Basics of the semiparametric frailty model Luc Duchateau Ghent University, Belgium

Fitting the semiparametric frailty model The EM approach The penalised partial likelihood approach The bayesian approach The laplacian integration approach

The EM approach For frailty models we have observed information Unobserved information

Solution based on marginal likelihood? Assuming a Weibull baseline hazard and a one-parameter gamma frailty model we obtained an explicit expression for the marginal log likelihood The marginal log likelihood contains the baseline hazard. So we run into trouble if we want to take a semiparametric approach, i.e., unspecified baseline hazard Heterogeneity Weibull Risk coeff

Splitting log likelihood in two parts Assume we have complete information observed o Denote by the joint ‘density’

Introducing partial likelihood Since -at the moment- we assume observed we can estimate using the partial likelihood idea, i.e., replace by maximize with respect to maximize with respect to Since is unobserved, approach needs to be adapted

Use Expectation-Maximisation Step 1: Initialisation Start with and (obtained by fitting the classical Cox model) as initial values Step 2: Expectation Given obtain

Use Expectation-Maximisation Step 1: Initialisation Step 2: Expectation Step 3: Maximisation Take with  Maximise to obtain

Use Expectation-Maximisation Step 1: Initialisation Step 2: Expectation Step 3: Maximisation Step 4: Maximisation Take with  Maximise to obtain

Expectation step in detail Can we get explicit expressions for the conditional expectations in Step 2? Yes for the gamma frailty density

Explicit expression Expectation with ordered event times number of events at

Illustration For k=1 We obtain and

Illustration For k=2 We obtain and

Schematic presentation of EM

The PPL approach We use random effects rather than the frailties The full data log likelihood:

The penalty term can be seen as penalty term For the reference value (mean) is zero or have small (close to zero) and therefore takes a large negative value: a decrease of the likelihood (act as a penalty) We therefore write with

PPL: partial likelihood - penalty term To apply semiparametric ideas, consider the ‘s in as ‘parameters’ (we put the ‘s and the components of at the same level Using partial likelihood ideas we replace by with

Conclusion For inference on and we will use the penalized partial likelihood We now consider two concrete examples: (McGilchrist and Aisbett (1991)) is loggamma, i.e., is a one-parameter gamma density

Zero-mean normal density Maximization of consists of an inner loop (k index) an outer loop (l index)

The loggamma density Maximization as before with inner and outer loop but more involved since REML estimator for is not available

Marginal partial likelihood We obtain a new value of by a golden section search on As stopping criterion we now use What is ?