Poisson random effect models with a shared random effect have been widely used in actuarial science for analyzing the number of claims. In particular, the random effect is a key factor in a posteriori risk classification. However, the necessity of the random effect may not be properly assessed due to the dual role of the random effect; it affects both the marginal distribution of the number of claims and the dependence among the numbers of claims obtained from an individual over time. In line with such observations, we explain that one should be careful in using the score test for the nullity of the variance of the shared random effect, as a sufficient condition for the existence of the posteriori risk classification. To safely perform the a posteriori risk classification, we propose considering an alternative random effect model based on the negative binomial distribution, and show that safer conclusions about the a posteriori risk classification can be made based on it. We also derive the score test as a sufficient condition for the existence of the a posteriori risk classification based on the proposed model.