examining the effects of time-varying treatments or predictors daniel almirall va medical center,...

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

GENERAL OVERVIEW

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)

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

WHAT IS THE CONTEXT?

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

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

MARGINAL STRUCTURAL MODELS

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

MOTIVATING EXAMPLE

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.

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

DATA STRUCTUREWHAT TYPE OF DATA ARE WE TALKING ABOUT?

Temporal Ordering of the DataTime, Time-varying treatments, Outcome

A1 A2

Y3

Time Interval 1 Time Interval 2 End of Study

met with health specialist or not =

1/0


1/0

outcome = end of study depression rating, continuous

Longitudinal Outcomes?Yes, they exist, but consider them…

A1 A2

Y3Y1 Y2



1/0


1/0

end of study

depression ratingbaseline depression intermediate depression

Longitudinal Outcomes?…time-varying covariates for now.

A1 A2

Y3Y1 Y2


X1 X2

baseline depression intermediate depression

Time-varying CovariatesAlong with other baseline covariates…

X1 X2

A1 A2

Y


baseline depression, age, race, …

intermediate depression


1/0


1/0

end of study

depression rating

Time-varying Covariates…and other time-varying covariates.

X1 X2

A1 A2

Y


baseline depression, age, race, suicidal id,…

intermediate depression, suicidal id, …


1/0


1/0

end of study

depression rating

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.)

FORMALIZING SCIENTIFIC QUESTIONS USING MSMs

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


• 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


• 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 ?

THE CHALLENGE OF TIME-VARYING CONFOUNDING

When does ordinary least squares regression analysis may work? How about “adjusted” OLS regression?

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).

Schematic for Effect(s) of InterestIn general: Want the effect of g(A1,A2) on EY

A1 A2

Y


g(A1,A2) may represent a multitude of effects of interest.


1/0


1/0

end of study

depression rating

Baseline Confounders

X1

A1 A2

Y



1/0


1/0

end of study

depression rating

Adjusting for X1 in ordinary regression is a legitimate strategy in this case.

spurious

spurious



X1

A1 A2

Y



1/0


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

spurious



X1

A1 A2

Y



1/0


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

spurious


Time-varying Confounders

X1 X2

A1 A2

Y



1/0


1/0

end of study

depression rating



spurious

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

First ProblemWith conditioning on (or “adjusting”) X2 in OLS.

X2

A1 A2

Y


Xcut o

ffmet with health specialist or not =

1/0

end of study

depression rating


Second Problem

X2

A1 A2


U

spurious non-causal path


1/0

end of study

depression rating


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

Second Problem

X2

A1 A2


U



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

Second Problem

X2

A1 A2


U



1/0

end of study

depression rating

outside therapy, …

income, social support, …

But …

+

-

-

Y

So what can we do to overcome?What is the alternative to “OLS adjustment” ?

X1 X2

A1 A2


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

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

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.

IPT Weighting Tutorial(non-time-varying setting)

• X is a confounder of the effect of the

effect of A on Y.

X

A

Y


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.

ORIGINALDATA

Met with HS = YES

Met with HS = NO

Sev. Base. Depression = YES

60 30Sev. Base. Depression = NO

20 40


• 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


y/n

end of study

depression


+

ORIGINALDATA

Met with HS = YES

Met with HS = NO



20 40

IPT Weighting TutorialPr(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


y/n

end of study

depression


+

the “propensity

score”

ORIGINALDATA

Met with HS = YES

Met with HS = NO



20 40

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


y/n

end of study

depression


+

WEIGHTED DATA

Met with HS = YES

Met with HS = NO



20 40

IPT Weighting TutorialPr(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)

X

A

Y


y/n

end of study

depression


+

the “propensity

score”

IPT Weighting TutorialPi = 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”:

X

A

Y


y/n

end of study

depression


WEIGHTED DATA

Met with HS = YES

Met with HS = NO


60*3/2= 90

30*3= 90

Sev. Base. Depression = NO

20*3= 60

40*3/2= 60

X

IPT Weighting TutorialPi = 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

X

A

Y


y/n

end of study

depression


WEIGHTEDDATA

Met with HS = YES

Met with HS = NO



60 60

X

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 modelX

A

Y


y/n

end of study

depression


X

β1


• Basic steps:– Calculate Pi = Pr(A=1|Xi)

– Assign Weights Wi = Ai / Pi + (1-Ai) / (1-Pi)

– 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


• 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.Pi bounded away from 0 and 1

3.Ignorability Assumption


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).

IPTW in the Time-varying Setting• Remember our Goal: Estimate the MSM

E(Y(A1,A2)) = β0 + β1 A1 + β2 A2

But…

X1 X2

A1 A2


Y

IPTW in the Time-varying SettingGoal: E(Y(A1,A2)) = β0 + β1 A1 + β2 A2• But … how do we eliminate the red

arrows? Using a IP weighting scheme.

X1 X2

A1 A2


XX X

Y

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


Key Assumption: Sequential Ignorability

X2

A1 A2


C


1/0

end of study

depression rating


unknown or unmeasured confounder

Y

X1



y/n

XX


Key Assumption: C (baseline or time-varying) does not exist.

X2

A1 A2


C


1/0

end of study

depression rating


unknown or unmeasured confounder

Y

X1



y/n

IPT WEIGHTING IN PRACTICE

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 Pt

5. Calculate Weights Wt and Examine Its Distribution

6. Re-Examine Balance at t Using the Wt Weighted Sample

7. Repeat Steps 4-6 Until Achieve Desired Balance

End loop over t .

8. Calculate Final Weights W = t Wt

9. Run Weighted Least Squares Analysis (Use Robust SEs)

10. Compare Results in 9 with Results in 2 and Comment/Discuss

A WORKED EXAMPLE USING SIMULATED (COMPUTER GENERATED) DATA

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).

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

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.

FINAL REMARKS

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.

Design RecommendationsWhat 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

Baseline Conditional MSMsCan 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.

Baseline Conditional MSMsE(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)

Structural Nested Mean ModelFor examining

Time-varying Causal Effect Moderation

X1 X2

A1 A2

Y



1/0


1/0

end of study

depression rating

baseline suicidal ideation, depression,…


Structural Nested Mean ModelWe will do this next time we meet…

X1 X2

A1 A2

Y



1/0


1/0

end of study

depression rating

baseline suicidal ideation, depression,…


References

• Robins. (1999). Association, causation, and marginal structural models. Synthese, 121:151-179.– 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.

• Robins, Hernán, Brumback. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, September 11(5):550-560. – 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. – This paper looks at the MSM and IPT Weighting when the primary

analysis model is a Discrete-time Survival Analysis.

http://philosophy.pdf/

http://joint-causal.pdf/




http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

References• McCaffrey, et al (2004).

Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychological Methods. 9(4)– 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. – 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”.– 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.

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

http://dx.doi.org/10.1007/s11121-005-0023-0

Thank you.

examining the effects of time-varying treatments or predictors daniel almirall va medical center,...

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