The local influence diagnostics, proposed by Cook (1986), provide a flexible way to assess the impact of minor model perturbations on key model parameters’ estimates. In this paper, we apply the local influence idea to the detection of test speededness in a model describing nonresponse in test data, and compare this local influence approach to the optimal person fit index proposed by Drasgow and Levine (1986), and the empirical Bayes estimate of the test speededness random effect. The performance of the methods is illustrated on the Chilean SIMCE mathematics test data. The data example indicates that the three statistics are promising when it comes to the detection of special profiles, and besides overlap to a considerable extent. Given that the statistics were developed for different purposes, they react of course differentially to the various characteristics of the response profiles, and hence also exhibit some specificity.