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雷达卡
by Nicholas T. Longford
About the Author
Nicholas T. Longford is a senior statistician at Imperial College, London, specialising in statistical methods for neonatal medicine. His interests include causal analysis of observational studies, decision theory, and the contest of modelling and design in data analysis. His longer-term appointments in the past include Educational Testing Service, Princeton, NJ, USA, de Montfort University, Leicester, England, and directorship of SNTL, a statistics research and consulting company. He is the author of over 100 journal articles and six other monographs on a variety of topics in applied statistics.
About this Book
Making decisions is a ubiquitous mental activity in our private and professional or public lives. It entails choosing one course of action from an available shortlist of options. Statistics for Making Decisions places decision making at the centre of statistical inference, proposing its theory as a new paradigm for statistical practice. The analysis in this paradigm is earnest about prior information and the consequences of the various kinds of errors that may be committed. Its conclusion is a course of action tailored to the perspective of the specific client or sponsor of the analysis. The author’s intention is a wholesale replacement of hypothesis testing, indicting it with the argument that it has no means of incorporating the consequences of errors which self-evidently matter to the client.
The volume appeals to the analyst who deals with the simplest statistical problems of comparing two samples (which one has a greater mean or variance), or deciding whether a parameter is positive or negative. It combines highlighting the deficiencies of hypothesis testing with promoting a principled solution based on the idea of a currency for error, of which we want to spend as little as possible. This is implemented by selecting the option for which the expected loss is smallest (the Bayes rule).
The price to pay is the need for a more detailed description of the options, and eliciting and quantifying the consequences (ramifications) of the errors. This is what our clients do informally and often inexpertly after receiving outputs of the analysis in an established format, such as the verdict of a hypothesis test or an estimate and its standard error. As a scientific discipline and profession, statistics has a potential to do this much better and deliver to the client a more complete and more relevant product.
Brief Contents
1 First steps 1
1.1 What shall we do? 1
Example1 3
1.2 The setting 6
1.2.1 Losses and gains 7
1.2.2 States,spaces and parameters 9
1.2.3 Estimation Fixed and random 10
1.3 Study design 11
1.4 Exercises 11
2 Statistical paradigms 15
2.1 Frequentist paradigm 16
2.1.1 Bias andvariance 17
2.1.2 Distributions 18
2.1.3 Sampling from finite populations 19
2.2 Bayesian paradigm 20
2.3 Computer-based replications 22
2.4 Design and estimation 24
2.5 Likelihood and fiducial distribution 25
2.5.1 Example.Variance estimation 26
2.6 From estimate to decision 27
2.7 Hypothesis testing 29
2.8 Hypothesis test and decision 32
2.9 Combining values and probabilities — Additivity 35
2.10 Further reading 36
2.11 Exercises 36
3 Positive or negative? 41
3.1 Constant loss 41
3.1.1 Equilibrium and critical value 43
3.2 The margin of error 44
3.3 Quadratic loss 46
3.4 Combining loss functions 47
3.5 Equilibrium function 48
Example 2 49
Example 3 51
3.6 Plausible values and impasse 53
3.7 Elicitation 55
3.7.1 Post-analysis elicitation 56
3.8 Plausible rectangles 57
Example 4 60
3.8.1 Summary 61
3.9 Further reading 62
3.10 Exercises 62
4 Non-normally distributed estimators 67
4.1 Student t distribution 67
4.1.1 Fiducial distribution for the t ratio 68
Example 5 70
Example 6 71
4.2 Verdicts for variances 72
4.2.1 Linear loss for variances 74
4.2.2 Verdicts for standard deviations 76
4.3 Comparing two variances 77
Example7 79
4.4 Statistics with binomial and Poisson distributions 81
4.4.1 Poisson distribution 86
Example 8 86
4.5 Further reading 87
4.6 Exercises 87
Appendix 90
5 Small or large? 91
5.1 Piecewise constant loss 92
5.1.1 Asymmetric loss 95
5.2 Piecewise linear loss 97
Example 9 101
5.3 Piecewise quadratic loss 101
Example 10 103
Example 11 105
5.4 Ordinal categories 106
5.4.1 Piecewise linear and quadratic losses 108
5.5 Multitude of options 109
5.5.1 Discrete options 109
5.5.2 Continuum of options 111
5.6 Further reading 113
5.7 Exercises 113
Appendix 116
A.Expected loss Ql in equation(5.3) 116
B.Continuation of Example 9 116
C.Continuation of Example 10 117
6 Study design 119
6.1 Design and analysis 119
6.2 How big a study? 121
6.3 Planning for impasse 127
6.3.1 Probability of impasse 128
Example12 131
6.4 Further reading 133
6.5 Exercises 134
Appendix.Sample size calculation for hypothesis testing 137
7 Medical screening 139
7.1 Separating positives and negatives 140
Example 13 142
7.2 Cut points specific to subpopulations 145
7.3 Distributionsotherthannormal 146
7.3.1 Normaland t distributions 147
7.4 Anearlyperfectbutexpensivetest 148
Example 14 149
7.5 Further reading 150
7.6 Exercises 151
8 Many decisions 153
8.1 Ordinary and exceptional units 154
Example 15 157
8.2 Extreme selections 158
Example 16 159
8.3 Greyzone 162
8.4 Actions in a sequence 163
8.5 Further reading 166
8.6 Exercises 167
Appendix 168
A.Moment-matching estimator 168
B.The potential outcomes framework 169
9 Performance of institutions 175
9.1 The setting and the task 176
9.1.1 Evidence of poor performance 176
9.1.2 Assessment as a classification 177
9.2 Outliers 177
9.3 As good as the best 180
9.4 Empirical Bayes estimation 181
9.5 Assessment based on rare events 185
9.6 Furtherreading 186
9.7 Exercises 186
Appendix 189
A.Estimation of theda and v2 189
B.Adjustment and matching on background190
10 Clinical trials 195
10.1 Randomisation 197
10.2 Analysis by hypothesis testing 199
10.3 Electing a course of action—approve or reject? 201
10.4 Decision about superiority 202
10.4.1 More complex loss functions 203
10.4.2 Trials for non-inferiority 204
10.5 Trials for bioequivalence 205
10.6 Crossover design 207
10.6.1 Composition of within-period estimators 209
10.7 Further reading 212
10.8 Exercises 213
11 Model uncertainty 217
11.1 Ordinary regression 218
11.1.1 Ordinary regression and model uncertainty 221
11.1.2 Some related approaches 224
11.1.3 Bounded bias 224
11.2 Composition 228
11.3 Composition of a complete set of candidate models 232
11.3.1 Summary 238
11.4 Further reading 238
11.5 Exercises 239
Appendix 241
A.Inverse of a partitioned matrix 241
B.Mixtures 243
EM algorithm 243
C.Linearloss 244
12 Postscript 247
References 251
Solutions to exercises 257
Index 287
Pages: 307
Language: English
Publisher: Chapman and Hall/CRC (March 3, 2021)
ISBN-10: 0367342677
ISBN-13: 9780367342678
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