1. Adjusting for Time- Varying Confounding in Survival Analysis Natalya Verbitsky Joint Work with Jennifer Barber Susan Murphy University of Michigan

2. 2 Outline Introduction Problem Possible Solutions Evaluation Data Simulation Statistical Models Results “Path Analysis” Rules Simulation Results Conclusions Future Work

3. 3 Introduction We are concerned with cause-and- effect relationship Q: If educational opportunities for children in poor countries are increased, would couples limit their total family size via contraception? Q: Does having a teenage birth curtail educational attainment?

4. Problem: How do we assess the effect of an exposure on a response in the presence of confounders in a time-varying setting? Def: Confounders are variables that affect both response and predictors either directly or indirectly.

5. 5 Example: Q: If educational opportunities for children in poor countries are increased, would couples limit their total family size via contraception? Exposure: schooling children Response: limiting family size via contraception Some Possible Confounders: political power, parents’ education, number of children in the family, proximity of school nearby

6. 6 Possible Solutions: Some Traditional Methods Naïve approach: regular logistic regression of response on the exposure Standard approach: regular logistic regression of response on exposure including confounders as covariates New Method: Weighted logistic regression

7. 7 Data Simulation Diagram

8. 8 Variables: Unmeas=unmeasured confounder Ex: political power Unmeas=2, high Unmeas=1, medium Unmeas=0, low Conf1 j = binary measured confounder 1 at time j, j=0 or 1 Ex: presence of school nearby

9. 9 Variables (Con’d) Conf2 j = binary measured confounder 2 at time j, j=0 or 1 Ex: having a small family Expos j = binary exposure at time j, j=0 or 1 Ex: any child in the family has attended school by time j Resp j = binary response at time j, j=0 or 1 initiating permanent contraception at time j

10. 10 Statistical Models: Model 1: Regular Logistic Regression (non- parametric) Logit Pr(Resp 0 = 1)=a 0 + a 2 Expos 0 Logit Pr(Resp 1 = 1)=a 0 + a 1 + a 3 Expos 0 + a 4 Expos 1 Model 2: Logistic Regression with Confounders Logit Pr(Resp 0 = 1)=a 0 + a 2 Expos 0 + a 5 Conf1 0 + a 6 Conf2 0 Logit Pr(Resp 1 = 1)=a 0 + a 1 + a 3 Expos 0 + a 4 Expos 1 + a 5 Conf1 1 + a 6 Conf2 1 Model 3 and 4: Regular Logistic Regression with Weights (do not include confounders as covariates) Model 3 weights: both confounders included Model 4 weights: only Conf1 is included

11. 11 “Path Analysis” Rules Find all possible path between your predictor and response To calculate the effect of a particular path: Multiply the effect within the path If you conditioned on a variable and the arrows of the path meet there, multiply the arrow effect by -1 Add the effect of all possible paths to get the total effect

12. 12 Path Analysis for Expos 0 on Resp 1 in Naïve Model: Logit Pr(Resp 0 = 1)=a 0 + a 2 Expos 0

13. 13 Path Analysis for Expos 0 on Resp 1 in Naïve Model: Logit Pr(Resp 1 = 1)=a 0 + a 1 + a 3 Expos 0 + a 4 Expos 1

14. 14 Results (general) The values inside the cells represent 1 st line: average estimate and average std. error 2 nd line: the proportion of the time you would reach the wrong conclusion All results are based on 1,000 samples (unless otherwise specified) of 1,000 observations each

15. 15 Results (Table 1) NaiveStandardWeighted (Conf1, Conf2) Prt. Wtd (Conf1) Effect of Expos 0 on Resp 0 0.31 (.15) 0.56 -0.03 (.16) 0.07 -.002 (.14) 0.04 0.13 (.14) 0.13 Effect of Expos 0 on Resp 1 0.22 (.20) 0.24 -0.43 (.23) 0.48 -.003 (.20) 0.05 0.09 (.20) 0.09 Effect of Expos 1 on Resp 1 0.33 (.30) 0.22 -0.14 (.39) 0.07 0.02 (.30) 0.05 0.15 (.30) 0.08 Note: intercepts are 0.0; alphas=etas=1.5; gammas=0.5 biased estimates in regular logistic regression (Model 1) biased estimates in Model 2 in the estimates of past values of predictors; no bias in the estimates of present values of predictors According to “path analysis” rules expect positive bias in 1 st and 3 rd cells of Model 1; and a negative bias in the middle cell of Model 2

16. 16 Path Analysis for Expos 0 on Resp 1 in Standard Model: Logit Pr(Resp 1 = 1)=a 0 + a 1 + a 3 Expos 0 + a 4 Expos 1 + a 5 Conf1 1 + a 6 Conf2 1

17. 17 Path Analysis for Weighted Model

18. 18 Results (Table 2) NaïveStandardWeightedPrt. Wtd Effect of Expos 0 on Resp 0 -0.29 (.13) 0.58 -.001 (.14) 0.04 -0.01 (.12) 0.02 -0.13 (.13) 0.14 Effect of Expos 0 on Resp 1 -0.18 (.18) 0.14 -1.17 (.24) 0.99 -.0004 (.18) 0.03 -0.08 (.18) 0.06 Effect of Expos 1 on Resp 1 -0.31 (.25) 0.24 0.10 (.31) 0.06 0.002 (.25) 0.04 -.13 (.25) 0.08 Note: intercepts are 0.0; alphas=etas=-1.5; gammas=0.5 According to “path analysis” rules of thumb, expect negative bias in the 1 st and 3 rd estimates of Model 1 and in the middle row of Model 2; on the other hand, expect unbiased estimates in Model 3

19. 19 Results (Table 3) NaïveStandardWeightedPrt. Wtd. Effect of Expos 0 on Resp 0 0.17 (.14) 0.21 0.01 (.14) 0.05 -0.01 (.14) 0.05 0.08 (.14) 0.09 Effect of Expos 0 on Resp 1 0.07 (.19) 0.06 -.004 (.20) 0.05 -0.01 (.20) 0.05 0.03 (.19) 0.06 Effect of Expos 1 on Resp 1 0.10 (.27) 0.07 -.04 (.28) 0.05 -.004 (.28) 0.04 0.04 (.28) 0.04 Note: intercepts are 0.0; alphas=etas=0.5; gammas=0.5 Compare with Table 1, degree of bias depends on the values of alphas and etas Including weights does not hurt your analysis

20. 20 Results (Table 4) Naive (parsim.) Standard (parsim.) Weighted (parsim.) Prt. Wtd. (parsim.) Effect of Expos 0 on Resp 0 0.31 (.15) 0.56 -0.09 (.16) 0.10 -.002 (.14) 0.04 0.13 (.14) 0.13 Effect of Expos 1 on Resp 1 0.49 (.25) 0.51 -0.45 (.34) 0.30 0.02 (.25) 0.04 0.22 (.26) 0.14 Note: intercepts=0.0; alphas=etas=1.5; gammas=0.5 Compare with Table 1, in parsimonious models 1 and 2 bias in the estimates of effect of Expos 1 on Resp 1 has increased Model 3 has unbiased estimates

21. 21 Results (Table 5) NaïveStandardWeightedPrt. Wtd. (Conf1) Prt. Wtd. (Conf2) Effect of Expos 0 on Resp 0.27 (.14) 0.47 -.03 (.15) 0.04 0.003 (.14) 0.03.09 (.14) 0.08.15 (.14) 0.15 Effect of Expos 0 on Resp 1.17 (.20) 0.16 -.51 (.23) 0.62 -0.03 (.20) 0.06.01 (.20) 0.06.11 (.20) 0.10 Effect of Expos 1 on Resp 1.30 (.29) 0.18 -.07 (.36) 0.05 -.002 (.29) 0.06.06 (.29) 0.06.20 (.30) 0.10 Note: intercepts=0.0; alpha1j=2.25, alpha2j=0.75 j=0 or 1; etas=1.5 Gammas=0.5 Model 3 is weighted logistic regression, weights use Conf1 and Conf2 Model 4 is weighted logistic regression, weights use only Conf1 (3/4) Model 5 is weighted logistic regression, weights use only Conf2 (1/4) Weighting Model is robust to missing confounders

22. 22 Conclusions Regular logistic regression (Model 1)—biased estimates Regular logistic regression with confounders (Model 2)—bias due to confounders affected by past exposure The sign of the bias can be found by using “path analysis” type rules of thumb Degree of the bias depends on the strength of correlations between Unmeas and Conf, Conf and Expos, and past Expos and Conf

23. 23 Conclusions (Con’d): Parsimonious Models have biased estimates of effect of Expos on Resp The use of weights incorporating all confounders eliminates this bias Weighting method is robust to missing confounders

24. 24 Future Work Diagnostics for situations with “bad” weights