A contaminated regression model for count health data

In medical and health research, investigators are often interested in countable quantities such as hospital length of stay (e.g., in days) or the number of doctor visits. Poisson regression is commonly used to model such count data, but this approach can't accommodate overdispersion-when the variance exceeds the mean. To address this issue, the negative binomial (NB) distribution (NB-D) and, by extension, NB regression provide a well-documented alternative. However, real-data applications present additional challenges that must be considered. Two such challenges are (i) the presence of (mild) outliers that can influence the performance of the NB-D and (ii) the availability of covariates that can enhance inference about the mean of the count variable of interest. To jointly address these issues, we propose the contaminated NB (cNB) distribution that exhibits the necessary flexibility to accommodate mild outliers. This model is shown to be simple and intuitive in interpretation. In addition to the parameters of the NB-D, our proposed model has a parameter describing the proportion of mild outliers and one specifying the degree of contamination. To allow available covariates to improve the estimation of the mean of the cNB distribution, we propose the cNB regression model. An expectation-maximization algorithm is outlined for parameter estimation, and its performance is evaluated through a parameter recovery study. The effectiveness of our model is demonstrated via a sensitivity analysis and on two health datasets, where it outperforms well-known count models. The methodology proposed is implemented in an R package which is available at https://github.com/arnootto/cNB.