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Examining the Effects of Time-varying Treatments or Predictors

Examining the Effects of Time-varying Treatments or Predictors. Daniel Almirall VA Medical Center, Health Services Research and Development Duke Medical Center, Department of Biostatistics. November 16, 2007 Association for Cognitive and Behavioral Therapies Orlando, Florida.

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Examining the Effects of Time-varying Treatments or Predictors

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  1. Examining the Effects of Time-varying Treatments or Predictors Daniel Almirall VA Medical Center, Health Services Research and Development Duke Medical Center, Department of Biostatistics November 16, 2007 Association for Cognitive and Behavioral Therapies Orlando, Florida

  2. General overview

  3. Overview • In this workshop we will discuss modern methods forconceptualizing and estimating the impact of treatments or predictors that vary over time • Impact of timing and sequencing of treatments • Two classes of longitudinal causal models (developed by James Robins, Harvard) will be discussed: • Marginal Structural Models • Structural Nested Mean Models (time permitting)

  4. Goals of this Workshop • Minimum Case Scenario (awareness) • Spur interest in these new methods • Direct you to further reading on the subjects • Understand your data’s potential • Hopeful Case Scenario (+ conceptual) • Understand conceptual issues & assumptions • How do these methods compare with traditional methods • Best Case Scenario (+ technical) • Understand the estimation techniques • Carry out estimation yourself with your data

  5. What is the context?

  6. Context: Data Source? • The context is any observational study. • This includes data from an RCT where initial treatment assignments are made, but patients fall into different (measured) “sequences” of treatments over time • We discuss secondary data analysis methods • Or a classic observational study (e.g., database or retrospective study) where patients happen to be observed switching in and out of treatment(s) over time

  7. Time-varying Treatments? • Treatment Sequencing: • CBT: weeks 1-6; Family Therapy: weeks 8-12 • CBT: weeks 1-6; no follow-up therapy • Timing of Treatment Discontinuation • CBT for 3 weeks and none thereafter • CBT for 5 weeks and none thereafter • Dosing of Treatment Over Time • Number of CBT “homework assignments” finished during the CBT treatment period • Adherence to a Full Suite of Treatments • Received full treatment during weeks 1-4 • Received full treatment for the full 8 weeks

  8. Marginal structural models

  9. Motivating Example(s) (in the RCT context) What is the Data Structure? Formalizing Questions using MSMs Primary Challenge for Data Analysis The Nuisance of Time-varying confounders Why traditional OLS does not work? Data Analysis using Inverse-probability of Treatment Weighting Miscellaneous Issues and Considerations Marginal Structural Models:Specific Outline

  10. Motivating example

  11. PROSPECT Study • RCT of a tailored primary care intervention (TPCI) for depression vs. treatment as usual (TAU) • Subjects in the TPCI group were to meet with a depression health specialist on a regular basis • Primary Goal of the Study: Assess the efficacy of the TPCI vs. TAU on depression and other outcomes • So-called intent to treat analysis (ITT) • However, not all patients in the TPCI group met with their depression health specialist throughout the full course of the “treatment period”. • Patients “switched off treatment” at different time points.

  12. PROSPECT Study • The variability in treatment received (in terms of meeting with health specialist) created an opportunity to ask the following question: • Among patients in the TPCI group, what is the impact of switching off of treatment early versus later on end of study depression outcomes? • This could also be phrased as a dosing/timing question

  13. Data structurewhat type of data are we talking about?

  14. Temporal Ordering of the DataTime, Time-varying treatments, Outcome Y3 outcome = end of study depression rating, continuous A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 Time Interval 1 Time Interval 2 End of Study

  15. Longitudinal Outcomes?Yes, they exist, but consider them… Y2 Y3 Y1 end of study depression rating baseline depression intermediate depression A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 Time Interval 1 Time Interval 2 End of Study

  16. Longitudinal Outcomes?…time-varying covariates for now. Y2 Y3 Y1 A1 A2 X1 X2 baseline depression intermediate depression Time Interval 1 Time Interval 2 End of Study

  17. Time-varying CovariatesAlong with other baseline covariates… Y end of study depression rating A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 X2 X1 baseline depression, age, race, … intermediate depression Time Interval 1 Time Interval 2 End of Study

  18. Time-varying Covariates…and other time-varying covariates. Y end of study depression rating A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 X2 X1 baseline depression, age, race, suicidal id,… intermediate depression, suicidal id, … Time Interval 1 Time Interval 2 End of Study

  19. In the PROSPECT Study Recall: In our PROSPECT data, once a patient stopped meeting with their health specialist, they never met with them again for the remainder of treatment. (In general, treatment patterns do not have to be monotonic for proper application of the methods described here.)

  20. Formalizing scientific questions using msms

  21. Motivating Example: PROSPECT • Question: Among patients in the TPCI group, what is the impact of switching off of treatment early versus later on end of study depression outcomes? • Consider Potential Outcomes: Yi (A1,A2) Yi (0, 0) = Y had patient i never met specialist Yi (1, 0) = Y had patient i met specialist once Yi (1, 1) = Y had patient i met specialist twice

  22. Motivating Example: PROSPECT • Question: What is the impact of switching off of treatment early versus later on end of study depression outcomes? • Formalize the Question Using a MSM: E( Y (A1, A2) ) = β0 + β1 A1 + β2 A2 • β0 = E( Y(0, 0) ) • β1 = E( Y(1, 0) - Y(0, 0) ) = causal effect 1 • β2 = E( Y(1, 1) - Y(1, 0) ) = causal effect 2

  23. Motivating Example: PROSPECT • Question: What is the impact of switching off of treatment early versus later on end of study depression outcomes? • Formalize the Question Using a MSM: E( Y (A1, A2) ) = β0 + β1 A1 + β2 A2 • Why not just OLS regression of Y ~ [A1,A2] ? • That is, why not just fit the regression model: E(Y | A1, A2) = β0* + β1* A1 + β2* A2 ?

  24. When does ordinary least squares regression analysis may work? How about “adjusted” OLS regression? The challenge of time-varying confounding

  25. Definition of a Confounder • Loosely, a confounder is a variable that impacts subsequent treatment adoption ( assignment or receipt) and also impacts subsequent outcomes. • However, this requires more careful thought in the time-varying setting. Why? • Because of the existence of baseline and/or time-varying confounders; and • Because time-varying confounders may also be outcomes of prior treatment (e.g., on the causal pathway for prior treatment).

  26. Schematic for Effect(s) of InterestIn general: Want the effect of g(A1,A2) on EY Y end of study depression rating A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 g(A1,A2) may represent a multitude of effects of interest. Time Interval 1 Time Interval 2 End of Study

  27. Baseline Confounders Adjusting for X1 in ordinary regression is a legitimate strategy in this case. Y spurious spurious end of study depression rating A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 X1 baseline depression, age, race, suicidal id,… Time Interval 1 Time Interval 2 End of Study

  28. Baseline Confounders Ex: Fit the following model by OLS E(Y | A1, A2, X1 ) = β0* + β1* A1 + β2* A2 +  X1 Y spurious spurious end of study depression rating A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 X1 baseline depression, age, race, suicidal id,… Time Interval 1 Time Interval 2 End of Study

  29. Baseline Confounders Ex: E(Y | A1, A2, X1 ) = β0* + β1* A1 + β2* A2 + 1 X1 As usual, note that this requires model to be correct. Y spurious spurious end of study depression rating A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 X1 baseline depression, age, race, suicidal id,… Time Interval 1 Time Interval 2 End of Study

  30. Time-varying Confounders However, adjusting for X2 in ordinary regression may be problematic in the time-varying treatment setting. Y Ex: E(Y | X1, A1, X2, A2 ) = β0* + β1* A1 + β2* A2 + 1 X1 + 2 X2 spurious spurious end of study depression rating Why? ... A1 A2 met with health specialist or not = 1/0 met with health specialist or not = 1/0 X2 X1 baseline depression, age, race, suicidal id,… intermediate depression, suicidal id, … Time Interval 1 Time Interval 2 End of Study

  31. First ProblemWith conditioning on (or “adjusting”) X2 in OLS. Y end of study depression rating A1 A2 met with health specialist or not = 1/0 cut off X X2 intermediate depression, suicidal id, … Time Interval 1 Time Interval 2 End of Study

  32. Second Problem But U is neither a confounder of A1, nor on the causal pathway for A1 or A2!! With conditioning on (or “adjusting” for) X2 in OLS. spurious non-causal path Y U social support, life event... end of study depression rating A1 A2 met with health specialist or not = 1/0 X2 intermediate depression, suicidal id, … Time Interval 1 Time Interval 2 End of Study

  33. Second Problem Given outside therapy, we will see that meeting with health specialist decreases end-of-study depression. - spurious non-causal path Y U + income, social support, … end of study depression rating A1 A2 - met with health specialist or not = 1/0 X2 outside therapy, … Time Interval 1 Time Interval 2 End of Study

  34. Second Problem But … - spurious non-causal path Y U + income, social support, … end of study depression rating A1 A2 - met with health specialist or not = 1/0 X2 outside therapy, … Time Interval 1 Time Interval 2 End of Study

  35. So what can we do to overcome?What is the alternative to “OLS adjustment” ? Weights: function of Pr(A1| X1) and Pr(A2| X1, A1, X2). That eliminate/reduce confounding in the sample. Y Requires that we have all confounders of A1 and A2. Does not require knowledge about U. A1 A2 X X X X2 X1 Time Interval 1 Time Interval 2 End of Study

  36. Now Entering … “doer of deeds” section of the workshop Estimating msms using Inverse-probability-of-treatment weighting

  37. Inverse-Probability Weighting? • Sometimes known as “propensity score weighting” methodology • Related to the Horvitz-Thompson Estimator • see the Survey Sampling / Demography literature • To make ideas concrete, we first consider how to do it in the one-time point setting. • Then we see how these ideas can be extended to the time-varying setting.

  38. IPT Weighting Tutorial(non-time-varying setting) • X is a confounder of the effect of the effect of A on Y. Y • Ex: Patients more depressed at • baseline may be more likely to • meet with their HS. end of study depression A + + • Ex: They may also be • more likely to be • depressed later. met with health specialist or not = y/n X severe baseline depression = y/n

  39. IPT Weighting Tutorial(non-time-varying setting) • X is a confounder of the effect of the effect of A on Y. • Suppose we have a data set with N = 150 subjects Y end of study depression A + met with health specialist or not = y/n X severe baseline depression = y/n

  40. IPT Weighting Tutorial the “propensity score” Pr(A=yes | X=yes) = 60/90 = 2/3 Pr(A=yes | X=no) = 20/60 = 1/3 • Odds Ratio = 4.0 > 1.0 • Risk Ratio = 2.0 > 1.0 • Risk Difference = 1/3 > 0.0 Y end of study depression A + met with health specialist or not = y/n X severe baseline depression = y/n

  41. IPT Weighting Tutorial • The basic idea behind IPT weighting is to use the information in the propensity score to undo the association between the confounder(s) X and the primary “treatment” variable A • How? Y end of study depression A + met with health specialist or not = y/n X severe baseline depression = y/n

  42. IPT Weighting Tutorial the “propensity score” Pr(A=yes | X=yes) = 60/90 = 2/3 Pr(A=yes | X=no) = 20/60 = 1/3 Pi = 2/3 Xi + 1/3 (1-Xi) = propensity score Assign the following weights Wi = Ai / Pi + (1-Ai) / (1-Pi) Y end of study depression A + met with health specialist or not = y/n X severe baseline depression = y/n

  43. IPT Weighting Tutorial Pi = 2/3 Xi + 1/3 (1-Xi) = propensity score Assign the weights Wi = Ai / Pi + (1-Ai) / (1-Pi) Does this really work? Yes. Take a look at the “weighted table”: Y end of study depression A X met with health specialist or not = y/n X severe baseline depression = y/n

  44. IPT Weighting Tutorial Pi = 2/3 Xi + 1/3 (1-Xi) = propensity score Wi = Ai / Pi + (1-Ai) / (1-Pi) = weights • “Weighted” Odds Ratio = 1.0 • “Weighted” Risk Ratio = 1.0 • “Weighted” Risk Diff = 0.0 Y end of study depression A X met with health specialist or not = y/n X severe baseline depression = y/n

  45. IPT Weighting Tutorial • The final step is to model the effect of A on Y just as you would (e.g., linear regression), but using the weighted sample. • One way to do this is weighted ordinary least squares. • Ex: E(Y | A) =W= β0* + β1*A • No need to adjust for X in the actual regression model Y β1 end of study depression A X met with health specialist or not = y/n X severe baseline depression = y/n

  46. IPT Weighting Tutorial(non-time-varying setting) • Basic steps: • Calculate Pi = Pr(A=1|Xi) • Assign Weights Wi = Ai / Pi + (1-Ai) / (1-Pi) • Run a weighted regression E(Y | A) =Wβ0* + β1*A • Have more than one confounder X? • No problem. Just model Pr(A=1|X) using your favorite model for binary outcomes: • Logistic regression model, probit models, or generalized boosting models (GBM) • GBM: see McCaffrey et al 2004, Psych Methods

  47. IPT Weighting Tutorial(non-time-varying setting) • Under what assumptions does the estimate of β1* in the weighted least squares regression E(Y | A) =W= β0* + β1*A identify the causal effect β1 from the MSM E(Y(A)) = β0 + β1A • SUTVA (Consistency): Y = Y(1)*A + Y(0)*(1-A) • Pi bounded away from 0 and 1 • Ignorability Assumption

  48. IPT Weighting Tutorial(non-time-varying setting) Ignorability Assumption • Also known as the No Unmeasured Confounders Assumption • Or, more precisely, No Unmeasured Direct Confounders Assumption. • Informally, this assumptions says that all confounders (measured or unmeasured, known or unknown) have been included in X (that is, accounted, or adjusted, for).

  49. IPTW in the Time-varying Setting • Remember our Goal: Estimate the MSM E(Y(A1,A2)) = β0 + β1A1 + β2A2 But… Y A1 A2 X2 X1 Time Interval 1 Time Interval 2 End of Study

  50. IPTW in the Time-varying Setting Goal: E(Y(A1,A2)) = β0 + β1A1 + β2A2 • But … how do we eliminate the red arrows? Using a IP weighting scheme. Y A1 A2 X X X X2 X1 Time Interval 1 Time Interval 2 End of Study

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