4 Computational challenges in survival models with random effects

The models I presented in Chapter 2 and 3 present significant computational challenges during the estimation process. I showed how frailty models with a Gamma frailty are analytically tractable, as it is possible to obtain closed-form expressions for the marginal survival function and therefore the likelihood; conversely, including a log-normal frailty (or, correspondingly, random effects) in a survival model yields a survival function - and likelihood - that does not have a closed form. Analogously, the joint likelihood of joint models for longitudinal and survival data \(\log L(\theta) = \log \int_{-\infty} ^ {+\infty} f(T_i, d_i, y_i, b_i; \theta) \ db_i\) requires evaluating an analytically intractable integral over a possibly multi-dimensional and infinite domain; it is therefore necessary to use some method to approximate it numerically.

Methods for approximating intractable integrals form the majority of this Chapter and are presented in Section 4.1. I will also be introducing numerical methods for differentiating a function and for root-finding in Sections 4.2 and 4.3.