Robit Regression: A Simple Robust Alternative to Logistic and Probit Regression

Logistic and probit regression models are commonly used in practice to analyze binary response data, but the maximum likelihood estimators of these models are not robust to outliers. This paper considers a robit regression model, which replaces the normal distribution in the probit regression model with a t-distribution with a known or unknown number of degrees of freedom. It is shown that (i) the maximum likelihood estimators of the robit model with a known number of degrees of freedom are robust; (ii) the robit link with about seven degrees of freedom provides an excellent approximation to the logistic link; and (iii) the robit link with a large number of degrees of freedom approximates the probit link. The maximum likelihood estimates can be obtained using efficient EM-type algorithms. EM-type algorithms also provide information that can be used to identify outliers, to which the maximum likelihood estimates of the logistic and probit regression coefficient would be sensitive. The EM algorithms for robit regression are easily modified to obtain efficient Data Augmentation (DA) algorithms for Bayesian inference with the robit regression model. The DA algorithms for robit regression model are much simpler to implement than the existing Gibbs sampler for the logistic regression model. A numerical example illustrates the methodology.

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