1. Marginal structural models: application Daniel Westreichdjw@unc.edu SPER Anaheim June 22 2009

2. Application of MSM: a brief introduction Our goal –To estimate the average causal effect of highly active antiretroviral therapy on time to the combined outcome of death or AIDS. First, the results. AnalysisHR (95% CL) Crude0.97 (0.76, 1.24) Adjusted (baseline)0.66 (0.50, 0.86) Adjusted (time-updated)0.81 (0.61, 1.06) Weighted (baseline-stratified)0.52 (0.35, 0.76)

3. The data 1498 individuals –506 men, from the Multicenter AIDS Cohort Study –992 women, from Women’s Interagency HIV Study Entered study between 1995 and 2000 At baseline, participants typically: –between 31 and 46 years old –not receiving any antiretroviral therapy –CD4 count >350 cells/mm3 (80%+ over 200) –HIV RNA viral load > 10000 copies/ml

4. Simplified causal model for these data CD4 count 0 HAART 1 AIDS/death 2 CD4 count 2 HAART 3 AIDS/death 4 …but this diagram is incomplete.

5. Simplified causal model for these data CD4 count 0 HAART 1 AIDS/death 2 CD4 count 2 HAART 3 AIDS/death 4 This additional arrow is the key problem.

6. Simplified causal model for these data CD4 count 0 HAART 1 AIDS/death 2 CD4 count 2 HAART 3 AIDS/death 4 Controlling for CD4 count…

7. Simplified causal model for these data CD4 count 0 HAART 1 AIDS/death 2 CD4 count 2 HAART 3 AIDS/death 4 …blocks these paths…

8. Simplified causal model for these data CD4 count 0 HAART 1 AIDS/death 2 CD4 count 2 HAART 3 AIDS/death 4 …which means that we cannot assess this causal pathway.

9. Analysis: statistical models Overall decision: Cox proportional hazards models Several contrasting models for illustration –Crude model –Adjusted model (baseline covariates only) –Adjusted model (time-updated covariates) –Weighted (marginal structural) model

10. Model specification: crude Crude Cox proportional hazards model Hazard is modeled as a product of –an unspecified baseline hazard AND –beta times the time-updated exposure What we want from this model (and from all following models) is

11. Model specification: baseline-adjusted Baseline-stratified Cox proportional hazards model Hazard is modeled as a product of –an unspecified baseline hazard AND –the sum of beta times the time-updated exposure AND the vector of beta coefficients X baseline covariates Z

12. Model specification: time-updated adjusted Fully-stratified Cox proportional hazards model Hazard is modeled as a product of –an unspecified baseline hazard AND –the sum of beta times the time-updated exposure AND the vector of beta coefficients X time-updated covariates L

13. Model specification: marginal structural Marginal structural Cox proportional hazards model, baseline stratified. Under the assumptions of no uncontrolled confounding or selection bias, adequate positivity, consistency, and correct model specification, we estimate this model as: …weighted by inverse probability of treatment and censoring weights Wit. If the above assumptions are met, then

14. Analysis of marginal structural model To accommodate time-varying IPTC weights, we approximate the above model using a pooled logistic regression (a discrete time hazards model). When risk per unit time is small (i.e., less than 10%), then So, with (a) low risk per time unit, and (b) assumptions met, Thus, we can estimate the quantity of interest from the MSM: the causal log hazard ratio.

15. Technical note Some of you have noticed that this model includes baseline covariates L i0 As a result, we are not estimating a purely marginal model, but instead a baseline-stratified marginal structural Cox model. This is because we plan on stabilizing the IPTC weights – by including baseline covariates in numerators – to reduce variance of the final estimator. As a result we must include covariates in the final model to ensure complete control of confounding. We could leave the weights unstabilized, and get a purely marginal estimate.

16. Estimation of the weights: theory These weights are where Denominator is: –probability of current treatment X given history of treatment, covariate history, and contingent on being uncensored. Numerator is: –similar, but contingent on baseline covariates, not all covariate history. Note that the π in front means that we’re estimating cumulative probabilities: cumulative through time. The censoring weights are modeled analogously.

17. Estimation of the weights: practice Used logistic regression models to estimate time-varying weights for –confounding of treatment AND –right-censoring by drop-out (that is, non-administrative censoring) We made a simplifying intent-to-treat assumption: once exposed, always exposed. –This assumption correctly classifies 94% of person-time in these data, so seems warranted. –As a result of this assumption, “exposure history” is simply “how long have you been unexposed?” which is the same as “how long have you been on study?”

18. Estimation of the weights: specifics We created weights using the following variables: –age (modeled as four-knot cubic spline) –race –sex –CD4 count (baseline and updated) –viral RNA (baseline and updated) –use of non-HAART ART –calendar date of study enrollment (modeled as four- knot cubic spline) –use of cotrimoxazole prophylaxis –presence of HIV symptoms reported persistent fever, diarrhea, night sweats, or weight loss Weights calculated in this way, with these variables, showed adequate positivity.

19. Results (for real this time) AnalysisHR (95% CL) Crude0.97 (0.76, 1.24) Adjusted (baseline)0.66 (0.50, 0.86) Adjusted (time-updated)0.81 (0.61, 1.06) Weighted (baseline-stratified)0.52 (0.35, 0.76)

20. Results: visual representation When we apply weights to the observed population, we create a pseudopopulation in which there is no confounding of the exposure- outcome relationship (in theory). –Equivalent to standardization under some, limited circumstances Visual contrasts between exposed and unexposed – for example, an extended Kaplan- Meier type cumulative incidence curve – drawn in the pseudopopulation will provide a theoretically unbiased causal contrast

21. Extended Kaplan-Meier curves

22. Discussion and limitations Of course, we can never know for sure whether we have controlled for all confounding or selection bias. –Sensitivity analysis can help. When we stabilize on baseline covariates, we estimate a baseline conditional association rather than a marginal association. Is that a problem? –We can explore this by not stabilizing on baseline covariates AnalysisHR (95% CL) Weighted (baseline-stratified)0.52 (0.35, 0.76) Weighted (purely marginal)0.59 (0.38, 0.92)

23. Next step: the role of time scale (teaser) Our marginal structural Cox model used time- on-study as the time scale for analysis. But most randomized trials of a drug treatment like antiretroviral therapy think about time since initiation of treatment (time-on-treatment) as the time scale. It is easy to adapt marginal structural models to a new time scale: when we do, risk sets for relative hazards estimation will change. Does this affect our analysis? (Yes. Come see my talk Wednesday.)

24. MSM: marginal structural (SER) meeting My talk (Timescale): Wed. 10.30 am, Session DEALING IN ABSOLUTES Steve Cole’s talk (Measurement error correction in MSM): Wed. 3.30pm, Session MEASUREMENT BIAS Other relevant talks –Lauren Cain –Claire Margerison –Anjum Hajat –Chanelle Howe Posters –Robert Platt –Daniel Westreich –And probably half a dozen more that I’ve missed.

25. Acknowledgements MACS and WIHS Daniel Westreich is funded by NIH/NIAID 5 T32 AI 07001-31 Training in Sexually Transmitted Diseases and AIDS Stephen R. Cole is supported by NIH/NIAID R03-AI- 071763; R01-AA-01759; P30-AI-50410

26. Marginal structural models: application Daniel Westreichdjw@unc.edu SPER Anaheim June 22 2009 Thank you.