MSc Stats Defence: A comparison of Cox and joint models for time-to-event data

CANDIDATE:  GEORGE STEFAN

ABSTRACT:

The Cox model (Cox, 1972) has traditionally been used to analyse the relationship between a set of covariates and a time-to-event outcome. However, it has been found to lead to biased estimates when fitting time-varying covariates subject to measurement error. Joint modelling procedures have been developed by Wulfsohn and Tsiatis (1997), Henderson et al. (2000) and Rizopoulos (2010), among others, with the purpose of alleviating this problem. These procedures use a linear mixed-effects model to estimate the trajectory of a time-varying covariate, followed by a Cox model to relate a time-to-event outcome to this trajectory and a set of time-fixed covariates. 

Austin (2012) derived an expression for generating exponentially distributed event times that depend on a continuous time-varying covariate directly proportional to time. We extended this procedure to generate exponentially distributed event times that depend on a time-varying covariate which follows a linear model with a subject-specific random intercept and fixed slope. Using this procedure, simulations were performed in which we compared bias and variability in parameter estimates between the Cox model and Rizopoulos joint model. The simulation study found that while the parameter associating the time-to-event outcome to the time-varying covariate in the joint model has very low bias, its Cox model analogue is heavily biased when the data is generated according to a joint model.

Furthermore, we analysed data collected by Duffy et al. (2018) which records the onset time of bipolar/major mood disorder (in years since birth) for subjects who were considered at risk due to at least one parent being affected with bipolar disorder. The Cox model and both joint models agreed that a time-varying covariate, a Hamilton anxiety score, has a significant effect (at a 5% level) on the risk of bipolar/major mood disorder. However, the magnitude of this effect was estimated to be more than twice as high in the joint model as compared to the Cox model. This result is supported by the simulation, which similarly estimated this association to be larger in the joint model. To further illustrate the difference between the two approaches, the liver cirrhosis drug trial data from Andersen et al. (1993) was also analysed using both the Cox and joint models.