Fitting Marginal Structural
Models
Eleanor M Pullenayegum
Asst Professor
Dept of Clin. Epi & Biostatistics
pullena@mcmaster.ca
Outline
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Causality and observational data
Inverse-Probability weighting and MSMs
Fitting an MSM
Goodness-of-fit
Assumptions/ Interpretation
Causality in Medical Research
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Often want to establish a causal association
between a treatment/exposure and an event
Difficult to do with observational data due to
confounding
Gold-standard for causal inferences is the
randomized trial
Randomize half the patients to receive the
treatment/exposure, and half to receive usual
care
Deals with measured and unmeasured
confounders
Randomized trials are not always
possible
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Sometimes, they are unethical
cannot do a randomized trial on the effects of
second-hand smoke on lung cancer
 or a randomized trial of the effects of living
near power stations
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Sometimes, they are not feasible
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Study of a rare disease (funding is an issue)
Observational Studies
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Observe rather than experiment (or interfere!)
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Recruit some people who are exposed to secondhand smoke and some who are not
Study communities living close to power lines vs.
those who don’t
Confounding is a major concern
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For 1st example, are workplace environment, home
environment, age, gender, income similar between
exposed and unexposed?
For 2nd example, are education, family income, air
pollution similar between cases and controls?
Handling Confounding
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Match exposed and unexposed on key
confounders
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Adjust for confounders
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e.g. for every family living close to a power station,
attempt to find a control family living in a similar
neighbourhood with a similar income
for the smoking example, adjust for age, gender,
level of education, income, type of work, family
history of cancer etc.
Cannot deal with unmeasured confounders
Causal Pathways
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There are some things we cannot adjust
for
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When studying the effect of a lipid-lowering
drug on heart disease, we can’t adjust for
LDL-cholesterol level
Causal Pathways
Drug
LDL-cholesterol
Heart Disease
LDL-cholesterol mediates the effect of the drug
Cannot adjust for variables that are on the
causal pathway between exposure and
outcome.
Motivating Example
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Juvenile Dermatomyositis (JDM) is a rare but
serious skin/muscle disease in children
Standard treatment is with steroids
(Prednisone), however these have unpleasant
side-effects
Intravenous immunoglobulin (IVIg) is a possible
alternative treatment
DAS measures disease activity
JDM Dataset
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81 kids, 7 on IVIg at baseline, 23 on IVIG later
Outcome is time to quiescence
Quiescence happens when DAS=0
IVIg tends to be given when the child is doing
particularly badly (high DAS)
DAS is a counfounder
Causal Pathway for JDM study
DASt
DASt+1
IVIgt
IVIgt+1
…
…
Time-to-Quiescence
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DAS confounds IVIg and outcome
DAS is on the causal pathway
A Thought Experiment
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Suppose that at each time t, we could create an
identical copy of each child i.
Then if the real child received IVIG, we would
give the copy control and vice versa
We could then compare the child to its copy
Solves confounding by matching: the child is
matched with the copy
If treatment varies on a monthly basis and we
follow for 5 years, we would have 260-1 copies
Counterfactuals
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Clearly, this is impossible.
But we can use the idea
Define the counterfactuals for child i to
be the outcomes for each of the 260-1
imaginary copies
Idea: treat the counterfactuals as missing
data
Inverse-Probability Weighting
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Inverse-Probability Weighting (IPW) is a way of
re-weighting the dataset to account for selective
observation
E.g. if we have missing data, then we weight the
observed data by the inverse of the probability
of being observed
Why does this work?
Suppose we have a response Yij, treatment
indicator xij and Rij=1 if Yij observed, 0 o/w
Inverse-Probability Weighting
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Suppose we want to fit the
marginal model
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Usually, we solve the GEE n
1
equation

x
V
 i i (Yi  x i )  0
E(Yij | x i )  x ij
i 1
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If we use just the observed data,
n
we solve
 1

x
i 1
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i
Vi i (Yi   x i )  0;  ijj  R ij
LHS does not have mean 0
Inverse-Probability Weighting
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If we replace  by  ijj  R ij pij with pij, the conditional
probability of observing Yij, then
E( ijj  (Yij  xij ) | x)
 E( E(Rij pij | x,Yij ,...,Yi1 )  (Yij  x ij ) | x)
 E(
1
pij
E(Rij | x,Yij ,...,Yi1 )  (Yij  x ij ) | x)
 E(Yij  xij | x)
 0 because pij  P(Rij  1| x,Yij ,...,Yi1 )
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What to condition on?
 Must condition on Yij
 If MAR, then conditionally independent given
previous Y
Marginal Structural Models
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MSMs use inverse-probability weighting to deal
with the unobserved (“missing”) counterfactuals
We cannot adjust for confounders…
…but using IPW, can re-weight the dataset so
that treatment and covariates are unconfounded
i.e. mean covariate levels are the sample
between treated and untreated patients
So can do a simple marginal analysis
Probability-of-Treatment Model
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Weighting is based on the Probability-ofTreatment model
Treatment is longitudinal
For each child at each time, need probability of
receiving the observed treatment trajectory
Probability is conditional on past responses and
confounders
Assume independent of current response
JDM Example
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Probability of being on IVIg at baseline (logistic
regression)
Probability of transitioning onto IVIg (Cox PH)
Probability of transitioning off IVIg (Cox PH)
Suppose a child initiates IVIG at 8 months and
is still on IVIG at 12 months.
What is the probability of the observed
treatment pattern?
P(no transition before month 8)
0
No IVIg
P(transition at month 8)
P(not on IVIg at baseline)
Trratment probability
P(no transition off before month 12)
8
Initiate IVIg
12
Still on IVIG
Model Fitting
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First identified covariates univariately
Then entered those that were sig. into model
and refined (by removing those that were no
longer sig.)
IVIg at baseline: Functional status (any vs.
none) OR 11.6, 95% CI 1.94 to 69.7; abnormal
swallow/voice OR 6.28, 95% CI 0.983 to 4.02.
IVIg termination: no covariates
Assessing goodness-of-fit
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If the IPT weights are correct, in the reweighted population, treatment and
covariates are unconfounded
This property is
crucial
 testable
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…so we should test it!
Goodness-of-fit in the JDM study
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Biggest concern is that kids are doing badly
when they start IVIg
If inverse-probability weights are correct, then at
each time t, amongst patients previously IVIgnaïve, IVIg is not associated with covariates.
Will look at differences in mean covariate values
by current IVIg status amongst patients
previously IVIg-naïve
Data are longitudinal, so use a GEE analysis,
adjusting for time
Model 1 – HRs for Treatment Initiation
Covariate
W1
Skin rash
3.48 (0.99 to 12.17)
CHAQ
1.99 (1.10 to 3.66)
Prednisone
4.01 (1.35 to 11.90)
Hazard Ratios and 95% confidence intervals for initiating
treatment
UW
UW
W1
W1
W2
W2
W3
W3
W4
W4
-1
0
1
2
3
-3
DAS
UW
W1
W1
W2
W2
W3
W3
W4
W4
-0.2
-1
0
1
Missing DAS
UW
-0.4
-2
0.0
Prednisone
0.2
-0.10
0.00
0.10
Methotrexate
0.20
Model 2 -Revised Treatment initiation
Covariate
W1
W2
Skin rash
3.48 (0.99 to 12.17)
3.33 (0.92 to 12.1)
CHAQ
1.99 (1.10 to 3.66)
1.97 (1.06 to 3.64)
Prednisone
4.01 (1.35 to 11.90)
3.96 (1.33 to 11.8)
DAS
1.03 (0.82 to 1.30)
Hazard Ratios and 95% confidence intervals for initiating treatment
New goodness-of-fit
UW
UW
W1
W1
W2
W2
W3
W3
W4
W4
-1
0
1
2
3
-3
DAS
UW
W1
W1
W2
W2
W3
W3
W4
W4
-0.2
-1
0
1
Missing DAS
UW
-0.4
-2
0.0
Prednisone
0.2
-0.10
0.00
0.10
Methotrexate
0.20
Back to basics
• Some patients start IVIg because they are steroidresistant (early-starters)
• Others start because they are steroid-dependent
(late-starters)
• Repeat model-fitting process separately for early and
late starters
Covariate
W3
Abnormal ALT & t < 230
5.44 (1.29 to 22.9)
CHAQ & t < 230
4.27 (1.70 to 10.7)
Prednisone & t > 230
4.92 (1.39 to 17.4)
UW
UW
W1
W1
W2
W2
W3
W3
W4
W4
-1
0
1
2
3
-3
DAS
UW
W1
W1
W2
W2
W3
W3
W4
W4
-0.2
-1
0
1
Missing DAS
UW
-0.4
-2
0.0
Prednisone
0.2
-0.10
0.00
0.10
Methotrexate
0.20
Refined two-stage model
Covariate
W3
W4
Abnormal ALT & t < 230
5.44 (1.29 to 22.9)
5.27 (0.98 to 28.3)
CHAQ & t < 230
4.27 (1.70 to 10.7)
4.22 (1.63 to 10.9)
Prednisone & t > 230
4.92 (1.39 to 17.4)
5.22 (1.44 to 19.0)
Missing DAS & t < 230
0.994 (0.77 to 1.28)
Missing DAS & t > 230
0.939 (0.80 to 1.11)
UW
UW
W1
W1
W2
W2
W3
W3
W4
W4
-1
0
1
2
3
-3
DAS
UW
W1
W1
W2
W2
W3
W3
W4
W4
-0.2
-1
0
1
Missing DAS
UW
-0.4
-2
0.0
Prednisone
0.2
-0.10
0.00
0.10
Methotrexate
0.20
Efficacy Results
Weighting Scheme
Hazard Ratio (95% CI)
Unweighted
0.646 (0.342, 1.22)
W1
0.825 (0.394, 1.73)
W2
0.851 (0.402, 1.80)
W3
0.703 (0.340, 1.44)
W4
0.756 (0.378, 1.51)
Other concerns with MSMs
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Format of treatment effect (e.g. constant over
time, PH etc.)
Unmeasured counfounders
Lack of efficiency
Experimental Treatment Assignment
Efficiency
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IPW reduces bias but also reduces efficiency
The further the weights are from 1, the worse
the efficiency
Can stabilise the weights:
Estimating equations will still be zero-mean if we
multiply ijj by a factor depending on j and treatment
 In JDM study, we used
ijj=RijP(Rx history)/P(Rx history|confounders)
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Efficiency – other techniques
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Doubly robust methods (Bang & Robins)
Could have used a more information-rich
outcome
Did a secondary analysis using DAS as the
outcome – got far more precise (and more
positive) results
Experimental Treatment Assignment
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In order for MSMs to work, there must be some
experimentality in the way treatment is assigned
Intuitively, if we can predict perfectly who will get
what treatment, then we have complete
confounding
Mathematically, if pij is 0 then we’re in trouble!
Actually, we get into trouble if pij = 0 or 1
Testing the ETA – simple checks
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At each time j, review the distribution of
covariates amongst those who are on treatment
vs. those who are not.
Review the distribution of the weights
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check p bounded away from 0/1
In the JDM example, also check distn of
transition probabilities
Testing the ETA – more advanced
methods
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Bootstrapping
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Wang Y, Petersen ML, Bangsberg D, van der Laan
MJ. Diagnosing bias in the inverse probability of
treatment weighted estimator resulting from violation
of experimental treatment assignment. UC Berkeley
Division of Biostatistics working paper series, 2006.
Implementing MSMs
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For time-to-event outcome, can do weighted PH
regression in R
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For continuous (or binary) outcome, use
weighted GEE
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Used the svycoxph function from the survey package
Used proc genmod in SAS with scgwt
Weighted GEEs are not straightforward in R
STATA could probably handle either type of
outcome
MSMs - potentials
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Often good observational databases exist
Should do what we can with them before using
large amounts of money to do trials
Can deal with a time-varying treatment
Conceptually fairly straightforward
Do not have to model correlation structure in
responses
MSMs - limitations
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There may always be unmeasured confounders
Relies heavily on probability-of-treatment model
being correct
Experimental ETA violations can often occur
(particularly with small sample sizes)
Somewhat inefficient
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Doubly robust methods may help
Not a replacement for an RCT
Key points
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MSMs can help to establish causal associations
from observational data
Make some strong assumptions
Need goodness-of-fit for measured confounders
Will never find the right model
Aim to find good models
References
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Robins JM, Hernan MA, Brumback B. Marginal structural models
and causal inference in epidemiology. Epidemiology 2000; 11: 550560.
Bang H, Robins JM (2005). Doubly Robust Estimation in Missing
Data and Causal Inference Models. Biometrics 61 (4), 962–973.
Pullenayegum EM, Lam C, Manlhiot C, Feldman BM. Fitting
Marginal Structural Models: Estimating covariate-treatment
associations in the re-weighted dataset can guide model fitting.
Journal of Clinical Epidemiology.
Wang Y, Petersen ML, Bangsberg D, van der Laan MJ. Diagnosing
bias in the inverse probability of treatment weighted estimator
resulting from violation of experimental treatment assignment. UC
Berkeley Division of Biostatistics working paper series, 2006.