Unconditional Exact Tests for Binomial Proportions in the Group Sequential Setting
Hakhu, Navneet Ram
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Exact inference for independent binomial outcomes in small samples is complicated by the presence of a mean-variance relationship that depends on nuisance parameters, discreteness of the outcome space, and departures from normality. Although large sample theory based on Wald, score, and likelihood ratio (LR) tests are well developed, suitable small sample methods are necessary when ``large'' samples are not feasible. Fisher's exact test, which conditions on an ancillary statistic to eliminate nuisance parameters, however its inference based on the hypergeometric distribution is ``exact'' only when a user is willing to base decisions on flipping a biased coin for some outcomes. Thus, in practice, Fisher's exact test tends to be conservative due to the discreteness of the outcome space. To address the various issues that arise with the asymptotic and/or small sample tests, Barnard (1945, 1947) introduced the concept of unconditional exact tests that use exact distributions of a test statistic evaluated over all possible values of the nuisance parameter. For test statistics derived based on asymptotic approximations, these ``unconditional exact tests'' ensure that the realized type 1 error is less than or equal to the nominal level. On the other hand, an unconditional test based on the conservative Fisher's exact test statistic can better achieve the nominal type 1 error. In fixed sample settings, it has been found that unconditional exact tests are preferred to conditional exact tests (Mehta and Senchaudhuri, 2003). In this thesis we first illustrate the behavior of candidate unconditional exact tests in the fixed sample setting, and then extend the comparisons to the group sequential setting. Adjustment of the fixed sample tests is defined as choosing the critical value that minimizes the conservativeness of the actual type 1 error without exceeding the nominal level of significance. We suggest three methods of using critical values derived from adjusted fixed sample tests to determine the rejection region of the outcome space when testing binomial proportions in a group sequential setting: (1) at the final analysis time only; (2) at analysis times after accrual of more than 50% of the maximal sample size; and (3) at every analysis time. We consider (frequentist) operating characteristics (Emerson, Kittelson, and Gillen, 2007) when evaluating group sequential designs: overall type 1 error; overall statistical power; average sample number (ASN); stopping probabilities; and error spending function. We find that using the fixed sample critical values is adequate provided they are used at each of the interim analyses. We find relative behavior of Wald, chi square, LR, and Fisher's exact test statistics all depend on the sample size and randomization ratio, as well as the boundary shape function used in the group sequential stopping rule. Owing to its tendency to behave well across a wide variety of settings, we recommend implementation of the unconditional exact test using the adjusted Fisher's exact test statistic at every analysis. Because the absolute behavior of that test varies according to the desired type 1 error, the randomization ratio, and the sample size, we recommend that the operating characteristics for each candidate stopping rule be evaluated explicitly for the chosen unconditional exact test.
- Biostatistics