Sample size determination: a review By C. J. ADCOCK University of Westminster, London, UK [Received November 1996. Final revision February 1997] SUMMARY This paper is concerned with methods of sample size determination. The approach is to cover a small number of simple problems, such as estimating the mean of a normal distribution or the slope in a regression equation, and to present some key techniques. The methods covered are in two groups: frequentist and Bayesian. Frequentist methods specify a null and alternative hypothesis for the parameter of interest and then find the sample size by controlling both size and power. These methods often need to use prior information but cannot allow for the uncertainty that is associated with it. By contrast, the Bayesian approach offers a wide variety of techniques, all of which offer the ability to deal with uncertainty associated with prior information. Keywords: Average coverage criterion; Average length criterion; Bayes factors; Bayesian methods; Binomial distribution; Coherence; Hypothesis testing; Maximum expected utility; McNemar’s test; Multinomial distribution; Multivariate analysis; Normal distribution; Pivots; Regression; Sample size determination; Tolerance intervals; Worst outcome criterion 1. Introduction This review is concerned with methods of sample size determination (SSD). The approach is to cover some simple problems, which may be regarded as the starting point for more complex SSD questions, and to present some key techniques. I regard the methods of SSD as falling into two broad groups: frequentist and Bayesian. However, this review does not seek to contribute to the debate between Bayesians and frequentists. It is more concerned with issues that affect users of SSD techniques. Following Smith (1976), the ideas are interpreted from a personal point of view and no attempt is made to be completely comprehensive. In any case, the subject is so large that any review could only address a subset of approaches and application areas. In re- viewing key papers, I have sought to draw out the main issues, but I should quote Smith (1976) and remind the reader that ‘no review can ever replace a thorough reading of a fundamental paper since what is left out is often just as important as what is included’. 2. Background This review is concerned with the question of determining the size of sample to take from a population. I consider the situation in which the problem or system of interest to the investigator is modelled by the supposition that there is a random quantity X which has a probability distribution function of known form dependent on an unknown parameter θ with corresponding density function p( x θ). It is possible to obtain n observations on X, x i ; i 1, , n say, which conditional on θ are independently and identically distributed with the same probability density. SSD is concerned with methods of choosing the sample size n so that inferences and decisions about the parameter θ may be made. 1997 Royal Statistical Society 0039–0526/97/46261 The Statistician (1997) 46, No. 2, pp. 261–283 Address for correspondence: School of Economic and Business Studies, University of Westminster, 309 Regent Street, London, W1R 8AL, UK. E-mail: adcockc@westminster.ac.uk