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

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

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

4. 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

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

6. 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

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

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

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

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

11. Explicit expression Expectation
with ordered event times
number of events at

12. Illustration
For k=1
We obtain
and

13. Illustration
For k=2
We obtain
and

14. Schematic presentation of EM

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

16. 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

17. 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

18. 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

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

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

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