Exact critical values for group sequential designs with small sample sizes

Group sequential clinical trial designs allow the sequential hypothesis testing as data is accumulated over time, while ensuring the control of type-1 error rate. These designs vary in how they split the overall type-1 error among analyses, but practically, all assume that: 1. The underlying data is normal or approximately so, and 2. the sample sizes are large, so the individual test statistics are sufficiently normal rather than Student's t. These two assumptions lead to the reliance on the multivariate normal distribution for calculation of the critical values. Several publications have pointed out that for small sample sizes, such an approach leads to an inflated type-1 error and proposed different sets of critical values from either simulations or by an ad-hoc adjustment to the asymptotic critical values. In this paper, we develop the exact joint distribution of the test statistics for any sample size. We show how to calculate exact critical values that conform to some well-known alpha-spending functions, such as the O'Brien-Fleming and Pocock critical values. We also compare the resulting type-1 error of these critical values with the asymptotic, as well as with other methods that have been proposed for small sample sizes.