Timevarying 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 ID: 441996
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Slide1
Examining the Effects of Time-varying Treatments or Predictors
Daniel AlmirallVA Medical Center, Health Services Research and DevelopmentDuke Medical Center, Department of Biostatistics
November 16, 2007
Association for Cognitive and Behavioral Therapies
Orlando, FloridaSlide2
General overviewSlide3
Overview
In this workshop we will discuss modern methods for conceptualizing and estimating the impact of treatments or predictors that vary over timeImpact 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)Slide4
Goals of this Workshop
Minimum Case Scenario (awareness)Spur interest in these new methodsDirect you to further reading on the subjectsUnderstand your data’s potentialHopeful 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 dataSlide5
What is the context?Slide6
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 timeWe discuss secondary data analysis methodsOr a classic observational study (e.g., database or retrospective study) where patients happen to be observed switching in and out of treatment(s) over timeSlide7
Time-varying Treatments?
Treatment Sequencing: CBT: weeks 1-6; Family Therapy: weeks 8-12CBT: weeks 1-6; no follow-up therapyTiming 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 weeksSlide8
Marginal structural modelsSlide9
Marginal Structural Models:Specific Outline
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 ConsiderationsSlide10
Motivating exampleSlide11
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 basisPrimary 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.Slide12
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 questionSlide13
Data structurewhat type of data are we talking about?Slide14
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
met with health specialist or not = 1/0
outcome = end of study depression rating, continuousSlide15
Longitudinal Outcomes?Yes, they exist, but consider them…
A1
A2
Y3
Y1
Y2
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression rating
baseline depression
intermediate depressionSlide16
Longitudinal Outcomes?…time-varying covariates for now.
A1
A2
Y3
Y1
Y2
Time Interval 1
Time Interval 2
End of Study
X1
X2
baseline depression
intermediate depressionSlide17
Time-varying CovariatesAlong with other baseline covariates…
X1
X2
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
baseline depression, age, race, …
intermediate depression
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression ratingSlide18
Time-varying Covariates…and other time-varying covariates.
X1
X2
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
baseline depression, age, race, suicidal id,…
intermediate depression, suicidal id, …
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression ratingSlide19
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.)Slide20
Formalizing scientific questions using msmsSlide21
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 twiceSlide22
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) ) =
β
0 +
β
1 A1 +
β
2 A2
β
0 =
E( Y(0, 0) )
β
1 =
E( Y(1, 0) - Y(0, 0) ) = causal effect 1
β
2 =
E( Y(1, 1) - Y(1, 0) ) = causal effect 2Slide23
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) ) =
β
0 +
β
1 A1 +
β
2 A2
Why not just OLS regression of Y ~ [A1,A2] ?
That is, why not just fit the regression model:
E(Y | A1, A2) =
β
0* +
β
1* A1 +
β
2* A2 ?Slide24
The challenge of time-varying confounding
When does ordinary least squares regression analysis may work? How about “adjusted” OLS regression?Slide25
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; andBecause time-varying confounders may also be outcomes of prior treatment (e.g., on the causal pathway for prior treatment).Slide26
Schematic for Effect(s) of InterestIn general: Want the effect of
g(A1,A2) on EY
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
g
(A1,A2) may represent a multitude of effects of interest.
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression ratingSlide27
Baseline Confounders
X1
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
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
spurious
baseline depression, age, race, suicidal id,…Slide28
Baseline Confounders
X1
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
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 ) =
β
0* +
β
1* A1 +
β
2* A2
+
X1
spurious
spurious
baseline depression, age, race, suicidal id,…Slide29
Baseline Confounders
X1
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression rating
Ex
: E(Y | A1, A2, X1 ) =
β
0* +
β
1* A1 +
β
2* A2
+
1 X1
As usual, note that this requires model to be correct.
spurious
spurious
baseline depression, age, race, suicidal id,…Slide30
Time-varying Confounders
X1
X2
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression rating
baseline depression, age, race, suicidal id,…
intermediate depression, suicidal id, …
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 ) =
β
0* +
β
1* A1 +
β
2* A2
+
1 X1 + 2 X2Slide31
First Problem
With conditioning on (or “adjusting”) X2 in OLS.
X2
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
X
cut off
met with health specialist or not = 1/0
end of study
depression rating
intermediate depression, suicidal id, …Slide32
Second Problem
X2
A1
A2
Time Interval 1
Time Interval 2
End 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.
YSlide33
Second Problem
X2
A1
A2
Time Interval 1
Time Interval 2
End 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.
+
-
-
YSlide34
Second Problem
X2
A1
A2
Time Interval 1
Time Interval 2
End 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 …
+
-
-
YSlide35
So what can we do to overcome?What is the alternative to “OLS adjustment” ?
X1
X2
A1
A2
Time Interval 1
Time Interval 2
End of Study
X
X
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.
YSlide36
Estimating msms using Inverse-probability-of-treatment weighting
Now Entering … “doer of deeds” section of the workshopSlide37
Inverse-Probability Weighting?
Sometimes known as “propensity score weighting” methodologyRelated to the Horvitz-Thompson Estimator see the Survey Sampling / Demography literatureTo 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.Slide38
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.Slide39
ORIGINAL
DATAMet with HS = YESMet with HS = NOSev.
Base. Depression = YES
60
30
Sev
.
Base. Depression = NO
20
40
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
+Slide40
ORIGINAL
DATAMet with HS = YESMet with HS = NOSev.
Base. Depression = YES
60
30
Sev
.
Base. Depression = NO
20
40
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”Slide41
ORIGINAL
DATAMet with HS = YESMet with HS = NOSev.
Base. Depression = YES
60
30
Sev
.
Base. Depression = 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
met with health specialist or not = y/n
end of study
depression
severe baseline depression = y/n
+Slide42
WEIGHTED
DATAMet with HS = YESMet with HS = NOSev.
Base. Depression = YES
60
30
Sev
.
Base. Depression = NO
20
40
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”Slide43
IPT Weighting Tutorial
Pi = 2/3 Xi + 1/3 (1-Xi) = propensity scoreAssign 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
XSlide44
IPT Weighting Tutorial
Pi = 2/3 Xi + 1/3 (1-Xi) = propensity scoreWi
= 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
90
Sev
.
Base. Depression = NO
60
60
XSlide45
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
=
β
0* +
β
1*
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
β
1Slide46
IPT Weighting Tutorial(non-time-varying setting)
Basic steps:Calculate Pi = Pr(A=1|Xi)Assign Weights W
i
= A
i
/ P
i
+ (1-A
i
) / (1-P
i
)
Run a weighted regression E(Y | A) =
W
β
0* +
β
1*
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 MethodsSlide47
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
=
β
0* +
β
1*
A
identify the
causal effect
β
1 from the MSM
E(Y(A)) =
β
0 +
β
1
A
SUTVA (Consistency): Y = Y(1)*A + Y(0)*(1-A)
P
i
bounded away from 0 and 1
Ignorability AssumptionSlide48
IPT Weighting Tutorial(non-time-varying setting)
Ignorability AssumptionAlso known as the No Unmeasured Confounders AssumptionOr, 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).Slide49
IPTW in the Time-varying Setting
Remember our Goal: Estimate the MSM E(Y(A1,A2)) = β0 + β1 A1 +
β
2
A2
But…
X1
X2
A1
A2
Time Interval 1
Time Interval 2
End of Study
YSlide50
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
A1
A2
Time Interval 1
Time Interval 2
End of Study
X
X
X
YSlide51
IPTW in the Time-varying Setting
Multiple Propensity Score Models (@ each t)Model P1 = Pr(A1=1|X1) andModel 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
=
β
0* +
β
1*
A1 +
β
2*
A2Slide52
IPTW in the Time-varying Setting
Key Assumption: Sequential Ignorability
X2
A1
A2
Time Interval 1
Time Interval 2
End 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
XSlide53
IPTW in the Time-varying Setting
Key Assumption: C (baseline or time-varying) does not exist.
X2
A1
A2
Time Interval 1
Time Interval 2
End 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/nSlide54
IPT Weighting in practiceSlide55
Actual Steps: IPT Weighting in the Time-Varying Setting
Specify the scientific question using the MSMRun Unadjusted Ordinary Least Squares Analysis At each time point t :
Examine Initial (Im)balance (Assess Measured Confounding)
Build Propensity Score P
t
Calculate Weights W
t
and Examine Its Distribution
Re-Examine Balance at
t
Using the W
t
Weighted Sample
Repeat Steps 4-6 Until Achieve Desired Balance
End loop over
t .
Calculate Final Weights W =
t
W
t
Run Weighted Least Squares Analysis (Use Robust SEs)
Compare Results in 9 with Results in 2 and Comment/DiscussSlide56
A worked example using simulated (computer generated) dataSlide57
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-baselineIn 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).Slide58
Setting up the Question (MSM)
Let Y = BDI8Let 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)) =
β
0 +
β
1
A1 +
β
2
A2 +
β
3
A1 A2Slide59
Setting up the Question (MSM)
Our MSM is a simple one: E(Y(A1,A2)) = β0 + β1 A1 + β2
A2 +
β
3
A1 A2
β
0 = E [Y(0,0)]
β
1 = E [Y(1,0) - Y(0,0)]
β
2 = E [Y(0,1) - Y(0,0)]
β
1 +
β
2 +
β
3 = E [Y(1,1) - Y(0,0)]
β
3 =
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.Slide60Slide61Slide62Slide63
Final remarksSlide64
Separability?
What if for particular levels of a covariate (or combination of covariates) all patients receive the same treatment?Think “regression discontinuity design” for intuitionIn 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.Slide65
Design Recommendations
What if you are planning a study like this?Key Step 1: Clear Sense of Scientific Question, MSMClear 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 frameworkSlide66
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) = β
0 +
β
1
A1 +
β
2
A2 +
β
3
A1 A2 +
V
For example: V = Age, race, gender, BDI0
Suppose V is a subset of X1
This is still a MSM.Slide67
Baseline Conditional MSMs
E(Y(A1,A2) | V) = β0 + β1 A1 + β2
A2 +
β
3
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)Slide68
Structural Nested Mean Model
For examiningTime-varying Causal Effect Moderation
X1
X2
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression rating
baseline suicidal ideation, depression,…
intermediate depression, suicidal id, …Slide69
Structural Nested Mean Model
We will do this next time we meet…
X1
X2
A1
A2
Y
Time Interval 1
Time Interval 2
End of Study
met with health specialist or not = 1/0
met with health specialist or not = 1/0
end of study
depression rating
baseline suicidal ideation, depression,…
intermediate depression, suicidal id, …Slide70
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.Slide71
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.Slide72
Thank you.