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An introduction to Marginal Structural Models and Inverse Probability Treatment Weights
Time Dependent Confounding
Marginal Structural Models and
Inverse Probability of Treatment Weighting
Patrick F. McArdle
Time Varying Confounding
When a variable changes over time and can be thought of as both a confounder and an intermediate.
It causes treatment and is effected by treatment.
E.g. high blood pressure may make one more likely to receive anti-hypertension medication; and blood pressure is effected by medication.
Time Modified Confounding
When the relationship between the confounder and the exposure or outcome changes over time.
E.g. compliance to treatment recommendation changes over time due to press coverage.
HIV Mortality Example of Time Dependant Confounding
In a study of the effect of zidovudine (AZT) treatment on mortality among human immunodeficiency virus (HIV)-infected subjects, the time-dependent covariate CD4 lymphocyte count is both an independent predictor of survival and initiation of therapy with AZT and is itself influenced by prior AZT treatment.
Can you create a DAG that describes that scenario?
HIV Mortality Example DAG
Estimate the effect on AZT treatment on survival?
Obesity Related Mortality Example of Time Dependant Confounding
In a study of the effect of obesity on mortality, the development of clinical cardiac or respiratory disease is an independent predictor of both mortality and subsequent weight loss and is influenced by prior weight gain.
Obesity Mortality Example DAG
Estimate the effect of obesity on mortality?
Ak = Dose of treatment on the kth
Y = Outcome of interest
We want to estimate the effect of treatment Ak on the outcome Y.
Lk = All measured risk factors for Y measured on day k
Uk = All unmeasured risk factors for Y measured on day k
Confounding in Point Treatment Studies
Confounding in Time Varying Studies
Treatment is confounded and unidentifiable
Treatment is confounded and identifiable
Treatment is unconfounded
Causal Estimates: A Review
Causal Effect: Prob(YA=1) – Prob(YA=0).
True risk difference is not identifiable.
Prob(Y|L, A=1) - Prob(Y|L, A=0) = True risk difference.
Prob(Y|A=1) - Prob(Y|A=0) = True risk difference.
Marginal Structural Models
Model the probabilities of counterfactual variables.
Pr[Ya = 1] = ß0 + ß1a
Log Pr[Ya = 1] = ß0 + ß1a
Logit Pr[Ya = 1] = ß0 + ß1a
Causal Risk Difference = ß1
Causal Risk Ratio = eß1
Causal Odds Ratio = eß1
These are models of the marginal distribution of the counterfactual variable and not observed associations.
Model the observed associations.
Pr[Y = 1|A=a] = ß'0 + ß'1a
Log Pr[Y = 1|A=a] = ß'0 + ß'1a
Logit Pr[Y = 1|A=a] = ß'0 + ß'1a
Crude Risk Difference = ß'1
Crude Risk Ratio = eß'1
Crude Odds Ratio = eß'1
The parameters from the crude regression model will match the parameters from the MSM only when treatment is unconfounded.
Controlling for Confounding
Option 1: Conditional analysis on covariates.
Works well for point treatment studies but can be problematic with time varying confounding since certain covariate values will be affected by previous treatment assignment.
Also suffers from non-collapsibility of the odds ratio
Option 2: Create a pseudo-population by weighting each observation
Works well for point treatment studies and time varying confounding.
This is called Inverse Probability of Treatment Weight (IPTW) approach since the weights are the inverse of the probability of being treated.
Inverse Probability-of-Treatment Weighting
Assign each observation a weight equal to the inverse conditional probability of receiving its treatment.
Wi = 1/pr[A = ai|L = li]
Where L are the measured confounders.
Practically, weights are obtained by regressing treatment on the set of confounders and taking the inverse of the expected value.
Sounds like propensity scoring?
The propensity score is the probability of being treated, and included as a covariate in the regression to “summarize” all measured confounders.
Propensity score = pr[A = 1|L = li]
The IPTW is the inverse of the probability of receiving whatever treatment was received. The weights then are used to create a pseudo-population and a crude regression is run.
IPTW = 1/pr[A = ai|L = li]
Each unit has a non-zero (i.e. positive) probability of receiving the treatment.
Both propensity scoring and IPTW require the positivity assumption.
Can you estimate risk factors without regression?
Assume there is no unmeasured confounding, what are the risks pr(YA=1 = 1) and pr(YA=0 = 1)?
Pr[Y=1|A=1,L=1] * Pr[L=1] + Pr[Y=1|A=1,L=0] * Pr[L=0]
(108/360) * (400/500) + (20/50) * (100/500)
pr(Ya=1 = 1) = 0.32
pr(Ya=0 = 1) = 0.64
What are the true causal parameters (RD, RR OR)?
What are the crude observed parameters?
True Causal Parameters
Crude Observed Parameters
RD = -0.32
RR = 0.50
OR = 0.26
RD = (128/410) – (64/90) = -0.40
RR = (128/410) / (64/90) = 0.44
OR = (128/410) – (64/90) = 0.18
What are the treatment weights?
Pr(A0=1|L0=1) = (360)/(360+40) = 0.9
Pr(A0=0|L0=1) = (40)/(360+40) = 0.1
Pr(A0=1|L0=0) = (50)/(50+50) = 0.5
Pr(A0=0|L0=0) = (50)/(50+50) = 0.5
Create a Pseudo-population
Estimate causal parameters using the pseudo population.
Estimate causal parameters in pseudo-population
Crude Parameters in pseudo-population
RD = (160/500) – (320/500) = -0.32
RR = (160/500) / (320/500) = 0.50
OR = (160/500) / (320/500) = 0.26
Crude parameters in pseudo-population equal true causal parameters.
Why MSM vs standard regression?
To summarize, standard regression methods adjust for covariates by including them in the model as regressors. These standard methods may fail to adjust appropriately for confounding due to measured confounders Lk when treatment is time varying, because
(1) Lk may be a confounder for later treatment and thus must be adjusted for, but
(2) may also be affected by earlier treatment and thus should not be adjusted for by standard methods.
A solution to this conundrum is to adjust for the time dependent covariates Lk by using them to calculate the weights swi rather than by adding the Lk to the regression model as regressors.
Robins et al (2000)
We will talk mostly about time-varying confounding (as introduced by Robins 2000) and not so much about time-modified confounding (per Platt 2009); though Marginal Structural Models can be used to estimate both.
Practically, weights are obtained by regressing treatment on the set of confounders and taking the inverse of the e