An Introduction to Medical Statistics Third Edition Martin Bland
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Contents
Sections marked * contain material usually found only in postgraduate courses
1 Introduction 1
1.1 Statistics and medicine 1
1.2 Statistics and mathematics 2
1.3 Statistics and computing 2
1.4 The scope of this book 3
2 The design of experiments 5
2.1 Comparing treatments 5
2.2 Random allocation 7
2.3 * Methods of allocation without random numbers 11
2.4 Volunteer bias 13
2.5 Intention to treat 14
2.6 Cross-over designs 15
2.7 Selection of subjects for clinical trials 16
2.8 Response bias and placebos 17
2.9 Assessment bias and double blind studies 19
2.10 * Laboratory experiments 20
2.11 * Experimental units / 21
2.12 * Consent in clinical trials 22
2M Multiple choice questions 1 to 6 24
2E Exercise: The ‘Know Your Midwife’ trial 25
3 Sampling and observational studies 26
3.1 Observational studies 26
3.2 Censuses 27
3.3 Sampling 27
3.4 Random sampling 29
3.5 Sampling in clinical and epidemiological studies 32
3.6 Cross-sectional studies 34
3.7 Cohort studies 36
3.8 Case-control studies 37
3.9 * Questionnaire bias in observational studies 40
3.10 * Ecological studies 42
3M Multiple choice questions 7 to 13 43
3E Exercise: Campylobacter jejuni infection 45
Summarizing data 47
4.1 Types of data 47'
4.2 Frequency distributions 47
4.3 Histograms and other frequency graphs 50
4.4 Shapes of frequency distribution 54
4.5 Medians and quantiles 56
4.6 The mean 59
4.7 Variance, range and interquartile range 59
4.8 Standard deviation 62
4A Appendix: The divisor for the variance 63
4B Appendix: Formulae for the sum of squares 64
4M Multiple choice questions 14 to 19 65
4E Exercise: Mean and standard deviation 66
Presenting data 68
5.1 Rates and proportions 68
5.2 Significant figures 69
5.3 Presenting tables 71
5.4 Pie charts 72
5.5 Bar charts 73
5.6 Scatter diagrams 75
5.7 Line graphs and time series 77
5.8 Misleading graphs 78
5.9 Logarithmic scales 81
5A Appendix: Logarithms 82
5M Multiple choice questions 20 to 24 84
5E Exercise: Creating graphs 86
Probability 87
6.1 Probability 87
6.2 Propert ies of probability 88
6.3 Probability distributions and random variables 88
6.4 The Binomial distribution 89
6.5 Mean and variance j 92
6.6 Propert ies of means and variances 93
6.7 * The Poisson distribution 95
6.8 * Conditional probability 96
6A- Appendix: Permutations and combinations 97
6B Appendix: Expected value of a sum of squares 98
6M Multiple choice questions 25 to 31 100
m Exercise: Probability and the life table 101
The Normal distribution 103
7.1 Probability for continuous variables 103
7.2 The Normal distribution ' 106
7.3 Properties of the Normal distribution 108
7.4 Variables which follow a Normal distribution 112
7.5 The Normal plot 114
7A Appendix: Chi-squared, £, and F 118
7M Multiple choice questions 32 to 37 120
7E Exercise: A Normal plot 121
8 Estimation
8.1 Sampling distributions
8.2 Standard error of a sample mean
8.3 Confidence intervals
8.4 Standard error and confidence interval for a proportion
8.5 The difference between two means
8.6 Comparison of two proportions
8.7 * Standard error of a sample standard deviation
8.8 * Confidence interval for a proportion when numbers are small
8.9 * Confidence interval for a median and other quantiles
8.10 What is the correct confidence interval?
8M Multiple choice questions 38 to 43
8E Exercise: Means of large samples
Significance tests 137
9.1 Testing a hypothesis 137
9.2 An example: The sign test 138
9.3 Principles of significance tests 139
9.4 Significance levels and types of error , 140
9.5 One- and two-sided tests of significance 141
9.6 Significant, real and important 142
9.7 Comparing the means of large samples 143
9.8 Comparison of two proportions 145
9.9 * The power of a test 147
9.10 * Multiple significance tests 148
9.11 * Repeated significance tests and sequential analysis 151
9M Multiple choice questions 44 to 49 152
9E Exercise: Crohn’s disease and cornflakes 153
10 Comparing the means of small samples 156
10.1 The t distribution 156
10.2 The one-sample t method 159
10.3 The means of two independent samples 162
10.4 The use of transformations 164
10.5 Deviations from the assumptions of t methods 167
10.6 What is a large sample? 168
10.7 * Serial data ,169
10.8 * Comparing two variances by the F test 171
10.9 * Comparing several means using analysis of variance 172
10.10 * Assumptions of the analysis of variance 175
10.11 * Comparison of means after analysis of variance 175
10.12 * Random effects in analysis of variance 177
10.13 * Units of analysis and cluster-randomized trials 179
10A Appendix: The ratio mean/standard error 1.81
10M Multiple choice questions 50 to 56 182
10E Exercise: The paired t method 183
11 Regression and correlation 185
11.1 Scatter diagrams 185
11.2 Regression 185
11.3 The method of least squares 187
11.4 * The regression of X on Y 190
11.5 The standard error of the regression coefficient 191
11.6 * Using the regression line for prediction 192
11.7 * Analysis of residuals 194
11.8 * Deviations from assumptions in regression 196
11.9 Correlation 197
11.10 Significance test and confidence interval for r 200
11.11 Uses of the correlation coefficient 202
11.12 * Using repeated observations 202
11.13 * Intraclass correlation 204
11A Appendix: The least squares estimates 205
11B Appendix: Variance about the regression line 205
11C Appendix: The standard error of b 206
11M Multiple choice questions 57 to 61 207
HE Exercise: Comparing two regression lines 208
12 Methods based on rank order 210
12.1 * .Non-parametric methods 210
12.2 * The Mann Whitney U testj , 211
12.3 * The Wilcoxon matched pahs test 217
12.4 * Spearman’s rank correlation coefficient p 220
12.5 * Kendall’s rank correlation coefficient, r 222
12.6 * Continuity corrections 225
12.7 * Parametric or non-parametric methods? 226
12M * Multiple choice questions 62 to 66 227
12E * Exercise: Application of rank methods 228
13 The analysis of cross-tabulations 230
13.1 The chi-squared test for association 230
13.2 Tests for 2 by 2 tables 233
13.3 The chi-squared test for small samples 234
13.4 Fisher’s exact test 236
13.5 Yates’ continuity correction for the 2 by 2 table- 238
13.6 * The validity of Fisher’s and Yates’ methods 239
13.7 Odds and odds ratios 240
13.8 * The chi-squared test for trend 243
13.9 * Methods for matched samples 245
13.10 * The chi-squared goodness of fit test 248
13A Appendix: Why the chi-squared test works 249
13B Appendix: The formula for Fisher’s exact test 251
13C Appendix: Standard error for the log odds ratio 252
13M Multiple choice questions 67 to 73 253
13E Exercise: Admissions to hospital in a heatwave 255
14 Choosing the statistical method 257
14.1 * Method oriented and problem oriented teaching 257
14.2 * Types of data 257
14.3 * Comparing two groups 258
14.4 * One sample and paired samples 260
14.5 * Relationship between two variables 261
14M Multiple choice questions 74 to 80 263
14E * Exercise: Choosing a statistical method 265
15 Clinical measurement 268
15.1 Making measurements 268
15.2 * Repeatability and measurement error 269
15.3 * Comparing two methods of measurement 272
15.4 Sensitivity and specificity / 275
15.5 Normal range or reference interval 279
15.6 * Survival data 281
15.7 * Computer aided diagnosis 288
15.8 * Number needed to treat 290
15M Multiple choice questions 81 to 86 291
15E Exercise: A reference interval 292
16 Mortality statistics and population structure 294
16.1 Mortality rates 294
16.2 Age standardization using the direct method 296
16.3 Age standardization by the indirect method 296
16.4 Demographic life tables 299
16.5 Vital statistics 302
16.6 The population pyramid 303
16M Multiple choice questions 87 to 92 305
16E Exercise: Deaths from volatile substance abuse 307
17 Multifactorial methods 308
17.1 * Multiple regression ' 308
17.2 * Significance tests and estimation in multiple
regression ЗЮ
17.3 * Interaction in multiple regression 313
17.4 * Polynomial regression 314
17.5 * Assumptions of multiple regression 315
17.6 * Qualitative predictor variables 316
17.7 * Multi-way analysis of variance 318
17.8 * Logistic regression 321
17.9 * Survival data using Cox regression 324
17.10 * Stepwise regression 326
17.11 * Meta-analysis: Data from several studies 326
17.12 * Other multifactorial methods 330
17M * Multiple choice questions 93 to 97 330
17E * Exercise: A multiple regression analysis 333
18 Determination of sample size 335
18.1 * Estimation of a population mean 335
18.2 * Estimation of a population proportion 336
18.3 * Sample size for significance tests 336
18.4 * Comparison of two means 339
18.5 * Comparison of two proportions 341
18.6 * Detecting a correlation 343
18.7 * Accuracy of the estimated sample size 344
18.8 * Trials randomized in clusters 344
18M * Multiple choice questions 98 to 100 346
18E * Exercise: Estimation of sample sizes 347
19 Solutions to exercises 348
References 381
Index 391
