Fuzzy p-values in latent variable problems

We consider the problem of testing a statistical hypothesiswhere the scientifically meaningful test statistic is a functionof latent variables. In particular, we consider detection ofgenetic linkage, where the latent variables are patterns ofinheritance at specific genome locations. Introduced by Geyer& Meeden (2005), fuzzy p-values are random variables, describedby their probability distributions, that are interpreted asp-values. For latent variable problems, we introduce the notionof a fuzzy p-value as having the conditional distribution ofthe latent p-value given the observed data, where the latentp-value is the random variable that would be the p-value ifthe latent variables were observed. The fuzzy p-value provides an exact test using two sets of simulationsof the latent variables under the null hypothesis, one unconditionaland the other conditional on the observed data. It providesnot only an expression of the strength of the evidence againstthe null hypothesis but also an expression of the uncertaintyin that expression owing to lack of knowledge of the latentvariables. We illustrate these features with an example of simulateddata mimicking a real example of the detection of genetic linkage.