Regression Models For Analyzing Clustered Multinomial and Continuous Outcomes Under The Assumption of Exchangeability
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We derive an expression for the joint distribution of exchangeable multinomial randomvariables and continous random variables for the purpose of analyzing clusteredmultinomial and continous data. In the past such clustered discrete and continous datacould be analyzed with the use of quasi-likelihood procedures and generalized estimatingequations to estimate marginal mean response parameters. Recently, the idea ofexchangeability has been introduced to handle such data but research has focusedprimarily on analysis of clustered binary and continous outcomes. In applications to areassuch as developmental toxicity studies, where discrete and contionus measures arerecorded on each fetus, the discrete data may not necessarily be binary. For example, wemay want to look at fetal death, malformation and normal fetuses as three possibleoutcomes separately. An impediment to a full likelihood-based analysis of such clusteredmultinomial data is the lack of a mathematically tractable representation of the jointdistribution. The assumption of exchangeability is often reasonable in these elds of studywhere outcomes are measured within clusters and cluster responses can be assumed to beexchangeable in the sense that their joint distribution is invariant to permutation. We usethis assumption to formulate fully parametric regression models for clusters of bivariatedata with multinomial and contionus components. Tractable expressions for likelihoodequations are derived and iterative schemes are given for computing efcient maximumlikelihood estimates of the marginal mean, correlations, variances and higher moments.Regression models are then proposed having marginal interpretations and reproduciblemodel structures. We demonstrate the use of the exchangeable procedure with anapplication to a developmental toxicity study involving fetal weight, malformation anddeath outcomes.