Upper bounds for type I and type II error rates in conditional power calculations

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Abstract
Stochastic curtailment procedures are used to monitor accumulating data in long term clinical trials. These procedures allow one to calculate the conditional probability under H0 (or alternative hypothesis, Ha) given current data, of rejecting (accepting) H0 at the end of the trial. One might stop early and reject (accept) H0 if this probability is equal to or greater than γ (γ'). Lan, Simon and Halperin (1982) have shown that if α and β are the type I and type II errors , respectively, then the type I (type II) error using a stochastic curtailment method is equal to or less than α/γ (β/γ'). This upper bound is based on an unlimited number of interim looks. In fact, there are usually a small number of looks. Lower upper bounds using only the standard normal distribution table are presented. A computer program to obtain these bounds is available from the authors.