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  • cox regression with shared frailty or vce cluster

    Hello,

    In running a cox regression model (stcox), I would like to look at within group correlation. I was wondering what the difference is between using vce(cluster x) as compared to shared (x).

    In the STATA manual is says:
    "One solution would be to fit a standard Cox model, adjusting the standard errors of the estimated hazard ratios to account for the possible correlation by specifying vce(cluster patient).
    We could instead model the correlation by assuming that the correlation is the result of a latent patient-level effect, or frailty. That is, rather than fitting a standard model and specifying vce(cluster patient), we could fit a frailty model by specifying shared(patient)".

    According to this either method is acceptable. But, what are the differences? Apart from the theta you get when using shared frailty, are there any other advantages to using shared rather than vce cluster?

    Thank you for any assistance/ clarification!

  • #2
    Hi @ashira menashe. In survival models -vce(cluster(x))- and -shared(x)- are fundamentally different models. Consider for simplicity a Cox proportional hazards model with a binary treatment where the baseline hazard is constant and the treatment halves the risk, from 2 to 1, so the situation is as in this graph:

    Click image for larger version

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    Suppose now that there is shared frailty, represented by a random effect -theta- which has mean one and variance > 0. Suppose that conditional on theta the risks are exactly as in the previous graph but multiplied by -theta-. In particular, the risks for the average group with theta=1 would be as before. Unconditionally, however, averaging over theta, the risks when shared frailty has a gamma distribution with mean 1 and variance 0.25 would look as follows:

    Click image for larger version

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    The explanation is selection. Members of the high-risk groups (theta > 1) tend to fail earlier than members of the low-risk groups (theta < 1). As time passes, we are left with people who are increasingly selected from lower-risk groups. Because the untreated group has higher risk, selection is faster there. This is why the hazards are no longer proportional, and the risk ratio is two only at the start.

    If the data are generated by this process and you fit a Cox proportional hazards model without shared frailty, you will be violating the proportionality assumption and underestimating the treatment effect. There isn't much point in correcting the standard errors when the estimate itself is biased.

    This is all in marked contrast with ordinary linear models, where unobserved heterogeneity doesn't affect the parameter estimates but simply inflates standard errors.

    You can visualize the effect of varying the variance from 0 to 0.5 at https://data.princeton.edu/pop509/frailtyApp. That website has other materials on survival analysis that you may find useful, including expressions for the unconditional hazard with gamma frailty.

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    • #3
      Hi German. Thank you for explaining! And for the link too.

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      • #4
        Hello, I would like to ask this: Lets assume I have survival single-event (exit from unemployment) single-spell per ID data. Next I have program participants (treatment=1) and control group(treatment=0). I would like to check directly assumption that treated and controls are homogenous with respect to any unobservables (e.g.motivation), because I dont want my treatment effect to be biased. And I do not want to use any observed proxies (e.g. unemployement history etc.). Can I check it using "shared (treatment)" in stcox or sstsreg? Or shall I use individual frailty "shared (ID)" instead?
        If I use "shared (treatment)" and frailty term is insignificant, which of the following is true (for simplicity lets assume that treatment and control group both consist of 3 people): a) motivation in both groups is for example 1,2,3 (within group they differ but across goups they are the same and also on average) or b) motivation in treatment is 1,3,5 while in control is 2,3,4 (difference both within and across, but the same on average) or c) motivation in treatment is 1,2,3 and in control is 4,5,6 (diffference both within and across, and also different on average).
        And on other hand, if shared frailty is significant (there is unobserved heterogeinty between both groups, i.e. they have different level of motivation) does it mean that a) motivation of tretment is 3,3,3 and controls is 2,2,2 or b) treatment motivation is 1,2,3 and controls 4,5,6.
        My point is that if in case of non-significant shared frailty c) is true, than controls have higher average motivation compared to treatments and thus treatment effect is underestimated, albeit there is not shared frailty.
        Thank you Miroslav
        Last edited by miroslav suchanec; 20 Apr 2020, 05:51.

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