Essential Statistics for the Pharmaceutical Sciences

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 T wo?]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 D escribing 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 405