Fitting Marginal Structural Models Eleanor M Pullenayegum Asst Professor Dept of Clin. Epi & Biostatistics [email protected].

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.