TITLE:
Empirical Bayesian Approach to Testing Homogeneity of Several Means of Inflated Poisson Distributions (IPD)
AUTHORS:
Mohamed M. Shoukri, Maha Aleid
KEYWORDS:
Distributions of Over-Dispersed Counts, Lagrange Class of Distributions, Knowledge Transfer, Gamma Prior, Posterior Inference, Wilson-Hilferty Transformation, RNA_SEQ Read Counts Data
JOURNAL NAME:
Open Journal of Statistics,
Vol.13 No.3,
June
16,
2023
ABSTRACT: Objectives: We introduce a special form of the Generalized Poisson Distribution. The
distribution has one parameter, yet it has a variance that is larger than the
mean a phenomenon known as “over dispersion”. We discuss potential applications
of the distribution as a model of counts, and under the assumption of
independence we will perform statistical inference on the ratio of two means,
with generalization to testing the homogeneity of several means. Methods: Bayesian methods depend on the choice of the prior distributions of
the population parameters. In this paper, we describe a Bayesian approach for estimation and inference on the parameters of
several independent Inflated Poisson (IPD) distributions with two
possible priors, the first is the reciprocal of
the square root of the Poisson parameter and the other is a conjugate Gamma
prior. The parameters of Gamma distribution are estimated in the
empirical Bayesian framework using the maximum likelihood (ML) solution using
nonlinear mixed model (NLMIXED) in SAS. With these priors we construct the highest
posterior confidence intervals on the ratio of two IPD parameters and test the
homogeneity of several populations. Results: We encountered convergence problem in estimating the
hyperparameters of the posterior distribution using the NLMIXED. However,
direct maximization of the predictive density produced solutions to the maximum
likelihood equations. We apply the methodologies to RNA-SEQ read count data of
gene expression values.