Interpreting HRs from stratified cox survival analysis in R

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.