### Article

## The correlated frailty model for paired event times

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Published: | September 20, 2011 |
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### Outline

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Bivariate data occurs in clinical research if the observations are pair wise correlated because they arise from the same patient or from close relatives. Bivariate observations are for example the event times for diabetic retinopathy, kidney failure or the survival time of twins or other siblings.

The bivariate correlated gamma frailty model [1] is an extension of the Cox model. It assumes that besides covariates there are unobservable random effects, possibly correlated within the pairs, influencing the survival times. As in the Cox model there is no parametric assumption for the baseline hazard.

Parametric correlated gamma frailty models with specific assumptions about the shape of the baseline hazard are easy to fit using traditional maximum likelihood estimation because the likelihood function has an explicit form. If information about the baseline hazard function is available, the parametric approach is efficient. But in real data applications the form of the baseline hazard is often unknown and distributional assumptions should be avoided as they are in the Cox model. Although desirable, there is no software package yet that can deal with correlated gamma frailty with nonparametric baseline hazards.

We estimate the parameters of the semi parametric correlated gamma frailty model by means of an EM algorithm that we implemented in the R software. It is an extension of the EM-algorithm used for the semi parametric shared gamma frailty model [2], [3].

We test the performance of the EM algorithm by means of simulated data. It can be seen that the choice of starting values has large influence on the convergence rate and speed. We also test the algorithm for a real data set.

### References

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- Yashin AI, Iachine I. How long can Humans live? Lower bound for biological limit of human longevity calculated from Danish twin data using correlated frailty model. Mechanisms of Ageing and Development. 1995;80:147-169.
- 2.
- Klein JP. Semi parametric Estimation of Random Effects Using the Cox Model Based on the EM Algorithm. Biometrics, International Biometric Society. 1992;48:795-806.
- 3.
- Nielsen GG. Gill RD, Andersen PK, Sorensen TI. A Counting Process Approach to Maximum Likelihood Estimation in Frailty Models. Scandinavian Journal of Statistics. 1992;19:25-43.