Marginal distribution test for checking proportional hazards model assumption

Let Z be the covariate vector and Y be the response variable with the joint cumulative distribution function F_Y,Z. Given a random sample from F_Y,Z, in order to analyze the data based on a certain proportional hazards (PH) model, say Θ₀, one needs to test the null hypothesis H₀: F_Y,Z∈Θ₀ first. The existing tests to achieve this task make use of the residuals and are invalid in certain situations, such as when F_Y,Z is not from any PH model. To overcome this disadvantage, we propose a valid model checking test of H₀. It is based on the weighted average of the difference between two estimators of the marginal distribution of the response variable: its non-parametric maximum likelihood estimator and its estimator under Θ₀. This test is called the marginal distribution (MD) test. We give the theoretical justification of the MD test. The simulation study suggests that the MD test is always valid, whereas the existing tests may be invalid and they are often unlikely to reject the wrong PH model assumption when they are not valid.