A note on checking the Clayton model assumption based on left-truncated bivariate data

We prove the following fact in this paper: assuming that the bivariate data follows some bivariate frailty model [Oakes, D., 1989. Bivariate survival models induced by frailties. J. Amer. Statist. Assoc. 84, 487-493], the original bivariate data follows the Clayton model if and only if one of its truncated samples follows the Clayton model. On the basis of this fact, we can establish simple conditions for checking the Clayton model assumption when the bivariate frailty model assumption is assumed for the original data.