1. Chapter 11 Survival Analysis Part 2

2. 2 Survival Analysis and Regression Combine lots of information Combine lots of information Look at several variables simultaneously Look at several variables simultaneously Explore interactions Explore interactions model interaction directly model interaction directly Control (adjust) for confounding Control (adjust) for confounding

3. 3 Proportional hazards regression (Cox Regression) Can we relate predictors to survival time? Can we relate predictors to survival time? We would like something like linear regression We would like something like linear regression Can we incorporate censoring too? Can we incorporate censoring too? Use the hazard function Use the hazard function

4. 4 Hazard function Given patient survived to time t, what is the probability they develop outcome very soon? Given patient survived to time t, what is the probability they develop outcome very soon? (t + small amount of time) Approximates proportion of patients having event around time t Approximates proportion of patients having event around time t

5. 5 Hazard function Hazard less intuitive than survival curve Conditional probability the event will occur between t and t+  given it has not previously occurred Rate per unit of time, as  goes to 0 get instant rate Tells us where the greatest risk is given survival up to that time (risk of the event at that time for an individual)

6. 6 Possible Hazard of Death from Birth Probability of dying in next year as function of age 0 6 17 23 80  t) At which age would the hazard be greatest?

7. 7 Possible Hazard of Divorce 0 2 10 25 35 50

8. 8 Why “proportional hazards”? Ratio of hazards measures relative risk If we assume relative risk is constant over time… The hazards are proportional!

9. 9 Proportional Hazard of Death from Birth Probability of dying in next year as function of age for two groups (women, men) 0 6 17 23 80  t) At which age would the hazard be greatest?

10. 10 Proportional Hazards and Survival Curves If we assume proportional hazards then If we assume proportional hazards then The curves should not cross. The curves should not cross.

11. 11 Proportional hazards regression model one covariate 0 (t) - unspecified baseline hazard (constant) (t) the hazard for subject with X=0 (cannot be negative)  1 = regression coefficient associated with the predictor (X)  1 positive indicates larger X increases the hazard Can include more than one predictor

12. 12 Interpretation of Regression Parameters For a binary predictor; X 1 = 1 if exposed and 0 if unexposed, exp(  1 ) is the relative hazard for exposed versus unexposed (  1 is the log of the relative hazard) exp(  1 ) can be interpreted as relative risk or relative rate with all other covariates held fixed.

13. 13 Example - risk of outcome for women vs. men For males; For females; Suppose X 1 =1 for females, 0 for males

14. 14 Example - Risk of outcome for 1 unit change in blood pressure For person with SBP = 114 Suppose X 1 = systolic blood pressure (mm Hg) Relative risk of 1 unit increase in SBP: For person with SBP = 113

15. 15 Example - Risk of outcome for 10 unit change in blood pressure For person with SBP = 110 Suppose X= systolic blood pressure (mmHg) Relative risk of 10 unit increase in SBP: For person with SBP = 100

16. 16 Parameter estimation How do we come up with estimates for  i ? How do we come up with estimates for  i ? Can’t use least squares since outcome is not continuous Can’t use least squares since outcome is not continuous Maximum partial-likelihood (beyond the scope of this class) Maximum partial-likelihood (beyond the scope of this class) Given our data, what are the values of  i that are most likely? Given our data, what are the values of  i that are most likely? See page 392 of Le for details See page 392 of Le for details

17. 17 Inference for proportional hazards regression Collect data, choose model, estimate  i s Collect data, choose model, estimate  i s Describe hazard ratios, exp(  i ), in statistical terms. Describe hazard ratios, exp(  i ), in statistical terms. How confident are we of our estimate? How confident are we of our estimate? Is the hazard ratio is different from one due to chance? Is the hazard ratio is different from one due to chance?

18. 18 95% Confidence Intervals for the relative risk (hazard ratio) Based on transforming the 95% CI for the hazard ratio Based on transforming the 95% CI for the hazard ratio Supplied automatically by SAS Supplied automatically by SAS “We have a statistically significant association between the predictor and the outcome controlling for all other covariates” Equivalent to a hypothesis test; reject H o : RR = 1 at alpha = 0.05 (H a : RR  1) Equivalent to a hypothesis test; reject H o : RR = 1 at alpha = 0.05 (H a : RR  1)

19. 19 Hypothesis test for individual PH regression coefficient Null and alternative hypotheses Null and alternative hypotheses Ho : B i = 0, Ha: B i  0 Ho : B i = 0, Ha: B i  0 Test statistic and p-values supplied by SAS Test statistic and p-values supplied by SAS If p<0.05, “there is a statistically significant association between the predictor and outcome variable controlling for all other covariates” at alpha = 0.05 If p<0.05, “there is a statistically significant association between the predictor and outcome variable controlling for all other covariates” at alpha = 0.05 When X is binary, identical results as log-rank test When X is binary, identical results as log-rank test

20. 20 Hypothesis test for all coefficients Null and alternative hypotheses Null and alternative hypotheses Ho : all B i = 0, Ha: not all B i  0 Ho : all B i = 0, Ha: not all B i  0 Several test statistics, each supplied by SAS Several test statistics, each supplied by SAS Likelihood ratio, score, Wald Likelihood ratio, score, Wald p-values are supplied by SAS p-values are supplied by SAS If p<0.05, “there is a statistically significant association between the predictors and outcome at alpha = 0.05” If p<0.05, “there is a statistically significant association between the predictors and outcome at alpha = 0.05”

21. 21 Example Myelomatosis: Tumors throughout the body composed of cells derived from hemopoietic(blood) tissues of the bone marrow. N=25 dur=>is time in days from the point of randomization to either death or censoring (which could occur either by loss to follow-up or termination of the observation). Status=>has a value of 1 if dead; it has a value of 0 if censored. Treat=>specifies a value of 1 or 2 to correspond to two treatments. Renal=>has a value of 0 if renal functioning was normal at the time of randomization; it has a value of 1 for impaired functioning. The MYEL Data set take from: Survival Analysis Using SAS, A Practical Guide by Paul D. Allison - page 269

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24. SAS- PHREG PROC PHREG DATA = myel; MODEL dur*status(0) =treat; MODEL dur*status(0) =treat; RUN; Fit proportional hazards model with time to death as outcome Fit proportional hazards model with time to death as outcome “ status(0)”; observations with status variable = 0 are censored “ status(0)”; observations with status variable = 0 are censored status= 1 means an event occurred status= 1 means an event occurred Look at effect of Treatment 2 vs. Treatment 1 on mortality. Look at effect of Treatment 2 vs. Treatment 1 on mortality. Same as LIFETEST

25. 25 PROC PHREG Output Analysis of Maximum Likelihood Estimates Analysis of Maximum Likelihood Estimates Parameter Standard Hazard Parameter Standard Hazard Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio treat 1 0.57276 0.50960 1.2633 0.2610 1.773 treat 1 0.57276 0.50960 1.2633 0.2610 1.773 77% increased risk of death for treatment 2 vs. treatment 1, But it is not significant? Why?

26. 26 Complications Complications competing risks (high death rate)– RENAL FUNCTION competing risks (high death rate)– RENAL FUNCTION Non proportional hazards -time dependent covariates (will show you later) Non proportional hazards -time dependent covariates (will show you later) Extreme censoring in one group Extreme censoring in one group

27. SAS- PHREG PROC PHREG DATA = myel; MODEL dur*status(0) = renal treat; MODEL dur*status(0) = renal treat; RUN; Look at effect of Treatment 2 vs. Treatment 1 on mortality adjusted for renal functioning at baseline. Look at effect of Treatment 2 vs. Treatment 1 on mortality adjusted for renal functioning at baseline. Same as LIFETEST

28. 28 Output with adjusted treatment effect Analysis of Maximum Likelihood Estimates Parameter Standard Hazard Parameter Standard Hazard Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio renal 1 4.10540 1.16451 12.4286 0.0004 60.667 renal 1 4.10540 1.16451 12.4286 0.0004 60.667 treat 1 1.24308 0.59932 4.3021 0.0381 3.466 treat 1 1.24308 0.59932 4.3021 0.0381 3.466

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