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I am confused about stratifying on a variable in Cox models. I think $e^{\beta_1}$ still represents the hazard ratio for somebody with $x=2$ vs $x=1$, but I don't quite understand how it's still a proportional hazards model. What if two people being compared are in a different stratum? Then how is $e^{\beta_1}$ valid? Shouldn't it be $\frac{h_1(t)}{h_2(t)} e^{\beta_1}$? But then $e^{\beta_1}$ no longer represents a hazard ratio, unless the two are in the same stratum. Is this right?

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  • $\begingroup$ The "hazard ratio" is constrained to be consistent across the multiple strata. $\endgroup$
    – AdamO
    Jun 2, 2022 at 18:44

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This image depicts a scenario where stratification is required to get a consistent estimate of the hazard ratio. S is a stratification variable, shown separately in black and red line type, and X is a binary adjustment variable, with the solid line type indicating the referent (negative) value (X=0) and the dashed line type indicating presence of condition or positive value (X=1).

The hazard ratio is simulated to be 1.5 consistently, but the baseline hazard function is linear for one stratum, and curvilinear for another stratum. You can see the hazard function for both X types is proportional within each stratum, but not between stratum. Therefore S cannot be adjusted as a covariate in the model, but should be handled as a stratification factor.

enter image description here

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  • $\begingroup$ Thanks for the answer. I'm still a bit confused though, it doesn't seem obvious how you compare somebody from stratum 1 to somebody from stratum 2. It's clear now that $e^{\beta}$ is the same regardless of which stratum somebody is in. But I don't understand how the math works with $\frac{h_1(t)}{h_2(t)} e^{\beta_1}$...the hazards do not cancel? $\endgroup$
    – fmtcs
    Jun 2, 2022 at 19:22

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