Abstract
The problem of choosing a sample size for a clinical trial is a very common one. In some settings, such as rare diseases or other small populations, the large sample sizes usually associated with the standard frequentist approach may be infeasible, suggesting that the sample size chosen should reflect the size of the population under consideration. Incorporation of the population size is possible in a decision‐theoretic approach either explicitly by assuming that the population size is fixed and known, or implicitly through geometric discounting of the gain from future patients reflecting the expected population size. This paper develops such approaches. Building on previous work, an asymptotic expression is derived for the sample size for single and two‐arm clinical trials in the general case of a clinical trial with a primary endpoint with a distribution of one parameter exponential family form that optimizes a utility function that quantifies the cost and gain per patient as a continuous function of this parameter. It is shown that as the size of the population, N, or expected size, N∗ in the case of geometric discounting, becomes large, the optimal trial size is O(N1/2) or O(N∗1/2). The sample size obtained from the asymptotic expression is also compared with the exact optimal sample size in examples with responses with Bernoulli and Poisson distributions, showing that the asymptotic approximations can also be reasonable in relatively small sample sizes.
Highlights
The problem of determining the sample size for a clinical trial is a very common one
We show that the result that the optimal sample size is O(N1/2) applies for any continuous utility function and for responses with a distribution of any one-parameter exponential family form assuming a conjugate prior distribution
The observed data for patients receiving treatment i are assumed to follow a distribution of one parameter exponential family form, with mean ξi assumed to have a conjugate prior distribution
Summary
The problem of determining the sample size for a clinical trial is a very common one. One way in which the size of the population can influence the sample size is to use a decision theoretic approach in which the benefits to future patients in the population, sometimes called the “patient horizon”, are explicitly considered so that future benefit depends on the size of this population Such an approach has been proposed and discussed by numerous authors over the last 50 years Anscombe, 1963; Colton, 1963; Sylvester, 1988; Berry et al, 1994; Cheng et al, 2003; Kikuchi and Gittins, 2009 and reviews by Pezeshk et al, 2013; Hee et al, 2016) This approach has very rarely been implemented in practice, it can provide important insight into an appropriate choice of sample size for a clinical trial. The results obtained depend on asymptotics, we show that, depending on the exact form of the utility function chosen, these may be reasonable even for extremely rare diseases, for example for patient populations of 1000 or less when the optimal sample size can be less than 50
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