Models whose associated likelihood functions fruitfully factorise are
an important minority allowing complete elimination of nuisance parameters via
partial likelihood, an operation that is valuable in both Bayesian and frequentist
inferences, particularly when the number of nuisance parameters is not small. After
some general discussion of partial likelihood for elimination of nuisance parameters,
we focus on marginal likelihood factorisations, which are particularly difficult to
ascertain from elementary calculations. We suggest a systematic approach for
deducing transformations of the data, if they exist, whose marginal likelihood
functions are free of the nuisance parameters. This is based on the solution to an
integro-differential equation constructed from aspects of the Laplace transform of
the probability density function, for which candidate solutions solve a simpler firstorder linear homogeneous differential equation. The approach is generalised to the
situation in which such factorisable structure is not exactly present. Examples are
used in illustration.