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Statistics for Making Decisions

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. Les mer
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Leveringstid: Sendes innen 7 virkedager
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Om boka

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.

Fakta

Innholdsfortegnelse

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

Om forfatteren

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.