1. Treat everyone with sincerity,
they will certainly appear likeable and friendly.
Survival Analysis

2. Parametric Regression Models
Survival Analysis
Parametric Regression Models
Survival Analysis

3. Abbreviated Outline Proportional hazards (PH) modeling
Accelerated failure time (AFT) modeling
Diagnosis for models/ model selection
Survival Analysis

4. Notation Y: survival time X: covariate vector
hx(y): the hazard function of Y given X
Sx(y): the survival function of Y given X
Yx: Y given X
Survival Analysis

5. Proportional Hazards Model
hx(y) = h0(y)*g(X)
Hazard function
of Y given X
Baseline hazard
function
A positive
function
Common choice of g(x):
Survival Analysis

6. Accelerated Failure Times Model
Yx * g(X) = Y0
Sx(y) = S0(yg(X))
Baseline survival
function
Common choice of g(x):
Survival Analysis

7. Notes
AFT model = PH model if and only if the survival time is Weibull distributed.
A more robust (semi-parametric) method has been developed for the PH model and so fitting the parametric PH model will not be demonstrated here.
Survival Analysis

8. Several AFT Models Weibull AFT model Lognormal AFT model
Survival Analysis

9. Model Diagnosis SAS reference: SAS textbook Chapter 4
Checking the parametric model for Y
Checking the AFT assumption
Residual analysis
Survival Analysis

10. Model Diagnosis Checking the model for Y:
If no censored observations, use Q-Q plots.
If with censored observations, compare to the K-M estimates.
Survival Analysis

11. Graphical Diagnosis for Parametric Models on Y
Exponential model
Weibull model
Lognormal model
Log logistic model (exercise)
Note: these methods do not take covariates into account; must be done by groups
Survival Analysis

12. Model Diagnosis Checking the AFT model:
Fit Kaplan-Meier estimator to each group separately
Compute a sequence of percentiles for each group
Draw the Q-Q plot of one group vs. another group
“almost linear” implies AFT model
Survival Analysis

13. Final Model Selection Parametric model comparisons:
Use likelihood ratio test (See SAS textbook p.89 for details and examples)
Use AIC (See Klein Sec. 12.4)
Survival Analysis

14. Residual Analysis Cox-Snell residual: and are i.i.d. exp(1).
Survival Analysis

15. Residual Analysis See SAS textbook p.95 for SAS code.
The residual analysis is NOT sensitive to the difference in model fit.
Survival Analysis

16. Summary
Fit AFT model including all covariates based on the Lognormal, Weibull and Generalized Gamma models for Y (totally 3 models)
Use LR tests/AIC to determine your initial model (either lognormal or weibull)
Do backward model selection and residual analysis
Survival Analysis