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

This article provides an overview of survival analysis, focusing on parametric regression models like the proportional hazards (PH) model and accelerated failure time (AFT) model. It discusses model diagnosis, selection, and residual analysis, using SAS as a reference.

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

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

  2. Survival Analysis Parametric Regression Models

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

  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

  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):

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

  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.

  8. Several AFT Models • Weibull AFT model • Log-logistic AFT model • Log-normal AFT model • Generalized Gamma AFT model

  9. Model Diagnosis SAS reference: SAS textbook Chapter 4 • Checking the parametric model for Y • Residual 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)

  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

  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

  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)

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

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

  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.

  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.

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