5. Discussion

In this chapter, two joint models using a penalized spline with a truncated polynomial basis have been proposed to model a non-linear longitudinal outcome and a time-to-event data. The use of a truncated polynomial basis gives us an intuitive and obvious way to model non-linear longitudinal outcome. By adding some penalties for the coefficients of the knots and using linear mixedeffects models, the smoothing is controlled and the individual curves are specified.

We have conducted a sensitivity analysis on the assumption of normality for either random effects or errors. The t-distribution with the degree of freedom 5 is applied for each of them. The results show that the estimates of parameters are sensitive when both of terms are not normally distributed.

The main findings we may derive from this chapter are, at least, threefold: (1) the ECM algorithm provides a reasonable quick convergence algorithm for the proposed models; (2) the fitted joint models are able to measure the association between the internal time-dependent covariates and the risk for an event and (3) the two models provide a good prediction for both the longitudinal and survival functions, as presented in empirical results.

The limitations of this study are, at least, threefold: (1) the number of internal knots is limited to three due to computational costs; (2) the polynomial power functions can form an illconditioned basis for the models and (3) the estimation results are sensitive when both random effects and error are not normally distributed.

Based on the limitations, our future work will focus on using new methods for approximating the integrals to reduce the computational problems or relaxing the normality assumption. Furthermore, we will apply a different basis for joint models, that is the penalized B-spline. In terms of parameter estimation, we are considering a different approach to estimate the parameters in the models using a Bayesian approach, via Markov chain Monte Carlo (MCMC) algorithms.
