Exact tests based on pre-estimation and second order pivotals: non-inferiority trials

Abstract

Pre-estimation is a technique for adjusting a standard approximate P-value to be close to exact. While conceptually simple, it can become computationally intensive. Second order pivotals [N. Reid, Asymptotics and the theory of inference, Ann. Statist. 31 (2003), pp. 1695–1731] are constructed to be closer to exact than standard approximate pivotals. The theory behind these pivotals is complex, and their properties are unclear for discrete models. However, since they are typically given in closed form they are easy to compute. For the special case of non-inferiority trials, we investigate Wald, Score, likelihood ratio and second order pivotals. Each of the basic pivotals are used to generate an exact test by maximising with respect to the nuisance parameter. We also study the effect of pre-estimating the nuisance parameter, as described in Lloyd [C.J. Lloyd, Exact P-values for discrete models obtained by estimation and maximisation, Aust. N. Z. J. Statist. 50 (2008), pp. 329–346]. It appears that second order methods are not as close to exact as might have been hoped. On the other hand, P-values, based on pre-estimation are very close to exact, are more powerful than competitors and are hardly affected by the basic generating statistic chosen.