Using strata instead of stratifying groups separately?

Can someone help me understand what strata(groups) does when it is used as a covariate? I am used to seeing stratified models where each level is run as a separate model, so that you can get estimates of main effects under each partition. When I run models using strata, you no longer get beta estimates for that variable.

> coxph(Surv(futime, fustat) ~ age + strata(rx), data=ovarian)
Call:
coxph(formula = Surv(futime, fustat) ~ age + strata(rx), data = ovarian)

coef exp(coef) se(coef)   z      p
age 0.1374    1.1472   0.0474 2.9 0.0038

Likelihood ratio test=12.7  on 1 df, p=0.000368
n= 26, number of events= 12
> o1 <- ovarian[which(ovarian$rx == 1), ] > o2 <- ovarian[which(ovarian$rx == 2), ]
> coxph(Surv(futime, fustat) ~ age, data=o1)
Call:
coxph(formula = Surv(futime, fustat) ~ age, data = o1)

coef exp(coef) se(coef)    z     p
age 0.1149    1.1218   0.0456 2.52 0.012

Likelihood ratio test=8.68  on 1 df, p=0.00321
n= 13, number of events= 7
> coxph(Surv(futime, fustat) ~ age, data=o2)
Call:
coxph(formula = Surv(futime, fustat) ~ age, data = o2)

coef exp(coef) se(coef)    z     p
age 0.351     1.420    0.183 1.92 0.055

Likelihood ratio test=6  on 1 df, p=0.0143
n= 13, number of events= 5


In a standard Cox model you assume that all subjects share the same hazard (which can vary as a function of time) except for a (multiplicative) effect of their covariates. This is the proportional hazard assumption,

$\lambda(t\mid X) = \lambda_0(t)\exp(X^T\beta),$

where $\lambda(t\mid X)$ is the hazard function for a subject with covariates $X$.

In a stratified model, you allow the baseline hazard to vary between strata, just like it would if you fitted separate models, but you restrict the effect of the covariates to be the same for each strata. For each subject in strata $i$, you have

$\lambda_i(t\mid X) = \lambda_{0i}(t)\exp(X^T\beta).$

Note that there is no $i$ on the $\beta$ vector. If you fit separate models for each straum, you would have

$\lambda_i(t\mid X) = \lambda_{0i}(t)\exp(X^T\beta_i),$

which is the most general model (of these three models).