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 Log-logistic AFT model
Log-normal AFT model
Generalized Gamma AFT model
Survival Analysis

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

10. Model Diagnosis Checking the model for Y:
If no censored observations, use Q-Q plots.
If with censored observations, use probability plot (SAS option PROBPLOT in PROC LIFEREG)
Survival Analysis

11. Graphical Methods for Model Diagnosis
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 method to each group separately
Compute a sequence of percentiles for each group
Draw the Q-Q plot, percentile of one group vs. that of another group
“almost linear” implies AFT model
Survival Analysis

13. Initial Model Selection
Which parametric AFT model (with all covariate) to start with:
For nested models: use likelihood ratio test (See SAS textbook p.89 for details and examples)
Otherwise, use AIC (See Klein Sec. 12.4)
Survival Analysis

14. Final Model Selection Fit the initial model
Conduct backward model selection by L-R tests
Survival Analysis

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

16. Residual Analysis
The Cox-Snell residuals form a right censored dataset and it must follow the exponential distribution with mean one if the model fits the data right
The residual analysis is NOT sensitive to the difference in model fit.
Survival Analysis

17. Summary
Fit AFT model including all covariates based on the Lognormal, log-logsitic, Weibull and Generalized Gamma models for Y (totally 3 models)
Use LR tests/AIC to determine your initial model
Do backward model selection to identify your final model
Conduct residual analysis
If it fits, write the fitted final model and interpret the model/describe the effects of covariates.
Survival Analysis