1. 2 December 2004PubH8420: Parametric Regression Models Slide 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

2. 2 December 2004PubH8420: Parametric Regression Models Slide 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)

3. 2 December 2004PubH8420: Parametric Regression Models Slide 3 Applications – PROC LIFEREG Log Likelihood –if all the responses are observed, where –If some of the responses are right censored,

4. 2 December 2004PubH8420: Parametric Regression Models Slide 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)

5. 2 December 2004PubH8420: Parametric Regression Models Slide 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;

6. 2 December 2004PubH8420: Parametric Regression Models Slide 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;

7. 2 December 2004PubH8420: Parametric Regression Models Slide 7 Applications – PROC LIFEREG 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. Output

8. 2 December 2004PubH8420: Parametric Regression Models Slide 8 Applications – PROC LIFEREG 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 Output Continued

9. 2 December 2004PubH8420: Parametric Regression Models Slide 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 ( )

10. 2 December 2004PubH8420: Parametric Regression Models Slide 10 Applications – PROC LIFEREG Estimated CDF of Residuals Vs. Observed Duration

11. 2 December 2004PubH8420: Parametric Regression Models Slide 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;

12. 2 December 2004PubH8420: Parametric Regression Models Slide 12 Applications – PROC LIFEREG Estimation Comparison Exponential RegressionCox Regression Para- meter Hazards Ratio 95% Confidence Limits Hazards Ratio 95% Confidence Limits age1.0221.0101.0341.0171.0061.029 ALG0.6510.4730.8970.5770.4170.798

13. 2 December 2004PubH8420: Parametric Regression Models Slide 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;

14. 2 December 2004PubH8420: Parametric Regression Models Slide 14 Applications – PROC LIFEREG Median Predicted Values Vs. AGE by the Use of ALG

15. 2 December 2004PubH8420: Parametric Regression Models Slide 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.

16. 2 December 2004PubH8420: Parametric Regression Models Slide 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=lntime type3; TITLE2 "Multiple Piecewise Exponential Regression"; RUN;

17. 2 December 2004PubH8420: Parametric Regression Models Slide 17 Applications – PROC LOGISTIC Dichotomized data DATA kidney1; SET kidney; DO month=1 TO duration; IF month=duration AND status=1 THEN fail=1; ELSE fail=0; OUTPUT; END; RUN;

18. 2 December 2004PubH8420: Parametric Regression Models Slide 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;

19. 2 December 2004PubH8420: Parametric Regression Models Slide 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;

20. 2 December 2004PubH8420: Parametric Regression Models Slide 20 Applications - SAS Comparison of Parameter Estimates –Hazards Ratio in Log Scale PHREGLIFEREGGENMODLOGISTIC MethodCox Reg. Exp. Reg ( -β ) Piecewise Exp. Reg LOGITCLOGLOG AGE0.01680.0216 0.02190.0217 ALG-0.549-0.429 -0.4346-0.431