Fitting Marginal Structural Models

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Fitting Marginal Structural Models

• Eleanor M Pullenayegum
• Asst Professor
• Dept of Clin. Epi Biostatistics
• pullena_at_mcmaster.ca

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Outline

• Causality and observational data
• Inverse-Probability weighting and MSMs
• Fitting an MSM
• Goodness-of-fit
• Assumptions/ Interpretation

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Causality 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

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Randomized trials are not always possible

• 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
• Sometimes, they are not feasible
• Study of a rare disease (funding is an issue)

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Observational 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 dont
• 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?

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Handling 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

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Causal Pathways

• There are some things we cannot adjust for
• When studying the effect of a lipid-lowering drug
  on heart disease, we cant adjust for
  LDL-cholesterol level

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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.
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Motivating 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

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JDM Dataset

• 81 kids, 7 on IVIg at baseline, 23 on IVIG later
• Outcome is time to quiescence
• Quiescence happens when DAS0
• IVIg tends to be given when the child is doing
  particularly badly (high DAS)
• DAS is a counfounder

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Causal Pathway for JDM study

DASt
DASt1

IVIgt
IVIgt1
Time-to-Quiescence

• DAS confounds IVIg and outcome
• DAS is on the causal pathway

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A 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

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Counterfactuals

• 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

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Inverse-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 Rij1 if Yij observed, 0 o/w

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Inverse-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

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Inverse-Probability Weighting

• If we replace D by with
  pij, the conditional probability of observing
  Yij, then
• What to condition on?
• Must condition on Yij
• If MAR, then conditionally independent given
  previous Y

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Marginal 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

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Probability-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

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JDM 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?

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Trratment probability
P(transition at month 8)
P(no transition before month 8)
P(no transition off before month 12)
P(not on IVIg at baseline)
0
8
12
No IVIg
Initiate IVIg
Still on IVIG
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Model 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

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Assessing goodness-of-fit

• If the IPT weights are correct, in the
  re-weighted population, treatment and covariates
  are unconfounded
• This property is
• crucial
• testable
• so we should test it!

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Goodness-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

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Model 1 HRs for Treatment Initiation
Hazard Ratios and 95 confidence intervals for
initiating treatment
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Model 2 -Revised Treatment initiation
Hazard Ratios and 95 confidence intervals for
initiating treatment
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New goodness-of-fit
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Back 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

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Refined two-stage model
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Efficacy Results
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Other concerns with MSMs

• Format of treatment effect (e.g. constant over
  time, PH etc.)
• Unmeasured counfounders
• Lack of efficiency
• Experimental Treatment Assignment

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Efficiency

• 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 Dijj by a factor depending on j and
  treatment
• In JDM study, we used
• DijjRijP(Rx history)/P(Rx historyconfounders)

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Efficiency 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

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Experimental 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 pij is 0 then were in
  trouble!
• Actually, we get into trouble if pij 0 or 1

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Testing 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 p bounded away from 0/1
• In the JDM example, also check distn of
  transition probabilities

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Testing the ETA more advanced methods

• 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.

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Implementing 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

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MSMs - 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

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MSMs - 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

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Key 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

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References

• 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), 962973.
• 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.