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The model is, however, dependent on the proportional hazards assumption. It assumes that the hazard ratios between groups remain constant. In other words, no matter how the hazard rates of the subjects change during the period of observation, the hazard rate of one group relative to the other will always stay the same.

Is "groups" referring to the groups within one variable, or between the variables themselves?

My instinct says that it's the former as the latter would fall through once two variables are categorical.

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  • $\begingroup$ The "group" means different levels of the same variable, with all other values held constant. So if $X$ is continuous, and $W$ is binary, modeling the log hazard additively, then we assume that, if $X=5$, then the hazard function is $\beta$ times the same hazard when $X=4$ for the same value of $W$. $\endgroup$
    – AdamO
    Dec 13, 2022 at 17:32

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If I understand the way you phrased the question correctly, it's the predictor variables that are important. The assumption is that the ratio of hazards associated with any two specified combinations of predictor variables is constant over time.

In a Cox model, the hazard of an event for patient $i$ over time, $\lambda_i(t)$, associated with a set of covariate values $X_i(t)$ is (following Section 3.1 of Therneau and Grambsch):

$$ \lambda_i(t) = \lambda_0(t) e^{X_i(t)\beta},$$

where $\lambda_0(t)$ is the baseline hazard and $\beta$ is the corresponding set of regression coefficients, both shared among all patients. If the covariate values are constant in time for each patient, the ratio of hazards between patients $i$ and $j$ is:

$$\frac{\lambda_i(t)}{\lambda_j(t)} = \frac{\lambda_0(t) e^{X_i\beta}}{\lambda_0(t) e^{X_j\beta}} = \frac{ e^{X_i\beta}}{ e^{X_j\beta}}.$$

That's the ratio assumed to hold for all combinations of time-constant predictor values, at all times. In simple cases with a single categorical predictor you can might about that as differences between "groups," but I'd suggest thinking of that as a special case.

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