Most crash severity studies ignored severity correlations between driver-vehicle units involved in the same crashes. Models without accounting for these within-crash correlations will result in biased estimates in the factor effects. This study developed a Bayesian hierarchical binomial logistic model to identify the significant factors affecting the severity level of driver injury and vehicle damage in traffic crashes at signalized intersections. Crash data in Singapore were employed to calibrate the model. Model fitness assessment and comparison using intra-class correlation coefficient (ICC) and deviance information criterion (DIC) ensured the suitability of introducing the crash-level random effects. Crashes occurring in peak time and in good street-lighting condition as well as those involving pedestrian injuries tend to be less severe. But crashes that occur in night time, at T/Y type intersections, and on right-most lane, as well as those that occur in intersections where red light cameras are installed tend to be more severe. Moreover, heavy vehicles have a better resistance on severe crash and thus induce less severe injuries, while crashes involving two-wheel vehicles, young or aged drivers, and the involvement of offending party are more likely to result in severe injuries.