On testing dependence between time to failure and cause of failure via conditional probabilities

Dependence structures between the failure time and the cause of failure are expressed in terms of the monotonicity properties of the conditional probabilities involving the cause of failure and the failure time. Further, these properties of the conditional probabilities are used for testing various dependence structures and several U-statistics are proposed. In the process, a concept of concordance and discordance between a continuous and a binary variable is introduced to propose an efficient test. The proposed tests are applied to two illustrative applications.