Cox’s Proportional Hazard Model
The goal is to evaluate simultaneously the effect of several covariates on survival.
Parametric Regression Model
\[\mu=\ln E[T]=\boldsymbol{\beta}^T\mathbf{z}=\sum_{i}\beta_iz_i\] where \(z_i=1\) if patient \(i\) receives treatment and \(0\) otherwise.
Under exponential distribution with mean \(1/\lambda\), the ratio \(\dfrac{\lambda(t;\mathbf{z}_1)}{\lambda(t;\mathbf{z}_0)}=\exp\left\{-\boldsymbol{\beta}^T(\mathbf{z}_1-\mathbf{z}_0)\right\}\).
Semi Parametric Model: Cox Proportional Hazard Model
\[\underbrace{\lambda(t;\mathbf{z}_i)}_{\text{nonparametric}}=\underbrace{\lambda_0(t)}_{\text{paramteric}}\quad\exp(\boldsymbol{\beta}^T\mathbf{z}_i)\]
Cox Partial Likelihood
\[L(\boldsymbol{\beta})=\prod_{j=1}^r\frac{\exp\{\boldsymbol{\beta}^T\mathbf{z}_{(j)}\}}{\sum_{l\in R_{(j)}}\exp\{\boldsymbol{\beta}^T\mathbf{z}_{(l)}\}}\] An MLE estimate of \(\beta\), \(\hat{\beta}\) is MVUE: \[E[\hat{\beta}]=\beta\quad\text{and}\quad V[\hat{\beta}]=I_n^{-1}(\beta)\overset{\text{asymptotically}}{\approx}-\left(\left.\frac{\partial^2}{\partial \beta^2}\ln L(\beta)\right|_{\beta=\hat{\beta}}\right)^{-1}\]