Abstract
Count data with excessive zeros are ubiquitous in healthcare, medical, and scientific studies. There are numerous articles that show how to fit Poisson and other models which account for the excessive zeros. However, in many situations, besides zero, the frequency of another count k tends to be higher in the data. The zero- and k-inflated Poisson distribution model (ZkIP) is appropriate in such situations The ZkIP distribution essentially is a mixture distribution of Poisson and degenerate distributions at points zero and k. In this article, we study the fundamental properties of this mixture distribution. Using stochastic representation, we provide details for obtaining parameter estimates of the ZkIP regression model using the Expectation–Maximization (EM) algorithm for a given data. We derive the standard errors of the EM estimates by computing the complete, missing, and observed data information matrices. We present the analysis of two real-life data using the methods outlined in the paper.
Highlights
Data that count the number of occurrences of certain events or the number of subjects or items that fall into certain categories arise in many scientific investigations, medical, and social science research
One could use the likelihood ratio test (LRT) to test the significance of the nested models, that is, whether the zero- and k-inflated Poisson (ZkIP) model could be replaced by the zero-inflated Poisson (ZIP) model or whether the ZIP model could be replaced by the Poisson model
Unlike Lambert [3], we obtain the standard errors of the parameters using the approach developed for the EM algorithm by Louis [4], which we believe is the right approach
Summary
Data that count the number of occurrences of certain events or the number of subjects or items that fall into certain categories arise in many scientific investigations, medical, and social science research. Bakouch et al [15] introduced COS-Poisson distribution and the corresponding regression model for zero-inflated count data. In. SAS, the finite mixture model (FMM) and count regression (COUNTREG) procedures can be used to study zero-inflated models. SAS, the finite mixture model (FMM) and count regression (COUNTREG) procedures can be used to study zero-inflated models They provide estimates, standard errors, and AIC values similar to glm procedure. Research questionnaire studies are examples with zero- and k-inflated count data sets typically resulting either in the way the questions were asked or the way the responses were provided. Lin and Tsai [1] proposed a zero- and k-inflated Poisson regression model (ZkIP) to analyze such data.
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