Applications - SAS

1. Applications - SAS • Parametric Regression in SAS • PROC LIFEREG • PROC GENMOD • PROC LOGISTIC • Reference: SAS ver. 8.0 SAS/STAT User’s Guide,SAS Institute, Inc., Cary, NC PubH8420: Parametric Regression Models

2. Applications – PROC LIFEREG • Mathematical Model where y is a vector of response values, (often the log of the failure times) X is a matrix of covariates variables (usually including an intercept term), β is a vector of unknown regression parameters σ is an unknown scale parameter, and ε is a vector of errors (assumed to come from any known distribution) PubH8420: Parametric Regression Models

3. Applications – PROC LIFEREG • Log Likelihood • if all the responses are observed , where • If some of the responses are right censored, PubH8420: Parametric Regression Models

4. Applications – PROC LIFEREG • Model & Estimation • Accelerated Failure Time (Life) Model • The effect of independent variables on an event time distribution is multiplicative on the event time • The effect of the covariates : change the scale of a baseline distribution of failure times, not the location • Estimation : MLE using a Newton-Raphson algorithm • Standard Errors of the parameter estimates : the inverse of the observed information matrix • Test : Normal based Test (e.g. chi-sq test, LRT) For the Exponential Distribution, ALM = PHM For the Weibull Distribution, ALM = PHM when having the same shape parameter (the parameterization for the covariates differs by a multiple of the scale parameter). PubH8420: Parametric Regression Models

5. Applications – PROC LIFEREG • Kidney Transplant Data PROC FORMAT; VALUE female 0='Male' 1='Female'; VALUE algfmt 0='Non-ALG' 1='ALG'; RUN DATA kidney; INFILE "surd01.dat"; INPUT id 1-4 age 5-6 sex 7 Alg 22 duration 25-27 status 28; lntime = log(duration); FORMAT sex female. Alg algfmt.; RUN; PubH8420: Parametric Regression Models

6. Applications – PROC LIFEREG • Exponential Regression TITLE1 "Kidney Transplants Data"; PROC LIFEREG DATA=kidney; CLASS ALG; MODEL DURATION*STATUS(0)= ALG/ DIST=EXPONENTIAL; OUTPUT OUT=out CDF=prob; TITLE2 "Simple Exponential Regression”; RUN; PubH8420: Parametric Regression Models

7. Applications – PROC LIFEREG Output Kidney Transplants Data 1 Simple Exponential Regression The LIFEREG Procedure Model Information Data Set WORK.KIDNEY Dependent Variable Log(duration) Censoring Variable status Censoring Value(s) 0 Number of Observations 469 Noncensored Values 192 Right Censored Values 277 Left Censored Values 0 Interval Censored Values 0 Name of Distribution Exponential Log Likelihood-645.2158149 Algorithm converged. PubH8420: Parametric Regression Models

8. Applications – PROC LIFEREG Output Continued Type III Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq ALG 1 6.7769 0.0092 Analysis of Parameter Estimates Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Intercept 1 4.2155 0.1400 3.9410 4.4899 906.28 Alg ALG 1 0.4254 0.1634 0.1051 0.7456 6.78 Alg Non-ALG 0 0.0000 0.0000 0.0000 0.0000 . Scale 0 1.0000 0.0000 1.0000 1.0000 Weibull Shape 0 1.0000 0.0000 1.0000 1.0000 PubH8420: Parametric Regression Models

9. Applications – PROC LIFEREG • Interpretation (Risk = λ exp(xβ) ) • λ = Exp(-β0) = exp(-4.215) = 0.015 • β1 = coefficient for ALG = 0.425 • RR(ALG=1:ALG=0) = exp(β1) = 0.654 • the risk of ALG group = λ exp(β1) = 0.015*0.654 = 0.0096 • the risk of Non-ALG group = λexp(0) = 0.015 • Testing & Conclusion • Using ALG decreased the risk 34.6% • Significant effect ( ) PubH8420: Parametric Regression Models

10. Applications – PROC LIFEREG Estimated CDF of Residuals Vs. Observed Duration PubH8420: Parametric Regression Models

11. Applications – PROC LIFEREG • Multiple Regression PROC LIFEREG DATA=kidney; CLASS ALG; MODEL DURATION*STATUS(0)= AGE ALG/ DIST=EXPONENTIAL; OUTPUT OUT=out QUANTILES=.5 STD=STD P=MED_DURATION; RUN; PubH8420: Parametric Regression Models

12. Applications – PROC LIFEREG • Estimation Comparison PubH8420: Parametric Regression Models

13. Applications – PROC LIFEREG • Predicted Values and Confidence Intervals DATA out1; SET out; ltime=log(med_duration); stde=std/med_duration; upper=exp(ltime+1.64*stde); lower=exp(ltime-1.64*stde); RUN; PubH8420: Parametric Regression Models

14. Applications – PROC LIFEREG Median Predicted Values Vs. AGE by the Use of ALG PubH8420: Parametric Regression Models

15. Applications – PROC LIFEREG • Other supported distributions • Generalized Gamma • Loglogistic • Lognormal • Weibull • Some relations among the distributions: • The Weibull with Scale=1 : exponential distribution • The gamma with Shape=1 : Weibull distribution. • The gamma with Shape=0 : lognormal distribution. PubH8420: Parametric Regression Models

16. Applications – PROC GENMOD • Piecewise exponential distribution (Poisson Regression) TITLE1 "Kidney Transplants Data"; PROC GENMOD DATA=kidney; CLASS ALG; MODEL STATUS = AGE ALG/ DIST=POISSON LINK=log OFFSET=lntimetype3; TITLE2 "Multiple Piecewise Exponential Regression"; RUN; PubH8420: Parametric Regression Models

17. Applications – PROC LOGISTIC • Dichotomized data Define a new data set, a record for each time point DATA kidney1; SET kidney; DO month=1 TO duration; IF month=duration AND status=1 THEN fail=1; ELSE fail=0; OUTPUT; END; RUN; PubH8420: Parametric Regression Models

18. Applications – PROC LOGISTIC • LOGISTIC REGRESSION with LOGIT LINK PROC LOGISTIC DATA=kidney1; CLASS month fail/ PARAM=reference REF=first; MODEL fail=age ALG; RUN; PubH8420: Parametric Regression Models

19. Applications – PROC LOGISTIC • LOGISTIC REGRESSION with CLOGLOG LINK PROC LOGISTIC DATA=kidney1 ; CLASS month fail/ PARAM=reference REF=first; MODEL fail=age ALG/ LINK=CLOGLOG; RUN; PubH8420: Parametric Regression Models

20. Applications - SAS • Comparison of Parameter Estimates • Hazards Ratio in Log Scale PubH8420: Parametric Regression Models