Joint Modelling Approaches to Survival Analysis via Likelihood-Based Boosting Techniques

Algorithm 1

lbbJM.

(i) Initialize starting values , , , , , , , and with variance-covariance-components according to Section 3.3. Choose and execute the following for every possible tuple with , :

           Longitudinal part

(ii) for to do

step1: Update fixed effects

For define with denoting the th component of . Compute the score vector and Fisher matrix

           ,

with respect to the current intercept , time effect , and the th linear effect . Obtain possible updates

and find the best performing component according to Section 2.2, yielding the update containing the update for the effect with corresponding updates for intercept and for the time effect. Receive , , and by updating

step2: Update random effects

Receive updates

           ,

for random intercepts and slopes in an additional Fisher scoring step on the penalized log-likelihood .

step3: Update variance-covariance components

Update variance-covariance components

following the description in Section 3.3.

end for

Proceed with estimates , , , , as fixed values.

           Survival part

for to do

Update survival effects

For define with denoting the th component of and . Compute the score vector and Fisher matrix

with respect to the current baseline hazard and the th linear effect or , respectively. Obtain possible updates

and find the best performing effect according to Section 2.2, yielding the update containing the update for the effect with corresponding baseline hazard update . Receive , , and by updating

end for

(iii) Determine the best performing tuple with respect to prediction based on the unpenalized joint log-likelihood as explained in more detail in Section 3.3.