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

Treat everyone with sincerity, they will certainly appear likeable and friendly. Survival Analysis. Parametric Regression Models. Abbreviated Outline. Proportional hazards (PH) modeling Accelerated failure time (AFT) modeling Diagnosis for models/ model selection. Notation.

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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 • Lognormal AFT model

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

  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

  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

  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)

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

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

  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

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