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Relative Efficiency of the Fuzzy p‐Value Approach to Hypothesis Testing

In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy p‐value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the “complete data” test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed “complete data” relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald‐type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy p‐value approach in latent variable or missing data problems and suggest some viable alternatives.