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.
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.
1 First steps
What shall we do?
Example
The setting
Losses and gains
States, spaces and parameters
Estimation Fixed and random
Study design
Exercises
2. Statistical paradigms
Frequentist paradigm
Bias and variance
Distributions
Sampling from finite populations
Bayesian paradigm
Computer-based replications
Design and estimation
Likelihood and fiducial distribution
Example Variance estimation
From estimate to decision
Hypothesis testing
Hypothesis test and decision
Combining values and probabilities Additivity
Further reading
Exercises
3. Positive or negative?
Constant loss
Equilibrium and critical value
The margin of error
Quadratic loss
Combining loss functions
Equilibrium function
Example
Example
Plausible values and impasse
Elicitation
Post-analysis elicitation
Plausible rectangles
Example
Summary
Further reading
Exercises
4. Non-normally distributed estimators
Student t distribution
Fiducial distribution for the t ratio
Example
Example
Verdicts for variances
Linear loss for variances
Verdicts for standard deviations
Comparing two variances
Example
Statistics with binomial and Poisson distributions
Poisson distribution
Example
Further reading
Exercises
Appendix
5. Small or large?
Piecewise constant loss
Asymmetric loss
Piecewise linear loss
Example
Piecewise quadratic loss
Example
Example
Ordinal categories
Piecewise linear and quadratic losses
Multitude of options
Discrete options
Continuum of options
Further reading
Exercises
Appendix
A Expected loss Ql in equation ()
B Continuation of Example
C Continuation of Example
6. Study design
Design and analysis
How big a study?
Planning for impasse
Probability of impasse
Example
Further reading
Exercises
Appendix Sample size calculation for hypothesis testing
7. Medical screening
Separating positives and negatives
Example
Cutpoints specific to subpopulations
Distributions other than normal
Normal and t distributions
A nearly perfect but expensive test
Example
Further reading
Exercises
8. Many decisions
Ordinary and exceptional units
Example
Extreme selections
Example
Grey zone
Actions in a sequence
Further reading
Exercises
Appendix
A Moment-matching estimator
B The potential outcomes framework
9. Performance of institutions
The setting and the task
Evidence of poor performance
Assessment as a classification
Outliers
As good as the best
Empirical Bayes estimation
Assessment based on rare events
Further reading
Exercises
Appendix
A Estimation of _ and _
B Adjustment and matching on background
10. Clinical trials
Randomisation
Analysis by hypothesis testing
Electing a course of action — approve or reject
Decision about superiority
More complex loss functions
Trials for non-inferiority
Trials for bioequivalence
Crossover design
Composition of within-period estimators
Further reading
Exercises
11. Model uncertainty
Ordinary regression
Ordinary regression and model uncertainty
Some related approaches
Bounded bias
Composition
Composition of a complete set of candidate models
Summary
Further reading
Exercises
Appendix
A Inverse of a partitioned matrix
B Mixtures
EM algorithm
C Linear loss
12. Postscript
References
Index
Solutions to exercises
Biography
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, U.S.A., 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.
