/* Parametric models: AFT modeling */ /* Data described in Chapter 3 of P. Allison, "Survival Analysis Using the SAS System." */

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1 /* Parametric models: AFT modeling */ /* Data described in Chapter 3 of P. Allison, "Survival Analysis Using the SAS System." */ options ls =79; data recidall; input week arrest fin age race wexp mar paro prio educ emp1-emp52; cards; 20 1 ; data recid; set recidall; drop emp1-emp52; /* Checking for the distribution of Y using strata fin */ proc lifetest data=recid outsurv=out; time week*arrest(0); strata fin; data b; set out; s=survival; logits=log((1-s)/s); /* for log-logistic model */ logneglog=log(-log(s)); /* for weibull model */ lnorm=probit(1-s); /* for lognormal model; probit: inverse function of cdf of N(0,1) */ lweek=log(week); proc gplot data=b; symbol1 value=circle i=join; plot logits*lweek=fin logneglog*lweek=fin lnorm*lweek=fin; /* Initial AFT model selection */ model week*arrest(0)=fin age race wexp mar paro prio educ / dist=gamma; /* generalized gamma distribution */ model week*arrest(0)=fin age race wexp mar paro prio educ / dist=lnormal; /* log-normal */ model week*arrest(0)=fin age race wexp mar paro prio educ / dist=llogistic; /* log-logistic */ 1 P age

2 model week*arrest(0)=fin age race wexp mar paro prio educ/dist=weibull; /* weibull */ /* backward model selection */ /* Weibull is selected by minimal AIC and L-R tests */ /* drop paro by Wald test p-value */ model week*arrest(0)=fin age race wexp mar prio educ/dist=weibull; PROBPLOT;/* check for goodness of fit of the initial model */ /* drop wexp */ model week*arrest(0)=fin age race mar prio educ/dist=weibull; /* drop educ */ model week*arrest(0)=fin age race mar prio/dist=weibull; /* drop race */ model week*arrest(0)=fin age mar prio/dist=weibull; /* drop mar */ /* the final model */ model week*arrest(0)=fin age prio/dist=weibull; /* Final model with covariance matrix, median survival times and residuals */ model week*arrest(0)=fin age prio /dist=weibull covb; output out=a cdf=f xbeta=xb p=median STD=se; probplot; proc print data=a; /* residual analysis */ data res; set a; e=-log(1-f); proc lifetest data=res plots=(ls) notable graphics; time e*arrest(0); symbol1 v=none; 2 P age

3 /* if use other baseline Y models */ model week*arrest(0)=fin age prio/dist=lognormal; PROBPLOT; model week*arrest(0)=fin age prio/dist=llogistic; PROBPLOT; SAS Outputs: Gamma -2 Log Likelihood AIC (smaller is better) Lognormal -2 Log Likelihood AIC (smaller is better) Weibull -2 Log Likelihood AIC (smaller is better) LLogistic -2 Log Likelihood AIC (smaller is better) Weibull is selected; check its probability plot: 3 P age

4 All points are within the confidence band, so it fits fine. Initial model is Weibull AFT full model. Backward model selection: Initial model: Type III Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq fin age race wexp mar paro prio educ Final model: It must be confirmed with L-R tests. Weibull -2 Log Likelihood AIC (smaller is better) P age

5 Type III Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq fin age prio Analysis of Parameter Estimates Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 fin age prio Scale Weibull Shape Estimated Covariance Matrix Intercept fin age prio Scale Intercept fin age prio Scale Residual analysis for final model: 5 P age

6 Probability plot for final model: Lognormal AFT model: Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq Intercept <.0001 fin age prio Scale Log-logistic AFT model: Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq Intercept <.0001 fin age prio Scale P age

7 Obs week arrest fin age race wexp mar paro prio educ _PROB_ median xb se f P age