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

## TL;DR

This paper improves a statistical model for analyzing irregular longitudinal data by estimating an intercept term, leading to better accuracy and efficiency in predicting outcomes.

## Contribution

The paper introduces an extended Sun estimator that estimates the intercept term in semi-parametric joint models, improving bias and standard error.

## Key findings

- The extended estimator outperforms the original Sun estimator in bias and standard error in simulations.
- The new estimator allows estimation of mean outcome trajectories, which the original model could not.
- The extended model is computationally more efficient and recommended for longitudinal data analysis when intercepts are modeled with splines.

## Abstract

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.