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In the R code below, binary_variable_2 has two levels (e.g. "a" and "b").

The code outputs only one HR for binary_variable_1. Is this HR a combination of two HRs (i.e. one from each strata of binary_variable_2) or something else? If it is a combination, how are the strata specific HRs combined (in simple terms)? If it is something else, what is it?

coxph(Surv(time_to,event)~binary_variable_1+strata(binary_variable_2),data=dat)
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A Cox model with a covariate $X$ is defined as $$ \lambda(t \mid X) = \lambda_0(t) \exp \left( \beta X \right) $$

where $\lambda_0(t)$ is the baseline risk and $e^\beta$ is the hazard-ratio.

A Cox model stratified upon a categorical variable $Y$ with $k$ modalities is a Cox model where a different baseline risk is used for each group:

$$ \lambda_k(t \mid X) :=\lambda(t \mid X, Y=k) = \lambda_{0k}(t) \exp(\beta X) $$

The assumption is that the effect of $X$ (the hazard-ratio) is same across each group but the baseline risks are different between those groups.

For example say $X$ is a binary treatment and $Y$ is also binary and encodes the age ($\geq 50$ vs $<50$).

Then the model assume that:

  • the effect of the treatment is the same no matter if the patient is older than 50 or not.
  • the baseline risk of getting an event is different between the two groups (e.g. people older than 50 have a higher risk)

Note that a stratified Cox model is different than a model where each group have its own hazard-ratio:

$$ \lambda(t \mid X, Y=k) = \lambda_{0}(t) \exp(\beta_k X) $$

because then since $\beta_k= \beta_j + \beta_k - \beta_j$

\begin{align*} \lambda(t \mid X, Y=k)&=\lambda_0(t)\exp(\beta_j)\exp(\beta_k-\beta_j) \\ &= \lambda(t \mid X, Y=j)\exp(\beta_k-\beta_j) \end{align*}

Thus the risks between two groups, $$\frac{ \lambda(t \mid X, Y=k)}{ \lambda(t \mid X, Y=j)}=\exp(\beta_k-\beta_j) $$ will be proportional, which may not be the case with the stratified Cox model since the baseline risks $\lambda_{0k}(t)$ and $\lambda_{0j}(t)$ are not assumed to have any particular proportional relashionship.

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