AbstractIn many biomedical applications, count data have a large proportion of zeros and the zero‐inflated Poisson regression (ZIP) model may be appropriate. A popular score test for zero‐inflation, comparing the ZIP model to a standard Poisson regression model, was given by van den Broek. Similarly, for count data that exhibit extra zeros and are simultaneously overdispersed, a score test for testing the ZIP model against a zero‐inflated negative binomial alternative was proposed by Ridout, Hinde and Demétrio. However, these test statistics are sensitive to anomalous cases in the data, and incorrect inferences concerning the choice of model may be drawn. In this paper, diagnostic measures are derived to assess the influence of observations on the score statistics. Two examples that motivated the application of zero‐inflated regression models are considered to illustrate the importance of sensitivity analysis of the zero‐inflation tests. Copyright © 2004 John Wiley & Sons, Ltd.