Non-parametric tests for distributional treatment effect for randomly censored responses

For a binary treatment ν=0, 1 and the corresponding 'potential response'Y⁰ for the control group (ν=0) and Y¹ for the treatment group (ν=1), one definition of no treatment effect is that Y ⁰ and Y¹ follow the same distribution given a covariate vector X. Koul and Schick have provided a non-parametric test for no distributional effect when the realized response (1-ν)Y⁰+ν Y¹ is fully observed and the distribution of X is the same across the two groups. This test is thus not applicable to censored responses, nor to non-experimental (i.e. observational) studies that entail different distributions of X across the two groups. We propose 'X-matched' non-parametric tests generalizing the test of Koul and Schick following an idea of Gehan. Our tests are applicable to non-experimental data with randomly censored responses. In addition to these motivations, the tests have several advantages. First, they have the intuitive appeal of comparing all available pairs across the treatment and control groups, instead of selecting a number of matched controls (or treated) in the usual pair or multiple matching. Second, whereas most matching estimators or tests have a non-overlapping support (of X) problem across the two groups, our tests have a built-in protection against the problem. Third, Gehan's idea allows the tests to make good use of censored observations. A simulation study is conducted, and an empirical illustration for a job training effect on the duration of unemployment is provided.