Survival Analysis1

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

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Survival Analysis

Parametric Regression Models

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Abbreviated Outline

Proportional hazards (PH) modeling Accelerated failure time (AFT)

modeling Diagnosis for models/ model selection

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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

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Proportional Hazards Model

hx(y) = h0(y)*g(X)

Hazard functionof Y given X

Baseline hazard function

A positive function

Common choice of g(x):

)exp()( XXg T

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Accelerated Failure Times Model

Yx * g(X) = Y0

Sx(y) = S0(yg(X))

Baseline survival function

Common choice of g(x):

)exp()( XXg T

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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.

Several AFT Models

Weibull AFT model

Lognormal AFT model

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Model Diagnosis

SAS reference: SAS textbook Chapter 4

1. Checking the parametric model for Y

2. Checking the AFT assumption

3. Residual analysis

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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.

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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

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Model Diagnosis

Checking the AFT model:

1. Fit Kaplan-Meier estimator to each group separately

2. Compute a sequence of percentiles for each group

3. Draw the Q-Q plot of one group vs. another group

4. “almost linear” implies AFT model

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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)

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Residual Analysis

Cox-Snell residual:

and

are i.i.d. exp(1).

)|(log iii xySe

sei

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Residual Analysis

See SAS textbook p.95 for SAS code.

The residual analysis is NOT sensitive to the difference in model fit.

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Summary

1. Fit AFT model including all covariates based on the Lognormal, Weibull and Generalized Gamma models for Y (totally 3 models)

2. Use LR tests/AIC to determine your initial model (either lognormal or weibull)

3. Do backward model selection and residual analysis