Bayesian Multilevel Models for Count Data

Olumide Sunday Adesina

doi:10.46481/jnsps.2021.168

Abstract

The traditional Poisson regression model for fitting count data is considered inadequate to fit over-or under-dispersed count data and new models have been developed to make up for such inadequacies inherent in the model. In this study, Bayesian Multi-level model was proposed using the No-U-Turn Sampler (NUTS) sampler to sample from the posterior distribution. A simulation was carried out for both over-and under-dispersed data from discrete Weibull distribution. Pareto k diagnostics was implemented, and the result showed that under-dispersed and over-dispersed simulated data has all its k value to be less than 0.5, which indicate that all the observations are good. Also all WAIC were the same as LOO-IC except for Poisson in the over-dispersed simulated data. Real-life data set from National Health Insurance Scheme (NHIS) was used for further analysis. Seven multi-level models were f itted and the Geometric model outperformed other model.

Highlights

The traditional Poisson regression model for fitting count data is considered inadequate to fit over- or under-dispersed count data and new models have been developed to make up for such inadequacies inherent in the model
Some of the improved techniques relative to Poisson regression model can be found in [5], [6], [3], amongst others. [7] carried out a on hidden markov model in multiple testing on dependent count data, [8] showed that the exponentiatedexponential Geometric distribution can be applied to fit underdispersed or over-dispersed count data, in the same manner [4] demonstrated that Dirichlet Process Mixture Prior of Generalized Linear Mixed Models (DPMglmm) can fit either overdispersed or under-dispersed count data well
The No-U-Turn Sampler (NUTS) sampler was used tional Health Insurance Scheme was used for further analysis

Summary

Multilevel Modelling

The multilevel modelling technique follows a similar process involved when fitting the Generalized Linear Model. [9] estimated multi-level parameters of ZIP regression in the generalized linear mixed models (GLMMs) context. Assuming two linear predictors are related in some ways, [16] provided a simplest form of (3) which is refers to the ZIP(τ)model as follows: log(λ) = Xβ, log (ω/1 − ω) = τXβ (6). Following equation (3) in multi-level case, [9] identified the extension of ZIP model to include random components wi and ui within logistic and Poisson linear predictors to take care of dependence of observations in given clusters. In a three-level hierarchical situation of Yi jk, the kth observation of the jth individual within the ith clusters is measured through random effects associated with the linear predictors as follows: log φi jk (1−φi jk). The covariates aTi jkand xiTjk are not always the same α and β are the corresponding vectors of regression coefficients. si jand vi jare variations at subject level

Hurdle Models

Prior Distributions

Data Description The data set was obtained from National Health Insurance