Graphical Validity Tests for Survival Model

Definition

If the selected model holds, a plot of the data resembles a straight line, and if model fails, a plot resembles a curved line.

There are two types of plots, survival plots $(\hat{S}(t),t)$ and hazard plots $(\hat{\Lambda }(t),t)$.

One Sample

Exponential Distribution

$\ln S(t)=-\lambda t$ $\Lambda (t)=\lambda t$

Weibull Distribution

$\ln \Lambda (t)=\ln \lambda +\alpha \ln t$

Log-Normal Distribution

${\Phi }^{-1}(1-S(t))=-\frac{\mu }{\sigma }+\frac{1}{\sigma }\ln t$ where ${\Phi }^{-1}$ is the Probit.

Two to K Samples

For parametric models, repeat one sample methods on each sample.

The validity of the Cox Proportional Hazards Model can be checked by the Lehmann alternatives property of the Survival Function. $\ln S_i(t)={\gamma }_{ij}\ln S_j(t)$

Regression

Linear Model

Ordinary residual $e_i=y_i-\hat{\beta }{x}_i$ may be used in model checking.

Cox Proportional Hazard Model

$\ln \hat{S}(t)\approx -\lambda t$ where $\hat{S}$ is the Kaplan-Meier Estimator based on $({\hat{\Lambda }}_i,{\delta }_i),i=1,2,\dots ,n$ and ${\hat{\Lambda }}_i=\exp ({\mathbf{x}}_i^{⊺}\hat{\mathbf{\beta }}){\int }_0^{Y_i}\widehat{{\lambda }_0}(u)du$.

Also, the estimated cumulative hazard function and the covariates shouldn’t have any systematic pattern.