University of Minnesota, Twin Cities School of Statistics

**Title:**
Fuzzy P-values in Latent Variable Problems

**Authors:**
Elizabeth A. Thompson and Charles J. Geyer

**Affiliations:**
Department of Statistics, University of Washington
and
School of Statistics, University of Minnesota

**Abstract:**
We consider the problem of testing a statistical hypothesis
where the scientifically meaningful test statistic is a function of latent
variables. In particular, we consider detection of
genetic linkage, where the latent variables are patterns of
inheritance at specific genome locations.
Fuzzy p-values, introduced by Geyer and Meeden (2005)
are random variables (described by their probability distributions)
that are interpreted as p-values.
For latent variable problems,
we introduce the notion of
a fuzzy p-value as the conditional distribution of the latent p-value given
the observed data, where the latent p-value is the random variable that
would be the p-value if the latent variables were observed.
The fuzzy p-value provides an exact test using two sets of simulations
of the latent variables under the null hypothesis,
one unconditional and the other conditional on the observed data.
It provides not only an expression of the strength of the evidence against
the null hypothesis but also an expression of the uncertainty in that
expression due to lack of knowledge of the latent variables.
We illustrate these features with an example of simulated data mimicking a
real example of the detection of genetic linkage.

**Key words and phrases:**
Allele sharing; Genetic linkage; Genetic mapping;
Identity by descent; Markov chain Monte Carlo; Randomised test.

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