# Which model should I use for Cox proportional hazards with paired data?

I am hoping someone can help me with which model (frailty, strata or cluster) I should use for my data. I have paired data so I need to take that into account when modelling the Cox PH and am unsure which model will give me a more accurate result.

My study was looking at the time it took for a person to become calm after being subjected to a particular stimulus. Each person was subjected to two different stimuli, on separate days. They were randomly assigned which stimulus was first. I have modeled this with survival analysis (time-to-event) but I now need to take into account that the data is paired.

Any help in regards to when you would use frailty, strata or cluster models would be great.

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It may be close enough to a prior question to be a duplicate of How to conduct conditional Cox regression for matched case-control study?. It has a very nice example of a frailty models for matched data in Stata. –  Andy W Jul 23 '12 at 12:40

This subject is covered by a number of papers including:

Here is a very brief (and non-exhaustive) summary of the differences between the two approaches.

Stratified approach

For each pair, there is an unspecified baseline hazard function. The partial likelihood idea is readily adapted by multiplying the partial likelihoods specific to each stratum.

Pros:

• Lack of structure.

Cons:

• It does not provide any information about heterogeneity between pairs;
• Pairs in which both members shared the same covariate information or which provide only censoring observations do not contribute to the likelihood; this is because no between-pair comparisons are attempted.

Frailty approach

Within-pair association is accounted for by a random effect common to both members from the same pair. Hence, there is again a different baseline hazard for each pair, but they are not totally unspecified; there is some structure. Estimation is based on the marginal likelihood.

Pros:

• Parsimony: heterogeneity is described by a single parameter;
• Summary measures about heterogeneity are available (Understanding heterogeneity...);
• It is possible to study the effect of variables common within the pairs.

Cons:

• software availability (in R, you can look at coxph() or parfm(); in SAS, you can look at proc phreg);
• research is still ongoing.



As a conclusion, the choice depends on your research. However, the last reference from the list gives some guideline:

For situations where group size is five or greater, it is difficult to justify use of the random effects model over that of the stratified model, this latter model being very much more readily implemented. The story changes for group sizes less than five and, for twin studies in particular, the efficiency gains are such that we would prefer to use a random effects model over a stratified model. The stratified model remains valid but can required from 20 per cent to 30 per cent more observations to achieve the same precision.

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Wow, awesome and concise answer. Thank you for the resources! –  Andy W Jul 23 '12 at 13:34
Thank you @Andy –  ocram Jul 23 '12 at 13:34
Thanks ocram, I appreciate the detail that you went into with your answer and the resources that you provided. You were a great help! –  Emma Jul 24 '12 at 7:19