Comparisons of Four Discrete Distributions in Count Regressions Using Elders’ Missing Teeth Data

Objectives: This present study aimed to select the best count distributions for missing teeth in elders and to investigate the relationship between missing teeth and the predictors. Materials and methods: Data were extracted from the biennial survey of 2015-2016 the U. S. National Health and Nutrition Examination Survey. Only adults aged 60 years or over who completed oral health examination and demographics interview were included. Descriptive statistics were used to demonstrate the basic information of this studied population. The performances of four different count regression models (Poisson regression, negative binomial regression with linear variance function (NB1), negative binomial regression with quadratic variance function (NB2), and zero-inflated negative binomial regression) were compared through different approaches including the values of model fit test statistics such as Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC), the magnitude of standard errors and a visual graph on the performances of fitted models. Results: The disparities on missing teeth existed in old adults by poverty and educational level and race/ethnicity. More missing teeth were found in participants who are Blacks (mean=13.89), with less education (<12 years) (mean: 13.11). Significance of t-test for “α” indicated that Poisson distribution is not appropriate for missing teeth due to overdispersion. NB1 is the best model with the smallest AIC and BIC and the smallest standard errors of parameter estimates compared to other three candidate models. Conclusion: The negative binomial distribution with linear variance function is the best distribution. Due to the fact of missing teeth which ranged from 0 to 28, the caution should be given when we interpreted the fitted model using NB1 as the missing teeth are close to 0 and 28. and ( ) , p i i i Var Y p μ αμ = + −∞ < < ∞ , where α is a constant [4], used to adjust the variance. Negative binomial distribution converges to Poisson distribution when α approaches to 0. In a binomial distribution, we can assume that a sequence of independent Bernoulli trials, and each trial has two possible outcomes called “success” with probability p and “failure” with probability 1-p. The random variable X is defined the number of success before a predetermined r of failures occurred, that is, X~NB(r, P). Two major forms of negative binomial distribution were known as linear (NB1) and quadratic (NB2) negative binomial distribution given by p=1 and p=2, respectively [5,6]. The outcome variable Yi given Xi is distributed as a negative binomial and its density function is denoted as: