Solution: Interpretation of Cox's regression model

We have

\[ \alpha_i(t)= \alpha_0(t)\exp\left\{\beta_1 x_{i1} +\cdots+\beta_p x_{ip}\right\}\]

so that

\begin{align*} \frac{\alpha_2(t)}{\alpha_1(t)} &= \frac{\alpha_0(t)\exp\left\{\beta_1 x_{21} +\cdots+\beta_p x_{2p}\right\}}{\alpha_0(t)\exp\left\{\beta_1 x_{11} +\cdots+\beta_p x_{1p}\right\}} \\ &= \exp\left\{\beta_1 (x_{21}-x_{11}) +\cdots+\beta_p (x_{2p}-x_{1p})\right\},\end{align*}

which does not depend on \(t\).

With \(x_{2j} = x_{1j} + 1\) and \(x_{2l} = x_{1l}\) for all \(l \neq j\) we get from the expression above that

\begin{align*} \frac{\alpha_2(t)}{\alpha_1(t)} &= \exp\left\{\beta_1 (x_{21}-x_{11}) +\cdots+\beta_p (x_{2p}-x_{1p})\right\} \\ &= \exp\left\{\beta_j (x_{2j}-x_{1j}) + \sum_{l\neq j}\beta_l (x_{2l}-x_{1l})\right\} \\ &= \exp\left\{\beta_j \cdot 1 + \sum_{l\neq j}\beta_l \cdot 0\right\}\\ &= e^{\beta_j}.\end{align*}

Hence we can interpret \(e^{\beta_j}\) as the factor with with the hazard rate increases if the covariate \(x_j\) increases by one unit while all other covariates remain constant.