Jointly modeling the relationship between longitudinal and survival data subject to left truncation with applications to cystic fibrosis.

Numerous methods for joint analysis of longitudinal measures of a continuous outcome y and a time to event outcome T have recently been developed either to focus on the longitudinal data y while correcting for nonignorable dropout, to predict the survival outcome T using the longitudinal data y, or to examine the relationship between y and T. The motivating problem for our work is in joint modeling of the serial measurements of pulmonary function (FEV1% predicted) and survival in cystic fibrosis (CF) patients using registry data. Within the CF registry data, an additional complexity is that not all patients have been followed from birth; therefore, some patients have delayed entry into the study while others may have been missed completely, giving rise to a left truncated distribution. This paper shows in joint modeling situations where y and T are not independent, that it is necessary to account for this left truncation to obtain valid parameter estimates related to both survival and the longitudinal marker. We assume a linear random effects model for FEV1% predicted, where the random intercept and slope of FEV1% predicted, along with a specified transformation of the age at death follow a trivariate normal distribution. We develop an expectation-maximization algorithm for maximum likelihood estimation of parameters, which takes left truncation and right censoring of survival times into account. The methods are illustrated using simulation studies and using data from CF patients in a registry followed at Rainbow Babies and Children's Hospital, Cleveland, OH.