Intercept Estimation of Semi‐Parametric Joint Models in the Context of Longitudinal Data Subject to Irregular Observations

Longitudinal data are often subject to irregular visiting times, with outcomes and visit times influenced by a latent variable. Semi-parametric joint models that account for this dependence have been proposed; among these, the Sun model is the most suitable for count data as it employs a multiplicative link function. Semi-parametric joint models define an intercept function as the mean outcome when all covariates are set to zero; this is differenced out in the course of estimation and is consequently not estimated. The Sun estimator thus provides estimates of relative covariate effects, but is unable to provide estimates of absolute effects or of longitudinal prognosis in the absence of covariates. We extend the Sun model by additionally estimating the intercept term, showing that our extended estimator is consistent and asymptotically Normal. In simulations, our estimator outperforms the original Sun estimator in terms of bias and standard error and is also more computationally efficient. We apply our estimator to a longitudinal study of tumor recurrence among bladder cancer patients. Provided the intercept term can be adequately captured using splines, we recommend that our extended Sun estimator be used in place of the original estimator, since it leads to smaller bias, smaller standard errors, and allows estimation of the mean outcome trajectories.