Traffic crash data sets may contain anomalous observations (i.e., outliers) which do not share the same distribution with the other observations. Regarding the most popular methods in the domain of traffic safety analysis, that is, logit and probit models, their sensitivity to outliers may lead to biased estimates, which has not been widely discussed. This paper introduces a robust Bayesian regression approach, namely the robit model, which substitutes the link function of these thin-tailed distributions with a heavy-tailed student’s t-distribution to down-weight the impacts of outliers. Additionally, a sandwich algorithm based on the data augmentation algorithm is proposed to improve the estimation efficiency of posteriors. The model is tested with a data set of tunnel crashes. Estimation results show that the robit model is efficient and robust and outperforms the baselines. Results also reveal that nine variables, that is, night, speeding, and fatigue driving, have significant effects on the severity of tunnel crashes.