Summary. Deletion diagnostics are developed for identifying observations that influence the estimates of regression parameters and the mixture parameter in the families of relative risk functions for failure time data. The diagnostic for the regression parameters is a generalization of Cain and Lange's (1984, Biometrics40, 493–499) measure of individual influence. The generalizations of martingale residuals, Schoenfeld's partial residuals (1982, Biometrika69, 239–241), and score residuals by Therneau, Grambsch, and Fleming (1990, Biometrika77, 147–160) are also obtained. The influence of some observations on regression parameters can be drastically modified as the mixture parameter changes, even for a very small change. In addition, adding or deleting some observations might result in choosing different models. The diagnostics are applied to a family proposed by Guerrero and Johnson (1982, Biometrika69, 309–314). One illustrative example is presented.