On partial likelihood and the construction of factorisable transformations

Abstract

Models whose associated likelihood functions fruitfully factorise are an important minority allowing 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, 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 first-order 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. Although motivated by inferential problems in statistics, the proposed construction is of independent interest and may find application elsewhere.

Minimum information divergence of Q-functions for dynamic treatment resumes

“…The result for the semiparametric consistency of the γ -power MDE is closely related to the theory of the partial likelihood for the proportional hazard model in survival analysis, in which the likelihood is factorized into the partial and remainder likelihood functions, cf Cox [8] and Battey et al [5]. The semiparametric models are both multiplicative, and hence the γ -power MDE for the survival analysis can be applied to with more discussion for a probabilistic mechanism of the time-to-event outcomes.…”

Inducement of population sparsity

“…How should the adequacy of such transformations be assessed? Are there systematic routes to deducing fruitful partial likelihood factorizations more generally, beyond the matched comparison problems of Section 3.1? This question was one of five posed by Cox (1975) and has remained open since, except for the modest progress by Battey, Cox & Lee (2022). The transformations of Section 3.2 used the composite of observed data, and the sparsity‐inducing transformations relied on the special structure of the linear regression model.…”

“…The transformations of Section 3.2 used the composite of observed data, and the sparsity‐inducing transformations relied on the special structure of the linear regression model. Is there a formulation broad enough to encompass the transformations of both Sections 3.1 and 3.2? In relation to points (4) and (5): a key example with a marginal likelihood component that does not appear to be recoverable through direct application of the ideas of Battey, Cox & Lee (2022) is a normal‐theory linear regression model with a coefficient vector $$$ \beta $$$ and an unknown error variance $$$ {\sigma}^2 $$$. The minimal sufficient statistic is $$$ \left(\hat{\beta},{S}^2\right) $$$, where $$$ \hat{\beta} $$$ is the least squares estimator and $$$ {S}^2 $$$ is the residual sum of squares divided by the residual degrees of freedom.…”

“…How does partial likelihood (including marginal and conditional likelihood as special cases) fit into this discussion? Can a connection be established between data‐based transformations for the elimination of nuisance parameters via marginal or conditional likelihood and the interest‐respecting reparameterizations of Cox & Reid (1987)? Some informal remarks in Battey, Cox & Lee (2022), which were based on Fraser (1964), made a connection between sample and parameter spaces through the notion of a local location model. McCullagh & Polson (2018) devised an approach to quantifying sparsity using ideas from extreme value theory. The connection to the present discussion is unclear.…”

“…Thus, the search for a transformation $$$ {S}_i=s\left({X}_i\right) $$$ that produces sparsity of $$$ {\nabla}_{\lambda }{f}_S $$$ at any set of evaluation points corresponds to a search for a factorizable transformation of $$$ {X}_i $$$. Battey, Cox & Lee (2022) used this sparsity as the basis for systematically deducing a factorizable transformation. This was achieved via integro‐differential equations constructed from the Laplace transform of $$$ {f}_S $$$ and convertible to standard forms of partial differential equations in some contexts.…”