1. Modeling clustered survival data The different approaches

2. Alternative approaches for modeling clustered survival data The fixed effects model The fixed effects model The stratified model The stratified model The frailty model The frailty model The marginal model The marginal model

3. Example of bivariate survival data Time to reconstitution of blood-milk barrier after mastitis Time to reconstitution of blood-milk barrier after mastitis Two quarters are infected with E. coli One quarter treated locally, other quarter not Blood milk-barrier destroyed Milk Na + increases Time to normal Na + level

4. Time to reconstitution data Cow number123…99100 Heifer110…10 Treatment1.96.50*4.78…0.664.93 Placebo0.416.50*2.62…0.986.50*

5. Time to reconstitution figurative

6. The parametric fixed effects model Introduce fixed cow effect Introduce fixed cow effect We parameterise baseline hazard We parameterise baseline hazard E.g. Weibull: E.g. Weibull: Baseline hazard Treatment effect Fixed cow effect, c 1 =0

7. The proportional hazards model From the model From the model it follows that the hazard ratio of two individuals is given by and this ratio is thus constant over time

8. Fixed effects model likelihood Survival likelihood: hazard and survival functions required Survival likelihood: hazard and survival functions required Maximise log likelihood to find estimates for , c i and  Maximise log likelihood to find estimates for , c i and 

9. Parameter estimates fixed effects model Parameter estimate for trt effect  with  =1 Parameter estimate for trt effect  with  =1 Additionally another 99 (!) parameters for the different cows Additionally another 99 (!) parameters for the different cows Model SE(  ) Fixed effects 0.1850.190

10. Disadvantages fixed effects model Estimates a large set of nuisance parameters Estimates a large set of nuisance parameters No estimate for the cow to cow variability No estimate for the cow to cow variability Only handles covariates that change within cluster Only handles covariates that change within cluster E.g. heifer effect can not be studied in fixed effects model E.g. heifer effect can not be studied in fixed effects model Less efficient than frailty model (see later) Less efficient than frailty model (see later)

11. The stratified model Different baseline hazard for each cow Different baseline hazard for each cow Baseline hazard is left unspecified Baseline hazard is left unspecified We use partial likelihood (Cox, 1972) We use partial likelihood (Cox, 1972) Baseline hazard Treatment effect

12. Stratified model likelihood Partial likelihood determined for each cow separately, then multiplied (independence) Partial likelihood determined for each cow separately, then multiplied (independence) Maximise partial log likelihood to find estimates for  alone Maximise partial log likelihood to find estimates for  alone

13. Parameter estimates stratified model Parameter estimate for trt effect  with  =1 Parameter estimate for trt effect  with  =1 Model SE(  ) Stratified0.1310.209 Fixed effects 0.1850.190

14. Disadvantages stratified model =disadvantages fixed effects model =disadvantages fixed effects model Even more inefficient Even more inefficient A cow only contributes to the partial likelihood if an event is observed for one quarter while the other quarter is still at risk A cow only contributes to the partial likelihood if an event is observed for one quarter while the other quarter is still at risk

15. The frailty model Different frailty term for each cow Different frailty term for each cow Baseline hazard is assumed to be parametric Baseline hazard is assumed to be parametric We make distributional assumptions for u i We make distributional assumptions for u i E.g. one parameter gamma frailty density E.g. one parameter gamma frailty density Baseline hazard Treatment effect Random cow effect

16. Frailty model likelihood Conditional (on frailty) survival likelihood Conditional (on frailty) survival likelihood Marginal survival likelihood: integrate out frailty Marginal survival likelihood: integrate out frailty

17. Parameter estimates frailty model Parameter estimate for trt effect  with  =1 Parameter estimate for trt effect  with  =1 Model SE(  ) Frailty0.1710.168 Stratified0.1310.209 Fixed effects 0.1850.190

18. Advantages frailty model Provides an estimate of the cow to cow variability,  or the variance of the random effect. Provides an estimate of the cow to cow variability,  or the variance of the random effect. In our example,  =0.286 In our example,  =0.286 It will also give estimates for covariates that are only changing from cluster to cluster, It will also give estimates for covariates that are only changing from cluster to cluster, E.g. the heifer variable changes from cow to cow E.g. the heifer variable changes from cow to cow It uses the available information in the most efficient way It uses the available information in the most efficient way It uses all information, even if within a cluster one observation is missing It uses all information, even if within a cluster one observation is missing Most efficient even for balanced bivariate survival data Most efficient even for balanced bivariate survival data

19. Undadjusted model Finally consider unadjusted model Finally consider unadjusted model Are observed results for our example coincidence or do they reflect a particular pattern? Are observed results for our example coincidence or do they reflect a particular pattern?Model SE(  ) Unadjusted0.1760.162 Frailty0.1710.168 Stratified0.1310.209 Fixed effects 0.1850.190

20. Asymptotic variance The asymptotic variance of the estimate of  is given as a diagonal element of the inverse of observed or expected information matrix The asymptotic variance of the estimate of  is given as a diagonal element of the inverse of observed or expected information matrix The expected (Fisher) information matrix is The expected (Fisher) information matrix is with H(  ) the Hessian matrix (  is parameter vector) with (q,r) th element

21. Asymptotic efficiency (1) Unadjusted model Unadjusted model Fixed effects model Fixed effects model Frailty model Frailty model

22. Asymptotic efficiency (2)

23. Small sample size efficiency by simulation Generate 2000 data sets with 100 pairs of two subjects with =0.23,  =0.18,  =0.3 Generate 2000 data sets with 100 pairs of two subjects with =0.23,  =0.18,  =0.3 Three different settings Three different settings 100 % balance 100 % balance 80 % balance 80 % balance 80 % uncensored 80 % uncensored Look at median and coverage Look at median and coverage

24. Simulation results

25. The marginal model Assume frailty model is true underlying model Assume frailty model is true underlying model Fitting model without taking clustering into account, likelihood contributions are based on Fitting model without taking clustering into account, likelihood contributions are based on Therefore, this is called the marginal model Therefore, this is called the marginal model

26. Marginal model parameter estimates The estimate is a consistent estimator for  The estimate is a consistent estimator for  See Wei, Lin and Weissfeld (1989) See Wei, Lin and Weissfeld (1989) Its asymptotic variance might not be correct because no adjustment done for correlation Its asymptotic variance might not be correct because no adjustment done for correlation We might use either We might use either Jackknife estimators Jackknife estimators Sandwich estimators Sandwich estimators

27. Jackknife estimator Generally given by Generally given by We use grouped jackknife technique We use grouped jackknife technique Left-out observations independent of remaining Left-out observations independent of remaining

28. Jackknife versus sandwich Lipsitz (1994) demonstrates correspondence between jaccknife and sandwich estimator Lipsitz (1994) demonstrates correspondence between jaccknife and sandwich estimator In the time to blood milk reconstitution In the time to blood milk reconstitution Unadjusted model: SE = 0.176 Unadjusted model: SE = 0.176 Grouped jackknife estimator: SE = 0.153 Grouped jackknife estimator: SE = 0.153 Grouped jackknife estimator leads to smaller variance!! Is this always so? Grouped jackknife estimator leads to smaller variance!! Is this always so?

29. Simulation results jackknife

30. Accelerated failure time models AFT model (for binary covariate) AFT model (for binary covariate)  is accelerator factor:  >1 accelerates process in treatment group  is accelerator factor:  >1 accelerates process in treatment group e.g.

31. Proportional hazards (PH) versus accelerated failure time (AFT) PH model (for binary covariate) PH model (for binary covariate) AFT model (for binary covariate) AFT model (for binary covariate)

33. Log-linear model representation In most packages (SAS, R) survival models (and their estimates) are parametrized as log linear models In most packages (SAS, R) survival models (and their estimates) are parametrized as log linear models If the error term e ij has extreme value distribution, then this model corresponds to If the error term e ij has extreme value distribution, then this model corresponds to PH Weibull model with PH Weibull model with AFT Weibull model with AFT Weibull model with