical significance 155
12.2 Starting from extreme scepticism 159
12.3 Bayesian statistics 160
12.4 Chapter summary 161
13 The paired t-test: Comparing two related sets of measurements 163
13.1 Paired data 163
13.2 We could analyse the data by a two-sample t?]test 165
13.3 Using a paired t-test instead 165
13.4 Performing a paired t-test 166
13.5 What determines whether a paired t-test will be significant? 169
13.6 Greater power of the paired t-test 170
13.7 Applicability of the test 170
13.8 Choice of experimental design 171
13.9 Requirement for applying a paired t-test 172
13.10 Sample sizes, practical significance and one-sided tests 173
13.11 Summarising the differences between paired and two-sample t-tests 175
13.12 Chapter summary 175
14 Analyses of variance: Going beyond t-tests 177
14.1 Extending the complexity of experimental designs 177
14.2 One-way analysis of variance 178
14.3 Two-way analysis of variance 188
14.4 Fixed and random factors 198
14.5 Multi-factorial experiments 204
14.6 Chapter summary 204
15 Correlation and regression - Relationships between measured values 207
15.1 Correlation analysis 208
15.2 Regression analysis 218
15.3 Multiple regression 225
15.4 Chapter summary 235
16 Analysis of Covariance 237
16.1 A clinical trial where ANCOVA would be appropriate 238
16.2 General interpretation of ANCOVA results 239
16.3 Analysis of the COPD trial results 241
16.4 Advantages of ANCOVA over a simple two?]sample t?]test 244
16.5 Chapter summary 249
PART 3: NOMINAL-SCALE DATA 251
17 Describing categorised data and the goodness of fit chi-square test 253
17.1 Descriptive statistics 254
17.2 Testing whether the population proportion might credibly be some pre-determined figure 258
17.3 Chapter summary 264
18 Contingency chi-square, Fisher's and McNemar's tests 265
18.1 Using the contingency chi?]square test to compare observed proportions 266
18.2 Extent of change in proportion with an expulsion - Clinically significant? 270
18.3 Larger tables - Attendance at diabetic clinics 270
18.4 Planning experimental size 273
18.5 Fisher's exact test 275
18.6 McNemar's test 277
18.7 Chapter summary 279
18.8 Appendix 280
19 Relative Risk, Odds Ratio and Number Needed to Treat 283
19.1 Measures of treatment effect - Relative Risk, Odds Ratio and Number Needed to Treat 283
19.2 Similarity between Relative Risk and Odds Ratio 287
19.3 Interpreting the various measures 288
19.4 95% confidence intervals for measures of effect size 289
19.5 Chapter summary 293
20 Logistic regression 295
20.1 Modelling a binary outcome 295
20.2 Additional predictors and the problem of confounding 304
20.3 Analysis by computer package 307
20.4 Extending logistic regression beyond dichotomous outcomes 308
20.5 Chapter summary 309
20.6 Appendix 309
PART 4: ORDINAL-SCALE DATA 311
21 Ordinal and non-normally distributed data. Transformations and non-parametric tests 313
21.1 Transforming data to a normal distribution 314
21.2 The Mann-Whitney test - a non?]parametric method 318
21.3 Dealing with ordinal data 323
21.4 Other non-parametric methods 325
21.5 Chapter summary 333
21.6 Appendix 334
PART 5: OTHER TOPICS 337
22 Measures of agreement 339
22.1 Answers to several questions 340
22.2 Several answers to one question - do they agree? 344
22.3 Chapter summary 358
23 Survival analysis 361
23.1 What special problems arise with survival data? 362
23.2 Kaplan-Meier survival estimation 363
23.3 Declining sample sizes in survival studies 369
23.4 Precision of sampling estimates of survival 369
23.5 Indicators of survival 371
23.6 Testing for differences in survival 374
23.7 Chapter summary 383
24 Multiple testing 385
24.1 What is it and why is it a problem? 385
24.2 Where does multiple testing arise? 386
24.3 Methods to avoid false positives 388
24.4 The role of scientific journals 392
24.5 Chapter summary 393
25 Questionnaires 395
25.1 Types of questions 396
25.2 Sample sizes and low return rates 398
25.3 Analysing the results 399
25.4 Problem number two: Confounded questionnaire data 401
25.5 Problem number three: Multiple testing with questionnaire data 401
25.6 Chapter summary 403
Index 000
