Robust inference in the multilevel zero-inflated negative binomial model

A popular way to model correlated count data with excess zeros and over-dispersion simultaneously is by means of the multilevel zero-inflated negative binomial (MZINB) distribution. Due to the complexity of the likelihood of these models, numerical methods such as the EM algorithm are used to estimate parameters. On the other hand, in the presence of outliers or when mixture components are poorly separated, the likelihood-based methods can become unstable. To overcome this challenge, we extend the robust expectation-solution (RES) approach for building a robust estimator of the regression parameters in the MZINB model. This approach achieves robustness by applying robust estimating equations in the S-step instead of estimating equations in the M-step of the EM algorithm. The robust estimation equation in the logistic component only weighs the design matrix (X) and reduces the effect of the leverage points, but in the negative binomial component, the influence of deviations on the response (Y) and design matrix (X) are bound separately. Simulation studies under various settings show that the RES algorithm gives us consistent estimates with smaller biases than the EM algorithm under contaminations. The RES algorithm applies to the data of the DMFT index and the fertility rate data.