Subgroup Analysis Based On Bayesian Hierarchical Model

In regression analysis,even after considering the influence of covariates,there may be subgroups within the response variable that are affected by unobserved latent factors.Therefore,identifying meaningful subgroups is crucial for better medical diagnosis or market segmentation.At present,statisticians have proposed several penalty methods to identify homogeneous subgroups based on regression models,in which heterogeneity is driven by unobserved latent factors and thus can be represented with subject-specific intercepts.When estimating these parameters with individual heterogeneity,those previous methods have applied the concave penalty function to the pairwise comparison of intercepts so that subjects with similar intercept values are assigned to the same subgroup.However,the procedure is very similar to that of penalty method for variable selection,which require certain data-driven methods(e.g.,AIC,BIC,etc.)to select the tuning parameters,requiring some optimization algorithms with large sample sizes.The wide application of the Bayesian method in variable selection and the fact that it can not only sufficiently consider prior information but also automatically adjust hyperparameters motivates us to consider subgroup analysis in a Bayesian framework.In this thesis,We consider the Bayesian approach for subgroup analysis under the mean regression model and quantile regression model,respectively.From the Bayesian perspective,firstly,we consider subgroup analysis of traditional mean regression models.We adopt a Laplace prior on the pairwise differences of the intercepts for mean regression model with subject-specific intercepts and prove the unimodality of the posterior distribution of the parameters under this prior.We construct a Bayesian hierarchical model using the hierarchical representation of the Laplace distribution and prove the propriety of the joint posterior distribution of the parameters.Based on the constructed hierarchical model,we propose an efficient Gibbs sampler to sample from the posterior distribution and introduce two inference methods based on the posterior sample.A large number of simulation studies show that the proposed Bayesian subgroup analysis method can not only identify subgroups accurately but also be efficient in the case of large sample data.Finally,the real data analysis also further demonstrates that the proposed Bayesian method can successfully identify homogeneous subgroups of the population.However,the mean regression model is more suitable for symmetric distributions.Statistical inferences based on the mean model is not reliable when distributions are asymmetrical,especially when heavy-tailed or skewed.At this point,the quantile regression model can be used to characterize the relationship between predictor variables and the response variable at arbitrary quantiles.Therefore,we consider subgroup analysis of quantile regression models from the Bayesian perspective.At arbitrary quantiles,first,for the quantile regression model with subject-specific intercepts,we also consider the Laplace prior on the pairwise differences of the intercepts and prove the unimodality of the posterior distribution of the parameters.Then,we construct a Bayesian hierarchical model using the hierarchical representation of the asymmetric Laplace distribution and the Laplace distribution,and prove the propriety of the joint posterior distribution of the parameters.Based on the proposed hierarchical model,we also propose an efficient Gibbs sampler to sample from posterior distributions,estimate intercepts and covariate coefficients,and identify homogeneous subgroups.Finally,a large number of simulation studies and real data analysis show that our proposed Bayesian method can identify subgroups accurately.