In observational studies where a survival outcome is of interest, treatment initiation
may be time-dependent, which is likely to be affected by both time-invariant and time-varying covariates. In certain situations, all subjects may be exposed to the treatment sooner or later. In this scenario, the causal effect of interest is the delay in treatment. We propose a propensity score matching strategy to estimate the treatment delay effect. The goal is to balance the covariate distribution between on-time treatment and delayed treatment groups at each time point under risk set matching. We apply this method to data from an EHR based study for the delayed treatment effect of progesterone therapy for patients with recurrent preterm birth.