The Cox proportional hazards model assumes that the hazard functions of two different groups are proportional to each other:

\(h_1(t) = h_0(t) \exp(\beta)\)

where \(h_1(t)\) and \(h_0(t)\) are the hazard functions for the two groups, and \(\beta\) is the coefficient representing the effect of a covariate (such as treatment).

Now, the survival function is related to the hazard function through:

\(S(t) = \exp\left(-\int_{0}^{t} h(u) \,du\right)\)

Applying this relationship to both groups:

• For the baseline group:

  \(S_0(t) = \exp\left(-\int_{0}^{t} h_0(u) \,du\right)\)

• For the other group:

  \(S_1(t) = \exp\left(-\int_{0}^{t} h_1(u) \,du\right) = \exp\left(-\int_{0}^{t} h_0(u)\exp(\beta) \,du\right)\)

Now, you can rewrite the expression for \(S_1(t)\) using the expression for \(S_0(t)\):

\(S_1(t) = \exp\left(\exp(\beta)\int_{0}^{t} h_0(u) \,du\right) = \exp\left(\exp(\beta) \cdot \left(-\log S_0(t)\right)\right) = S_0(t)^{\exp(\beta)}\)

This shows the relationship between the survival functions under the assumption of proportional hazards. It encapsulates the idea that the effect of the covariate (as captured by \(\beta\) has a multiplicative effect on the hazard, which leads to an exponential effect on the survival function.