1. Chapter 11 Survival Analysis Part 3

2. 2 Considering Interactions Adapted from "Anderson" leukemia data as presented in Survival Analysis: A Self-Learning Text. Adapted from "Anderson" leukemia data as presented in Survival Analysis: A Self-Learning Text. Survival time in weeks in remission for 42 leukemia patients. Survival time in weeks in remission for 42 leukemia patients. This was a randomized clinical trial where 21 patients were randomized to a treatment group and 21 to a placebo group (standard of care). This was a randomized clinical trial where 21 patients were randomized to a treatment group and 21 to a placebo group (standard of care).

3. 3 The Data Outcome Outcome status=1 for event, status=0 for censored status=1 for event, status=0 for censored Time in remission Time in remission time=time until event time=time until event White Blood Cell Count(WBC) White Blood Cell Count(WBC) logwbc=Log(WBC) logwbc=Log(WBC) Treatment Treatment trt=0 for new treatment trt=0 for new treatment trt=1 of standard of care trt=1 of standard of care

4. Survival Plot by Treatment

5. 5 SAS coding - Lifetest proc lifetest data=leuk plots=(s); time time*status(0); strata trt; symbol1 v=none color=blue line=1; symbol2 v=none color=red line=2; run;

6. 6 SAS coding - PHREG proc phreg data=leuk; model time*status(0)=trt/risklimits; run; proc phreg data=leuk; model time*status(0)=logwbc/risklimits; run; proc phreg data=leuk; model time*status(0)=trt logwbc/risklimits; run; proc phreg data=leuk; model time*status(0)=trt logwbc trt_logwbc/risklimits; run;

7. 7 Testing Equality over Strata Test Chi-Square DF Pr Chi-Square Test Chi-Square DF Pr Chi-Square Log-Rank 16.7929 1 <.0001 Log-Rank 16.7929 1 <.0001 Wilcoxon 13.4579 1 0.0002 Wilcoxon 13.4579 1 0.0002 Conclude there is a difference in the survival curves for the two treatments.

8. 8 OUTPUT from PHREG The PHREG Procedure Analysis of Maximum Likelihood Estimates Analysis of Maximum Likelihood Estimates Parameter Standard Hazard 95% Hazard Ratio Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits trt 1 1.50919 0.40956 13.5783 0.0002 4.523 2.027 10.094 trt 1 1.50919 0.40956 13.5783 0.0002 4.523 2.027 10.094 The risk of remission for standard of care is 4.523 time the risk for the new treatment. There is a 77.8% reduction in risk of remission for the new treatment compared to the standard of care. How did I get the second sentence?

9. 9 OUTPUT from PHREG The PHREG Procedure Analysis of Maximum Likelihood Estimates Analysis of Maximum Likelihood Estimates Parameter Standard Hazard 95% Hazard Ratio Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits logwbc 1 1.59375 0.29673 28.8480 <.0001 4.922 2.752 8.805 logwbc 1 1.59375 0.29673 28.8480 <.0001 4.922 2.752 8.805 The log of the white blood cell count is significant in predicting time until event. We want to include this in our analysis.

10. 10 OUTPUT from PHREG The PHREG Procedure Analysis of Maximum Likelihood Estimates Analysis of Maximum Likelihood Estimates Parameter Standard Hazard 95% Hazard Ratio Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits trt 1 1.29405 0.42210 9.3987 0.0022 3.648 1.595 8.342 trt 1 1.29405 0.42210 9.3987 0.0022 3.648 1.595 8.342 logwbc 1 1.60432 0.32933 23.7318 <.0001 4.974 2.609 9.486 logwbc 1 1.60432 0.32933 23.7318 <.0001 4.974 2.609 9.486 The risk of remission for standard of care is 3.648 time the risk for the new treatment adjusted for log(wbc). There is a 72.6% reduction in risk of remission for the new treatment compared to the standard of care adjusted for log(wbc).

11. 11 Interactions in proportional hazards regression Effect modification leads to complications in interpreting parameter estimates (the B i s) Example; (t) = 0 (t) exp(B 1 trt + B 2 log(wbc)+ B 3 trt*log(wbc)) (t) = 0 (t) exp(B 1 trt + B 2 log(wbc)+ B 3 trt*log(wbc)) What is relative risk for those on the new treatment vs. the standard treatment? How does log(wbc) effect this RR?

12. 12 OUTPUT from PHREG The PHREG Procedure Analysis of Maximum Likelihood Estimates Analysis of Maximum Likelihood Estimates Parameter Standard Hazard 95% Hazard Ratio Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits trt 1 2.35494 1.68102 1.9625 0.1612 10.537 0.391 284.201 trt 1 2.35494 1.68102 1.9625 0.1612 10.537 0.391 284.201 logwbc 1 1.80279 0.44672 16.2864 <.0001 6.067 2.528 14.561 logwbc 1 1.80279 0.44672 16.2864 <.0001 6.067 2.528 14.561 trt_logwbc 1 -0.34220 0.51974 0.4335 0.5103 0.710 0.256 1.967 trt_logwbc 1 -0.34220 0.51974 0.4335 0.5103 0.710 0.256 1.967 There does not appear to be an interaction between treatment and the log(WBC), we can leave this out of the model.

13. 13 Interactions in proportional hazards regression for two binary variables. Outcome Outcome status=1 for event, status=0 for censored status=1 for event, status=0 for censored Time in remission Time in remission time=time until event time=time until event White Blood Cell Count(WBC) White Blood Cell Count(WBC) logwbc=Log(WBC) logwbc=Log(WBC) Prior treatment Prior treatment Prior=1 if they have had prior treatment; 0 otherwise Prior=1 if they have had prior treatment; 0 otherwise Treatment Treatment trt=0 for new treatment trt=0 for new treatment trt=1 of standard of care trt=1 of standard of care

14. 14 Interactions in proportional hazards regression for two binary variables. (t) = 0 (t) exp(B 1 trt + B 2 prior + B 3 trt*prior) (t) = 0 (t) exp(B 1 trt + B 2 prior + B 3 trt*prior) For those with no prior treatment (prior = 0); Relative risk for standard of care vs. new treatment for no prior trt group;

15. 15 Interactions in proportional hazards regression for two binary variables. (t) = 0 (t) exp(B 1 trt + B 2 prior + B 3 trt*prior) (t) = 0 (t) exp(B 1 trt + B 2 prior + B 3 trt*prior) For those with prior treatment (prior = 0); Relative risk for standard of care vs. new treatment for prior trt group;

16. 16 Interactions in proportional hazards regression Question: Is the treatment effect the same for those who have had prior treatment vs. those who have not had prior treatment? Question: Is the treatment effect the same for those who have had prior treatment vs. those who have not had prior treatment? What do I look for? What do I look for?

17. 17 Tied event times Observations with outcome on same time are called “ties” Observations with outcome on same time are called “ties” They complicate the calculations They complicate the calculations Doesn’t happen too much in practice Doesn’t happen too much in practice Different methods for dealing with them Different methods for dealing with them Breslow, discrete, Efron, exact Breslow, discrete, Efron, exact Not many ties; all methods similar Not many ties; all methods similar Exact best, but computer intensive Exact best, but computer intensive Efron probably the next best Efron probably the next best Breslow default in SAS Breslow default in SAS

18. Methods for ties in PROC PHREG PROC PHREG; MODEL time*status(0) = trt / ties = exact; MODEL time*status(0) = trt / ties = exact; RUN; Use “exact” method for handling ties Use “exact” method for handling ties Other options “efron”, “breslow”, and “discrete” Other options “efron”, “breslow”, and “discrete”

19. Getting Confidence Intervals PHREG PROC PHREG DATA = vet; MODEL SurvTime*death(0) MODEL SurvTime*death(0) = treatment/risklimits; = treatment/risklimits; RUN; Provides 95% CI for HR

20. 20 Testing for Proportionality Assumption Main assumption in Cox Proportional Hazards model is proportionality. Main assumption in Cox Proportional Hazards model is proportionality. Look at Kaplan-Meier Survival Curves: Look at Kaplan-Meier Survival Curves: This method does not work well for continuous predictor This method does not work well for continuous predictor or categorical predictors that have many levels because the graph becomes to "cluttered". or categorical predictors that have many levels because the graph becomes to "cluttered". Univariate analysis: does not show whether a predictor will be proportional with other predictors in the model. Univariate analysis: does not show whether a predictor will be proportional with other predictors in the model.

21. 21 Testing for Proportionality Assumption Including time dependent covariates in the Cox model is another way to test proportionality Including time dependent covariates in the Cox model is another way to test proportionality Generate the time dependent covariates by creating interactions of the predictors and a function of survival time and include in the model. Generate the time dependent covariates by creating interactions of the predictors and a function of survival time and include in the model. If any of the time dependent covariates are significant then those predictors are not proportional. If any of the time dependent covariates are significant then those predictors are not proportional.

22. Testing Proportionality Assumption in SAS PROC PHREG DATA = myel; MODEL dur*status(0) = treat renal treatt renalt; MODEL dur*status(0) = treat renal treatt renalt;treatt=treat*log(time);renalt=renal*log(time); test_proportionality: test treatt, renalt; RUN; *Using log(time) because it is the most common function of time used in time dependant covariates, but any function of time could be used. *Using log(time) because it is the most common function of time used in time dependant covariates, but any function of time could be used.Test: Ho: Proportionality vs. Ha: Not Proportional

23. Testing Proportionality Assumption in SAS Output: The SAS System 13:33 Tuesday, April 25, 2006 4 The PHREG Procedure Analysis of Maximum Likelihood Estimates Parameter Standard Hazard Variabl DF Estimate Error Chi-Square Pr > ChiSq Ratio treat 1 -1.30384 1.96212 0.4416 0.5064 0.271 renal 1 -64.44484 8356 0.0001 0.9938 0.000 treatt 1 0.70127 0.52411 1.7903 0.1809 2.016 renalt 1 31.45503 4019 0.0001 0.9938 4.579E13 Independently the time covariates are not significant.

24. Testing Proportionality Assumption in SAS Output: Collectively the time covariates are not significant. Linear Hypotheses Testing Results Wald Label Chi-Square DF Pr > ChiSq test_proportionality 1.7903 2 0.4085

25. 25 If Proportionality Fails Choose a parametric regression model Choose a parametric regression modelOR Include a time-dependent variable for non- proportional predictors in Cox PH Include a time-dependent variable for non- proportional predictors in Cox PHOR Stratify on non-proportional predictors in Cox PH Stratify on non-proportional predictors in Cox PH

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