Using the posterior distribution of deviance to measure evidence of association for rare susceptibility variants

Aitkin recently proposed an integrated Bayesian/likelihood approach that he claims is general and simple. We have applied this method, which does not rely on informative prior probabilities or large-sample results, to investigate the evidence of association between disease and the 16 variants in the KDR gene provided by Genetic Analysis Workshop 17. Based on the likelihood of logistic regression models and considering noninformative uniform prior probabilities on the coefficients of the explanatory variables, we used a random walk Metropolis algorithm to simulate the distributions of deviance and deviance difference. The distribution of probability values and the distribution of the proportions of positive deviance differences showed different locations, but the direction of the shift depended on the genetic factor. For the variant with the highest minor allele frequency and for any rare variant, standard logistic regression showed a higher power than the novel approach. For the two variants with the strongest effects on Q1 under a type I error rate of 1%, the integrated approach showed a higher power than standard logistic regression. The advantages and limitations of the integrated Bayesian/likelihood approach should be investigated using additional regions and considering alternative regression models and collapsing methods

Finite mixtures of generalized linear regression models

Finite mixture models have now been used for more than hundred years (Newcomb (1886), Pearson (1894)). They are a very popular statistical modeling technique given that they constitute a flexible and-easily extensible model class for (1) approximating general distribution functions in a semi-parametric way and (2) accounting for unobserved heterogeneity. The number of applications has tremendously increased in the last decades as model estimation in a frequentist as well as a Bayesian framework has become feasible with the nowadays easily available computing power. The simplest finite mixture models are finite mixtures of distributions which are used for model-based clustering. In this case the model is given by a convex combination of a finite number of different distributions where each of the distributions is referred to as component. More complicated mixtures have been developed by inserting different kinds of models for each component. An obvious extension is to estimate a generalized linear model (McCullagh and Nelder (1989)) for each component. Finite mixtures of GLMs allow to relax the assumption that the regression coefficients and dispersion parameters are the same for all observations. In contrast to mixed effects models, where it is assumed that the distribution of the parameters over the observations is known, finite mixture models do not require to specify this distribution a-priori but allow to approximate it in a data-driven way. In a regression setting unobserved heterogeneity for example occurs if important covariates have been omitted in the data collection and hence their influence is not accounted for in the data analysis. In addition in some areas of application the modeling aim is to find groups of observations with similar regression coefficients. In market segmentation (Wedel and Kamakura (2001)) one kind of application among others of finite mixtures of GLMs aims for example at determining groups of consumers with similar price elasticities in order to develop an optimal pricing policy for a market segment