Increasing power with the unconditional maximization enumeration test in small samples – a detailed study of the MAX3 test statistic

We present and compare statistical methods for calculating p-values for discrete distributions in the presence of nuisance parameters in small samples. The methods we consider are asymptotic, conditional, and unconditional, and combinations thereof, where the p-value from one method is used in sequel as a test statistic in another method. We consider tests were for a given significance level we reject the null hypothesis if the p-value is not greater than the significance level. It is well known that the unconditional maximization method yields a valid p-value. This implies that when the unconditional maximization method is applied in sequel to a p-value that is not valid, the new maximization p-value will always be valid. Statistical methods for calculating valid p-values can be ranked according to the power of the corresponding test. When the unconditional maximization method is applied in sequel to a p-value that is valid (refer to the corresponding test as the original test), the new maximization p-value will always be as least as small as the starting p-value. This is true for all possible outcomes, and will thus give a new test with uniformely at least as high power as the original test. The unconditional maximization method can therefore be seen as a post processing tool to increase the power of an existing test. We elaborate on these general findings in a detailed study of the robust MAX3 test statistics for testing associations between a genotype and a phenotype in case–control data for sample sizes on the order of tens. 2× 3 contingency table; Cochrane–Armitage trend test; P -value; Robust test statistic; Polymorphism.