  This paper examines the statistical properties of a distributional form that
arises from pooled testing for the prevalence of a binary outcome. Our base
distribution is a two-parameter distribution using a prevalence and excess
intensity parameter; the latter is included to allow for a dilution or
intensification effect with larger pools. We also examine a generalised form of
the distribution where pools have covariate information that affects the
prevalence through a linked linear form. We study the general pooled binomial
distribution in its own right and as a special case of broader forms of
binomial GLMs using the complementary log-log link function. We examine the
information function and show the information content of individual sample
items. We demonstrate that pooling reduces information content of sample units
and we give simple heuristics for choosing an "optimal" pool size for testing.
We derive the form of the log-likelihood function and its derivatives and give
results for maximum likelihood estimation. We also discuss diagnostic testing
of the positive pool probabilities, including testing for
intensification/dilution in the testing mechanism. We illustrate the use of
this distribution by applying it to pooled testing data on virus prevalence in
a mosquito population.
