We investigate diagnostic measures for assessing the influence of observations and model misspecification on the Cox regression model when there are missing covariate data. Our diagnostics include case-deletion measures, conditional martingale residuals, and score residuals. The Q-distance is introduced to examine the effects of deleting individual observations on the estimates of finite- and infinite-dimensional parameters. Conditional martingale residuals are used to construct goodness-of-fit statistics for testing misspecification of the model assumptions. A resampling method is developed to approximate the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$p$\end{document} -values of the goodness-of-fit statistics. We conduct simulation studies to evaluate our methods, and analyse a real dataset to illustrate their use.