fitting marginal structural models
DESCRIPTION
Fitting Marginal Structural Models. Eleanor M Pullenayegum Asst Professor Dept of Clin. Epi & Biostatistics [email protected]. Outline. Causality and observational data Inverse-Probability weighting and MSMs Fitting an MSM Goodness-of-fit Assumptions/ Interpretation. - PowerPoint PPT PresentationTRANSCRIPT
Fitting Marginal Structural Fitting Marginal Structural ModelsModels
Eleanor M PullenayegumAsst Professor
Dept of Clin. Epi & [email protected]
OutlineOutline
Causality and observational data Inverse-Probability weighting and MSMs Fitting an MSM Goodness-of-fit Assumptions/ Interpretation
Causality in Medical ResearchCausality in Medical Research
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 Randomized trials are not always possiblepossible
Sometimes, they are unethicalcannot do a randomized trial on the effects of
second-hand smoke on lung canceror a randomized trial of the effects of living
near power stations Sometimes, they are not feasible
Study of a rare disease (funding is an issue)
Observational StudiesObservational Studies
Observe rather than experiment (or interfere!) Recruit some people who are exposed to second-
hand smoke and some who are not Study communities living close to power lines vs.
those who don’t Confounding is a major concern
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 ConfoundingHandling Confounding
Match exposed and unexposed on key confounders 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
Adjust for confounders 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 PathwaysCausal Pathways
There are some things we cannot adjust forWhen studying the effect of a lipid-lowering
drug on heart disease, we can’t adjust for LDL-cholesterol level
Causal PathwaysCausal 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 ExampleMotivating Example
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 DatasetJDM Dataset
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 studyCausal Pathway for JDM study
DAS confounds IVIg and outcome DAS is on the causal pathway
DASt
IVIgt IVIgt+1
DASt+1
Time-to-Quiescence
…
…
A Thought ExperimentA Thought Experiment
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
CounterfactualsCounterfactuals
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 WeightingInverse-Probability Weighting
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 WeightingInverse-Probability Weighting
Suppose we want to fit the marginal model
Usually, we solve the GEE equation
If we use just the observed data, we solve
LHS does not have mean 0
n1
i i i ii 1
x V (Y x ) 0
ij i ijE(Y | x ) x
n1
i i i i i ijj iji 1
x V (Y x ) 0; R
Inverse-Probability WeightingInverse-Probability Weighting
If we replace by with ij, the conditional probability of observing Yij, then
What to condition on?Must condition on Yij
If MAR, then conditionally independent given previous Y
ijj ij ijR
ij
ijj ij ij
ij ij ij i1 ij ij
1ij ij i1 ij ij
ij ij
ij ij ij i1
E( (Y x ) | x)
E( E(R | x,Y ,...,Y ) (Y x ) | x)
E( E(R | x,Y ,...,Y ) (Y x ) | x)
E(Y x | x)
0 because P(R 1| x,Y ,...,Y )
Marginal Structural ModelsMarginal Structural Models
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 ModelProbability-of-Treatment Model
Weighting is based on the Probability-of-Treatment 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 ExampleJDM Example
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?
Trratment probabilityTrratment probability
No IVIg
0 8
Initiate IVIg
12
Still on IVIG
P(n
ot o
n IV
Ig a
t ba
selin
e)
P(no transition before month 8)
P(t
rans
ition
at
mon
th 8
)
P(no transition off before month 12)
Model FittingModel Fitting
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-fitAssessing goodness-of-fit
If the IPT weights are correct, in the re-weighted population, treatment and covariates are unconfounded
This property iscrucial testable
…so we should test it!
Goodness-of-fit in the JDM studyGoodness-of-fit in the JDM study
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 IVIg-naïve, 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 InitiationModel 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
-1 0 1 2 3
DAS
W4
W3
W2
W1
UW
-3 -2 -1 0 1
Missing DAS
W4
W3
W2
W1
UW
-0.4 -0.2 0.0 0.2
Prednisone
W4
W3
W2
W1
UW
-0.10 0.00 0.10 0.20
Methotrexate
W4
W3
W2
W1
UW
Model 2 -Revised Treatment initiationModel 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
-1 0 1 2 3
DAS
W4
W3
W2
W1
UW
-3 -2 -1 0 1
Missing DAS
W4
W3
W2
W1
UW
-0.4 -0.2 0.0 0.2
Prednisone
W4
W3
W2
W1
UW
-0.10 0.00 0.10 0.20
Methotrexate
W4
W3
W2
W1
UW
New goodness-of-fitNew goodness-of-fit
Back to basicsBack to basics
•Some patients start IVIg because they are steroid-resistant (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)
-1 0 1 2 3
DAS
W4
W3
W2
W1
UW
-3 -2 -1 0 1
Missing DAS
W4
W3
W2
W1
UW
-0.4 -0.2 0.0 0.2
Prednisone
W4
W3
W2
W1
UW
-0.10 0.00 0.10 0.20
Methotrexate
W4
W3
W2
W1
UW
Refined two-stage modelRefined 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)
-1 0 1 2 3
DAS
W4
W3
W2
W1
UW
-3 -2 -1 0 1
Missing DAS
W4
W3
W2
W1
UW
-0.4 -0.2 0.0 0.2
Prednisone
W4
W3
W2
W1
UW
-0.10 0.00 0.10 0.20
Methotrexate
W4
W3
W2
W1
UW
Efficacy ResultsEfficacy 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 MSMsOther concerns with MSMs
Format of treatment effect (e.g. constant over time, PH etc.)
Unmeasured counfounders Lack of efficiency Experimental Treatment Assignment
EfficiencyEfficiency
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)
Efficiency – other techniquesEfficiency – other techniques
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 AssignmentExperimental Treatment Assignment
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 ij is 0 then we’re in trouble! Actually, we get into trouble if ij = 0 or 1
Testing the ETA – simple checksTesting the ETA – simple checks
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 check bounded away from 0/1
In the JDM example, also check distn of transition probabilities
Testing the ETA – more advanced Testing the ETA – more advanced methodsmethods
Bootstrapping 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 MSMsImplementing MSMs
For time-to-event outcome, can do weighted PH regression in R Used the svycoxph function from the survey package
For continuous (or binary) outcome, use weighted GEE Used proc genmod in SAS with scgwt Weighted GEEs are not straightforward in R
STATA could probably handle either type of outcome
MSMs - potentialsMSMs - potentials
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 - limitationsMSMs - limitations
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
Doubly robust methods may help Not a replacement for an RCT
Key pointsKey points
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
ReferencesReferences
Robins JM, Hernan MA, Brumback B. Marginal structural models and causal inference in epidemiology. Epidemiology 2000; 11: 550-560.
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.