Internal pilot studies I: Type I error rate of the naive t-test

When sample size is recalculated using unblinded interim data, use of the usual t-test at the end of a study may lead to an elevated type I error rate. This paper describes a numerical quadrature investigation to calculate the true probability of rejection as a function of the time of the recalculation, the magnitude of the detectable treatment effect, and the ratio of the guessed to the true variance. We consider both 'restricted' designs, those that require final sample size at least as large as the originally calculated size, and 'unrestricted' designs, those that permit smaller final sample sizes than originally calculated. Our results indicate that the bias in the type I error rate is often negligible, especially in restricted designs. Some sets of parameters, however, induce non-trivial bias in the unrestricted design.