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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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GENERAL OVERVIEW

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Overview In this workshop we will discuss modern methods for conceptualizing 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)

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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

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WHAT IS THE CONTEXT?

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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

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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

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MARGINAL STRUCTURAL MODELS

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Marginal Structural Models: Specific Outline 1.Motivating Example(s) (in the RCT context) 2.What is the Data Structure? 3.Formalizing Questions using MSMs 4.Primary Challenge for Data Analysis The Nuisance of Time-varying confounders Why traditional OLS does not work? 5.Data Analysis using Inverse-probability of Treatment Weighting 6.Miscellaneous Issues and Considerations

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MOTIVATING EXAMPLE

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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.

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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

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DATA STRUCTURE WHAT TYPE OF DATA ARE WE TALKING ABOUT?

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Temporal Ordering of the Data Time, Time-varying treatments, Outcome A1A2 Y3 Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 outcome = end of study depression rating, continuous

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Longitudinal Outcomes? Yes, they exist, but consider them… A1A2 Y3 Y1 Y2 Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating baseline depression intermediate depression

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Longitudinal Outcomes? …time-varying covariates for now. A1A2 Y3 Y1 Y2 Time Interval 1Time Interval 2End of Study X1X2 baseline depression intermediate depression

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Time-varying Covariates Along with other baseline covariates… X1 X2 A1A2 Y Time Interval 1Time Interval 2End of Study baseline depression, age, race, … intermediate depression met with health specialist or not = 1/0 end of study depression rating

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Time-varying Covariates …and other time-varying covariates. X1 X2 A1A2 Y Time Interval 1Time Interval 2End of Study baseline depression, age, race, suicidal id,… intermediate depression, suicidal id, … met with health specialist or not = 1/0 end of study depression rating

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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.)

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FORMALIZING SCIENTIFIC QUESTIONS USING MSMs

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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: Y i (A1,A2) Y i (0, 0) = Y had patient i never met specialist Y i (1, 0) = Y had patient i met specialist once Y i (1, 1) = Y had patient i met specialist twice

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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

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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 ?

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THE CHALLENGE OF TIME- VARYING CONFOUNDING When does ordinary least squares regression analysis may work? How about “adjusted” OLS regression?

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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).

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Schematic for Effect(s) of Interest In general: Want the effect of g(A1,A2) on EY A1A2 Y Time Interval 1Time Interval 2End of Study g(A1,A2) may represent a multitude of effects of interest. met with health specialist or not = 1/0 end of study depression rating

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Baseline Confounders X1 A1A2 Y Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating Adjusting for X1 in ordinary regression is a legitimate strategy in this case. spurious baseline depression, age, race, suicidal id,…

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Baseline Confounders X1 A1A2 Y Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating Ex: Fit the following model by OLS E(Y | A1, A2, X1 ) = β0* + β1* A1 + β2* A2 + X1 spurious baseline depression, age, race, suicidal id,…

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Baseline Confounders X1 A1A2 Y Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating Ex: E(Y | A1, A2, X1 ) = β0* + β1* A1 + β2* A2 + 1 X1 As usual, note that this requires model to be correct. spurious baseline depression, age, race, suicidal id,…

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Time-varying Confounders X1 X2 A1A2 Y Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating baseline depression, age, race, suicidal id,… intermediate depression, suicidal id, … spurious However, adjusting for X2 in ordinary regression may be problematic in the time-varying treatment setting. Why?... Ex: E(Y | X1, A1, X2, A2 ) = β0* + β1* A1 + β2* A2 + 1 X1 + 2 X2

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First Problem With conditioning on (or “adjusting”) X2 in OLS. X2 A1A2 Y Time Interval 1Time Interval 2End of Study X cut off met with health specialist or not = 1/0 end of study depression rating intermediate depression, suicidal id, …

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Second Problem X2 A1A2 Time Interval 1Time Interval 2End of Study U spurious non-causal path met with health specialist or not = 1/0 end of study depression rating intermediate depression, suicidal id, … social support, life event... 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. Y

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Second Problem X2 A1A2 Time Interval 1Time Interval 2End of Study U spurious non-causal path met with health specialist or not = 1/0 end of study depression rating outside therapy, … income, social support, … Given outside therapy, we will see that meeting with health specialist decreases end-of-study depression. + - - Y

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Second Problem X2 A1A2 Time Interval 1Time Interval 2End of Study U spurious non-causal path met with health specialist or not = 1/0 end of study depression rating outside therapy, … income, social support, … But … + - - Y

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So what can we do to overcome? What is the alternative to “OLS adjustment” ? X1 X2 A1A2 Time Interval 1Time Interval 2End of Study XX That eliminate/reduce confounding in the sample. Requires that we have all confounders of A1 and A2. Weights: function of Pr(A1| X1) and Pr(A2| X1, A1, X2). X Does not require knowledge about U. Y

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ESTIMATING MSMs USING INVERSE-PROBABILITY-OF- TREATMENT WEIGHTING Now Entering … “doer of deeds” section of the workshop

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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.

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

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ORIGINAL DATA Met with HS = YES Met with HS = NO Sev. Base. Depression = YES 6030 Sev. Base. Depression = NO 2040 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 X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n +

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ORIGINAL DATA Met with HS = YES Met with HS = NO Sev. Base. Depression = YES 6030 Sev. Base. Depression = NO 2040 IPT Weighting Tutorial 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 X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n + the “propensity score”

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ORIGINAL DATA Met with HS = YES Met with HS = NO Sev. Base. Depression = YES 6030 Sev. Base. Depression = NO 2040 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? X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n +

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WEIGHTED DATA Met with HS = YES Met with HS = NO Sev. Base. Depression = YES 6030 Sev. Base. Depression = NO 2040 IPT Weighting Tutorial Pr(A=yes | X=yes) = 60/90 = 2/3 Pr(A=yes | X=no) = 20/60 = 1/3 P i = 2/3 X i + 1/3 (1-X i ) = propensity score Assign the following weights W i = A i / P i + (1-A i ) / (1-P i ) X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n + the “propensity score”

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IPT Weighting Tutorial P i = 2/3 X i + 1/3 (1-X i ) = propensity score Assign the weights W i = A i / P i + (1-A i ) / (1-P i ) Does this really work? Yes. Take a look at the “weighted table”: X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n WEIGHTED DATA Met with HS = YES Met with HS = NO Sev. Base. Depression = YES 60*3/2 = 90 30*3 = 90 Sev. Base. Depression = NO 20*3 = 60 40*3/2 = 60 X

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IPT Weighting Tutorial P i = 2/3 X i + 1/3 (1-X i ) = propensity score W i = A i / P i + (1-A i ) / (1-P i ) = weights “Weighted” Odds Ratio = 1.0 “Weighted” Risk Ratio = 1.0 “Weighted” Risk Diff = 0.0 X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n WEIGHTED DATA Met with HS = YES Met with HS = NO Sev. Base. Depression = YES 90 Sev. Base. Depression = NO 60 X

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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 X A Y met with health specialist or not = y/n end of study depression severe baseline depression = y/n X β1β1

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IPT Weighting Tutorial ( non-time-varying setting) Basic steps: –Calculate P i = Pr(A=1|X i ) –Assign Weights W i = A i / P i + (1-A i ) / (1-P i ) –Run a weighted regression E(Y | A) = W β0* + 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

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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 + β1 A 1.SUTVA (Consistency): Y = Y(1)*A + Y(0)*(1-A) 2.P i bounded away from 0 and 1 3.Ignorability Assumption

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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).

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IPTW in the Time-varying Setting Remember our Goal: Estimate the MSM E(Y(A1,A2)) = β0 + β1 A1 + β2 A2 But… X1 X2 A1A2 Time Interval 1Time Interval 2End of Study Y

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IPTW in the Time-varying Setting Goal: E(Y(A1,A2)) = β0 + β1 A1 + β2 A2 But … how do we eliminate the red arrows? Using a IP weighting scheme. X1 X2 A1A2 Time Interval 1Time Interval 2End of Study XXX Y

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IPTW in the Time-varying Setting Multiple Propensity Score Models (@ each t) Model P1 = Pr(A1=1|X1) and Model P2 = Pr(A2=1|X1,A1,X2) Assign Inverse Prob. Weights (@ each t) Assign W1 = A1/P1 + (1-A1) / (1-P1) Assign W2 = A2/P2 + (1-A2) / (1-P2) Assign Overall Weights W = W1 * W2 (each person has 1 weight) Run a weighted least squares regression: E(Y | A1,A2) = W = β0* + β1* A1 + β2* A2

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IPTW in the Time-varying Setting Key Assumption: Sequential Ignorability X2 A1A2 Time Interval 1Time Interval 2End of Study C met with health specialist or not = 1/0 end of study depression rating intermediate depression, suicidal id, … unknown or unmeasured confounder Y X1 baseline depression, age, race, suicidal id,… met with health specialist or not = y/n X X

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IPTW in the Time-varying Setting Key Assumption: C (baseline or time-varying) does not exist. X2 A1A2 Time Interval 1Time Interval 2End of Study C met with health specialist or not = 1/0 end of study depression rating intermediate depression, suicidal id, … unknown or unmeasured confounder Y X1 baseline depression, age, race, suicidal id,… met with health specialist or not = y/n

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IPT WEIGHTING IN PRACTICE

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Actual Steps: IPT Weighting in the Time-Varying Setting 1.Specify the scientific question using the MSM 2.Run Unadjusted Ordinary Least Squares Analysis At each time point t : 3.Examine Initial (Im)balance (Assess Measured Confounding) 4.Build Propensity Score P t 5.Calculate Weights W t and Examine Its Distribution 6.Re-Examine Balance at t Using the W t Weighted Sample 7.Repeat Steps 4-6 Until Achieve Desired Balance End loop over t. 8.Calculate Final Weights W = t W t 9.Run Weighted Least Squares Analysis (Use Robust SEs) 10.Compare Results in 9 with Results in 2 and Comment/Discuss

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A WORKED EXAMPLE USING SIMULATED (COMPUTER GENERATED) DATA

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Setting up the Question (MSM) Consider the following hypothetical study: Patients meet with their clinician for CBT at baseline, 4 weeks and 8 weeks post-baseline In between visits to the clinic, patients are assigned various CBT “homework assignments” Suppose depression severity (BDI) is measured at the three clinic visits (base, 4wk, 8wk) Suppose we have measured whether or not patients completed their homework in the two intervals between clinic visits (0-4wk, 4-8wk).

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Setting up the Question (MSM) Let Y = BDI8 Let A1 and A2 denote the binary variables indicating whether HW was completed (0/1=n/y) Our goal is to understand the impact of patterns of CBT homework completion (over the two intervening intervals) on depression severity outcomes at 8 weeks. Our MSM is a simple one: E(Y(A1,A2)) = β0 + β1 A1 + β2 A2 + β3 A1 A2

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Setting up the Question (MSM) Our MSM is a simple one: E(Y(A1,A2)) = β0 + β1 A1 + β2 A2 + β3 A1 A2 β0 = E [Y(0,0)] β1 = E [Y(1,0) - Y(0,0)] β2 = E [Y(0,1) - Y(0,0)] β1 + β2 + β3 = E [Y(1,1) - Y(0,0)] β3 = E [Y(1,1) - Y(1,0)] - E [Y(0,1) - Y(0,0)] The most important confounder is previous levels of depression; that is, previous BDI scores.

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FINAL REMARKS

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Separability? What if for particular levels of a covariate (or combination of covariates) all patients receive the same treatment? –Think “regression discontinuity design” for intuition In this case, inverse-probability of treatment weighting does not work. –E.g., Cannot create the propensity score models. In this case, we must rely on models for the outcome for covariate “adjustment” and propensity score methods are less useful.

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Design Recommendations What if you are planning a study like this? Key Step 1: Clear Sense of Scientific Question, MSM Clear definition of time-varying treatment How time is defined becomes important Alignment of time, time-varying treatments, and Y Key Step 2: Make Sequential Ignorability Plausible Brainstorm and measure most important factors affecting your time-varying predictor or treatment –What are all baseline and time-varying variables that determine whether patient will meet with Health Specialist? Both of these informed heavily by a well-developed conceptual model or theoretical framework

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Baseline Conditional MSMs Can we condition on X1 (and/or other baseline variables) in the MSM? Yes. For example, the following MSM: E(Y(A1,A2) | V) = β0 + β1 A1 + β2 A2 + β3 A1 A2 + V For example: V = Age, race, gender, BDI0 Suppose V is a subset of X1 This is still a MSM.

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Baseline Conditional MSMs E(Y(A1,A2) | V) = β0 + β1 A1 + β2 A2 + β3 A1 A2 + V Model specification (model fit) is important Adjusting for baseline covariates may increase precision = smaller standard errors Use “stabilized weights” with a numerator that reflects adjustment for baseline covariates –Stabilized Weights (recall V is a subset of X1) W1 = P(A1 | V) / P(A1| X1) W2 = P(A2 | V,A1) / P(A2| X1,A1,X2)

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Structural Nested Mean Model For examining Time-varying Causal Effect Moderation X1 X2 A1A2 Y Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating baseline suicidal ideation, depression,… intermediate depression, suicidal id, …

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Structural Nested Mean Model We will do this next time we meet… X1 X2 A1A2 Y Time Interval 1Time Interval 2End of Study met with health specialist or not = 1/0 end of study depression rating baseline suicidal ideation, depression,… intermediate depression, suicidal id, …

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References Robins. (1999). Association, causation, and marginal structural models. Synthese, 121:151-179.Association, causation, and marginal structural models –A classic, well-written, paper introducing the MSM and IPT Weighting Hernán, Brumback, Robins. (2001). Marginal structural models to estimate the joint causal effect of nonrandomized treatments. Journal of the American Statistical Association, 96(454):440-448.Marginal structural models to estimate the joint causal effect of nonrandomized treatments Robins, Hernán, Brumback. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, September 11(5):550-560.Marginal structural models and causal inference in epidemiology –Two excellent papers by describing the MSM and IPT Weighting: the primary motivation here are epidemiologic studies Bray, Almirall, Zimmerman, Lynam & Murphy(2006). Assessing the Total Effect of Time-varying Predictors in Prevention Research. Prevention Science 7(1):1-17. Assessing the Total Effect of Time-varying Predictors in Prevention Research. –This paper looks at the MSM and IPT Weighting when the primary analysis model is a Discrete-time Survival Analysis.

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References McCaffrey, et al (2004). Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychological Methods. 9(4)Propensity score estimation with boosted regression for evaluating causal effects in observational studies. –This is an excellent paper describing propensity score weighting in one time point. The authors describe a modern method, boosting, for calculating the propensity score. Substance abuse application. Almirall, Ten Have, Murphy(2006). Structural nested mean models for time-varying effect moderation. Forthcoming.Structural nested mean models for time-varying effect moderation. –This paper describes the SNMM for assessing time-varying causal effect moderation and introduces a simple to use 2-stage regression estimator for the SNMM and compares it to the classic estimator, the G-Estimator. The motivating application in this paper is the PROSPECT study mentioned earlier in these slides. Almirall, Coffman, Yancy, Murphy(2006). Maximum likelihood estimation of the structural nested mean model using SAS PROC NLP. Forthcoming in a book entitled “Analysis of Observational Health-Care Data Using SAS”.Maximum likelihood estimation of the structural nested mean model using SAS PROC NLP. –This book chapter describes how to implement a maximum likelihood estimator of the SNMM using SAS PROC NLP. In this chapter we examine time-varying moderators (e.g., compliance to diet, exercise) of the impact of weight loss (time-varying) on health-related quality of life.

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Assessing the Effects of Time-varying Predictors or Treatments: A Conceptual Discussion Daniel Almirall VA Medical Center, HSRD Duke Medical Center, Dept.

Assessing the Effects of Time-varying Predictors or Treatments: A Conceptual Discussion Daniel Almirall VA Medical Center, HSRD Duke Medical Center, Dept.

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