1st Edition

Statistics for Making Decisions




ISBN 9780367342678
Published March 3, 2021 by Chapman and Hall/CRC
307 Pages

USD $120.00

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Book Description

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.

Table of Contents

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

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Author(s)

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.