Bivariate distributional copula regression for mixed non-time-to-event and time-to-event responses

We propose a distributional copula regression modelling approach for bivariate responses comprised of non-commensurate (i.e. mixed) variables. In our case, the margins are a right-censored time-to-event outcome and a non-time-to-event variable. The underlying hazard rate of the time-to-event margin is modelled using discrete-time-to-event (DT) or piecewise-exponential (PW) methods. A flexible statistical model is achieved by relying on the correspondence of the likelihood of the aforementioned time-to-event approaches with well-known univariate distributions. We construct joint bivariate distributions for these mixed responses by means of parametric bivariate copulas. This allows for separate specification of the dependence structure between the margins and their individual distribution functions. All coefficients of the distributional copula regression models considered here are estimated simultaneously via penalized maximum likelihood. We showcase the versatility of our proposed approach in an analysis of red-light running behaviour of E-cyclists by modelling the joint distribution of a mixed response comprised of a binary response and a time-to-event outcome that indicates the time of red traffic light running.