This article develops influence diagnostics for log-Birnbaum–Saunders (LBS) regression models with censored data based on case-deletion model (CDM). The one-step approximations of the estimates in CDM are given and case-deletion measures are obtained. Meanwhile, it is shown that CDM is equivalent to mean shift outlier model (MSOM) in LBS regression models and an outlier test is presented based on MSOM. Furthermore, we discuss a score test for homogeneity of shape parameter in LBS regression models. Two numerical examples are given to illustrate our methodology and the properties of score test statistic are investigated through Monte Carlo simulations under different censoring percentages.