Kulinskaya / Morgenthaler / Staudte - Meta Analysis

Meta Analysis

A Guide to Calibrating and Combining Statistical Evidence

Medium: Buch

ISBN: 978-0-470-02864-3

Verlag: Wiley

Erscheinungstermin: 01.04.2008

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Preface xiii

Part I The Methods 1

1 What can the reader expect from this book? 3

1.1 A calibration scale for evidence 4

1.1.1 T-values and p-values 4

1.1.2 How generally applicable is the calibration scale? 6

1.1.3 Combining evidence 7

1.2 The efficacy of glass ionomer versus resin sealants for prevention of caries 8

1.2.1 The data 8

1.2.2 Analysis for individual studies 9

1.2.3 Combining the evidence: fixed effects model 10

1.2.4 Combining the evidence: random effects model 10

1.3 Measures of effect size for two populations 11

1.4 Summary 13

2 Independent measurements with known precision 15

2.1 Evidence for one-sided alternatives 15

2.2 Evidence for two-sided alternatives 18

2.3 Examples 19

2.3.1 Filling containers 19

2.3.2 Stability of blood samples 20

2.3.3 Blood alcohol testing 20

3 Independent measurements with unknown precision 23

3.1 Effects and standardized effects 23

3.2 Paired comparisons 26

3.3 Examples 27

3.3.1 Daily energy intake compared to a fixed level 27

3.3.2 Darwin’s data on Zea mays 28

4 Comparing treatment to control 31

4.1 Equal unknown precision 31

4.2 Differing unknown precision 33

4.3 Examples 35

4.3.1 Drop in systolic blood pressure 35

4.3.2 Effect of psychotherapy on hospital length of stay 37

5 Comparing K treatments 39

5.1 Methodology 39

5.2 Examples 42

5.2.1 Characteristics of antibiotics 42

5.2.2 Red cell folate levels 43

6 Evaluating risks 47

6.1 Methodology 47

6.2 Examples 49

6.2.1 Ultrasound and left-handedness 49

6.2.2 Treatment of recurrent urinary tract infections 49

7 Comparing risks 51

7.1 Methodology 51

7.2 Examples 54

7.2.1 Treatment of recurrent urinary tract infections 54

7.2.2 Diuretics in pregnancy and risk of pre-eclamsia 54

8 Evaluating Poisson rates 57

8.1 Methodology 57

8.2 Example 60

8.2.1 Deaths by horse-kicks 60

9 Comparing Poisson rates 63

9.1 Methodology 64

9.1.1 Unconditional evidence 64

9.1.2 Conditional evidence 65

9.2 Example 67

9.2.1 Vaccination for the prevention of tuberculosis 67

10 Goodness-of-fit testing 71

10.1 Methodology 71

10.2 Example 74

10.2.1 Bellbirds arriving to feed nestlings 74

11 Evidence for heterogeneity of effects and transformed effects 77

11.1 Methodology 77

11.1.1 Fixed effects 77

11.1.2 Random effects 80

11.2 Examples 81

11.2.1 Deaths by horse-kicks 81

11.2.2 Drop in systolic blood pressure 82

11.2.3 Effect of psychotherapy on hospital length of stay 83

11.2.4 Diuretics in pregnancy and risk of pre-eclamsia 84

12 Combining evidence: fixed standardized effects model 85

12.1 Methodology 86

12.2 Examples 87

12.2.1 Deaths by horse-kicks 87

12.2.2 Drop in systolic blood pressure 88

13 Combining evidence: random standardized effects model 91

13.1 Methodology 91

13.2 Example 94

13.2.1 Diuretics in pregnancy and risk of pre-eclamsia 94

14 Meta-regression 95

14.1 Methodology 95

14.2 Commonly encountered situations 98

14.2.1 Standardized difference of means 98

14.2.2 Difference in risk (two binomial proportions) 99

14.2.3 Log relative risk (two Poisson rates) 99

14.3 Examples 100

14.3.1 Effect of open education on student creativity 100

14.3.2 Vaccination for the prevention of tuberculosis 101

15 Accounting for publication bias 105

15.1 The downside of publishing 105

15.2 Examples 107

15.2.1 Environmental tobacco smoke 107

15.2.2 Depression prevention programs 109

Part II The Theory 111

16 Calibrating evidence in a test 113

16.1 Evidence for one-sided alternatives 114

16.1.1 Desirable properties of one-sided evidence 115

16.1.2 Connection of evidence to p-values 115

16.1.3 Why the p-value is hard to understand 116

16.2 Random p-value behavior 118

16.2.1 Properties of the random p-value distribution 118

16.2.2 Important consequences for interpreting p-values 119

16.3 Publication bias 119

16.4 Comparison with a Bayesian calibration 121

16.5 Summary 123

17 The basics of variance stabilizing transformations 125

17.1 Standardizing the sample mean 125

17.2 Variance stabilizing transformations 126

17.2.1 Background material 126

17.2.2 The Key Inferential Function 127

17.3 Poisson model example 128

17.3.1 Example of counts data 129

17.3.2 A simple vst for the Poisson model 129

17.3.3 A better vst for the Poisson model 132

17.3.4 Achieving a desired expected evidence 132

17.3.5 Confidence intervals 132

17.3.6 Simulation study of coverage probabilities 134

17.4 Two-sided evidence from one-sided evidence 134

17.4.1 A vst based on the chi-squared statistic 135

17.4.2 A vst based on doubling the p-value 137

17.5 Summary 138

18 One-sample binomial tests 139

18.1 Variance stabilizing the risk estimator 139

18.2 Confidence intervals for p 140

18.3 Relative risk and odds ratio 142

18.3.1 One-sample relative risk 143

18.3.2 One-sample odds ratio 144

18.4 Confidence intervals for small risks p 145

18.4.1 Comparing intervals based on the log and arcsine transformations 145

18.4.2 Confidence intervals for small p based on the Poisson approximation to the binomial 146

18.5 Summary 147

19 Two-sample binomial tests 149

19.1 Evidence for a positive effect 149

19.1.1 Variance stabilizing the risk difference 149

19.1.2 Simulation studies 151

19.1.3 Choosing sample sizes to achieve desired expected evidence 151

19.1.4 Implications for the relative risk and odds ratio 153

19.2 Confidence intervals for effect sizes 153

19.3 Estimating the risk difference 155

19.4 Relative risk and odds ratio 155

19.4.1 Two-sample relative risk 155

19.4.2 Two-sample odds ratio 157

19.4.3 New confidence intervals for the RR and OR 157

19.5 Recurrent urinary tract infections 157

19.6 Summary 158

20 Defining evidence in t-statistics 159

20.1 Example 159

20.2 Evidence in the Student t-statistic 159

20.3 The Key Inferential Function for Student’s model 162

20.4 Corrected evidence 164

20.4.1 Matching p-values 164

20.4.2 Accurate confidence intervals 166

20.5 A confidence interval for the standardized effect 167

20.5.1 Simulation study of coverage probabilities 169

20.6 Comparing evidence in t- and z-tests 170

20.6.1 On substituting s for s in large samples 170

20.7 Summary 171

21 Two-sample comparisons 173

21.1 Drop in systolic blood pressure 173

21.2 Defining the standardized effect 174

21.3 Evidence in the Welch statistic 175

21.3.1 The Welch statistic 175

21.3.2 Variance stabilizing the Welch t-statistic 176

21.3.3 Choosing the sample size to obtain evidence 177

21.4 Confidence intervals for d 177

21.4.1 Converting the evidence to confidence intervals 177

21.4.2 Simulation studies 178

21.4.3 Drop in systolic blood pressure (continued) 179

21.5 Summary 179

22 Evidence in the chi-squared statistic 181

22.1 The noncentral chi-squared distribution 181

22.2 A vst for the noncentral chi-squared statistic 182

22.2.1 Deriving the vst 182

22.2.2 The Key Inferential Function 183

22.3 Simulation studies 184

22.3.1 Bias in the evidence function 184

22.3.2 Upper confidence bounds; confidence intervals 185

22.4 Choosing the sample size 188

22.4.1 Sample sizes for obtaining an expected evidence 188

22.4.2 Sample size required to obtain a desired power 190

22.5 Evidence for l > l 0 190

22.6 Summary 191

23 Evidence in F-tests 193

23.1 Variance stabilizing transformations for the noncentral F 193

23.2 The evidence distribution 197

23.3 The Key Inferential Function 200

23.3.1 Refinements 203

23.4 The random effects model 203

23.4.1 Expected evidence in the balanced case 205

23.4.2 Comparing evidence in REM and FEM 206

23.5 Summary 206

24 Evidence in Cochran’s Q for heterogeneity of effects 207

24.1 Cochran’s Q: the fixed effects model 208

24.1.1 Background material 208

24.1.2 Evidence for heterogeneity of fixed effects 210

24.1.3 Evidence for heterogeneity of transformed effects 211

24.2 Simulation studies 211

24.3 Cochran’s Q: the random effects model 214

24.4 Summary 218

25 Combining evidence from K studies 219

25.1 Background and preliminary steps 219

25.2 Fixed standardized effects 220

25.2.1 Fixed, and equal, standardized effects 220

25.2.2 Fixed, but unequal, standardized effects 221

25.2.3 Nuisance parameters 221

25.3 Random transformed effects 222

25.3.1 The random transformed effects model 222

25.3.2 Evidence for a positive effect 223

25.3.3 Confidence intervals for k and d: K small 224

25.3.4 Confidence intervals for k and d: K large 224

25.3.5 Simulation studies 225

25.4 Example: drop in systolic blood pressure 227

25.4.1 Inference for the fixed effects model 229

25.4.2 Inference for the random effects model 230

25.5 Summary 230

26 Correcting for publication bias 231

26.1 Publication bias 231

26.2 The truncated normal distribution 233

26.3 Bias correction based on censoring 235

26.4 Summary 238

27 Large-sample properties of variance stabilizing transformations 239

27.1 Existence of the variance stabilizing transformation 239

27.2 Tests and effect sizes 240

27.3 Power and efficiency 243

27.4 Summary 247

References 249

Index 253

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