Nonparametric estimation of risk ratios for bivariate data

Abstract

Inspired by the cross-ratio proposed by Clayton, we study a new risk ratio to describe the relation between the components of the random vector (T-1, T-2). It is the ratio of the conditional hazard rate function of T-1 at t(1), given that T-2 >= t(2) and the conditional hazard rate function of T-1 at t(1), given that T-2 >= t(2). A nonparametric estimator is proposed and its asymptotic distribution is obtained using Bernstein smoothing for the survival copula of (T-1, T-2) and its derivatives. The finite sample performance of the estimator is studied via simulations. The practical use of the risk ratio is illustrated in two real datasets, one on food expenditure and net income and one on the relation between maximum heart rate and age, for patients suffering from heart disease versus control patients (no heart disease). Extensions of the proposed risk ratio are given in the discussion section.