Theory
IP Weighting and Marginal Structural Models
Standardization and Parametric GFormula
Theory
Individual Causal Effects
 Zeus is a patient waiting for a heart transplant
 on Jan 1, he receives a new heart
 five days later, he dies  Another patient, Hera
 she also received a heart transplant on Jan 1
 five days later she was alive
Treatment: A (1: treated, 0: untreated)
Outcome: Y (1: death, 0: survival)
Potential outcomes or counterfactual outcomes:
Y ^{a=0} : the outcome variable that would have been observed under the treatment value a=0.
Y ^{a=1} : the outcome variable that would have been observed under the treatment value a=1.
Definition of individual causal effects: the treatment A has a causal effect on an individual's outcome Y if Y ^{a=1} ≠ Y ^{a=0} for the individual.
Y ^{a=1} =1
Y^{a=1}=0
In general, individual causal effects cannot be identified since only one of potential outcomes is observed for each individual.
Average Causal Effects
Definition of average causal effects: an average causal effect of treatment A on outcome Y is present if E[ Y ^{a=1} ] ≠ E[ Y ^{a=0} ]
Pr[ Y ^{a=1} =1]=10/20=0.5
Absence of an average causal effect does not imply absence of individual effects.
Pr[ Y ^{a=0} =1]=10/20=0.5
Average causal effects can sometimes be identified from data even if individual causal effects cannot.
Causation vs. Association
Pr[ Y ^{a} =1]
Pr[ Y=1A=a]
Marginal probability
Conditional probability
Causation vs. Association
 Causal inference requires data like the hypothetical first table, but all we can ever expect to have is real world data like those in the second table.
 Under which conditions real world data can be used for causal inference?
Randomization
 Randomization ensures that those missing values occurred by chance
 Ideal randomized experiment
 no loss to followup
 full adherence to treatment
 a single version of treatment
 double blind assignment  Exchangeability:
 which particular group received the treatment is irrelevant for the value of Pr[ Y^{a}=1A=1] and Pr[ Y^{a}=1A=0]
Randomization is so highly valued because it is expected to produce exchangeability.
Randomization
 Does exchangeability hold in our heart transplant study?

 Since the counterfactual data are not available in the real world, we are generally unable to determine whether exchangeability holds
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0
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Can we conclude that our study is not a randomized experiment?
 random variability
 conditional randomization
Conditional Randomization
 Prognosis factor L (1: critical condition, 0 otherwise)
 Design 1
 randomly assign treatments to 65% of the individuals
 Design 2
 randomly assign treatments to 75% of the individuals with critical condition
 randomly assign treatments to 50% of those in noncritical condition
Marginal randomization
Conditional randomization
Exchangeability holds
Exchangeability does not hold, but conditional exchangeability holds
How to compute the causal effect?
 Stratumspecific causal effects
 Average causal effects
Standardization
Inverse Probability Weighting
Since
Similarly,
Causal Inference for Observational Study
What randomized experiment are you trying to emulate?
Observational Study
 An observational study can be conceptualized as a conditionally randomized experiment under the following 3 conditions (also called identifiability conditions):
 exchangeability: the conditional probability of receiving every value of treatment, though not decided by the investigators, depends only on the measured covariates
 positivity: the conditional probability of receiving every value of treatment is greater than zero
 consistency: the values of treatment under comparison correspond to welldefined interventions that, in turn, correspond to the versions of treatment in the data
 When any of these conditions does not hold, another possible approach can be used:
 instrumental variable: a predictor of treatment, which was hoped to be randomly assigned conditional on the measured covariates
Exchangeability
 The crucial question for the observational study is whether L is the only outcome predictor that is unequally distributed between the treated and the untreated
 In the absence of randomization, there is no guarantee that conditional exchangeability holds
 When we analyze an observational study under the assumption of conditional exchangeability, we must hope that the assumption is at least approximately true
Positivity
 With all the subjects receiving the same treatment level, computing the average causal effect would be impossible
 In observational studies, positivity is not guaranteed, but it can sometimes be empirically verified
Consistency
 For observational studies, we assume that the treatment of interest either does not have multiple versions, or has multiple versions with identical effects
 An example:
 an obese individual (R=1) who died (Y=1)
 multiple versions A(r=0) of the treatment R=0 (e.g. more exercise, less food intake, more cigarette smoking, genetic modification, bariatric surgery, etc.)
 the counterfactual outcome Y is not well defined
 a nonobese individual might have died if he had been nonobese through cigarette smoking (lung cancer), and might have survived if through a better diet  In settings with illdefined interventions, the consistency condition does not hold because the counterfactual outcome Y may not equal the observed outcome Y in some people with R=r
r=0
r
IP Weighting and Marginal Structural Models

National Health and Nutrition Examination Survey Data I Epidemiologic Followup Study
 To estimate the average causal effect of smoking cessation A on weight gain Y
 1,566 cigarette smokers aged 2574 years
 with a baseline visit and a followup visit 10 years later
 A=1 if reported having quit smoking before the followup visit, and A=0 otherwise
 weight gain Y was measured as the body weight at the followup visit minus the body weight at the baseline visit
Data
Assume these 9 baseline variables are sufficient to adjust for confounding
Estimating IP weights via modeling
 IP weighting creates a pseudopopulation in which the arrow from the confounders L to the treatment A is removed
 The pseudopopulation is created by weighting each individual by the inverse of the conditional probability of receiving the treatment level that one indeed received

 Fit a logistic regression model for the probability of quitting smoking with all 9 confounders included as covariates
 Compute the difference in the pseudopopulation created by the estimated IP weights
= {
quitters
nonquitters
Estimating IP weights via modeling
Fit the linear model by weighted least squares, with individuals weighted by their estimated IP weights
To obtain a 95% confidence interval:
 to use statistical theory to derive the corresponding variance estimator (not available in most software)
 approximate the variance by bootstrapping (timeconsuming)
 use the robust variance estimator (widely available, quick, but generate conservative CI)
Stabilized IP Weights
 The goal of IP weighting is to create a pseudopopulation in which there is no association between the covariates L and treatment A
 all individuals have the same probability of receiving A=1 and the same probability of receiving A=0  The pseudopopulation was twice as large as the study population because all individuals were included both under treatment and under no treatment
 Equivalently, the expected mean of the weights was 2
 There are other ways to create a pseudopopulation:
f(A) equals to Pr[A=1] for treated, and Pr[A=0] for untreated
Marginal Structural Models
 The outcome variable of this model is counterfactual
 Models for mean counterfactual outcomes are referred to as structural mean models
 when it does not include any covariates, we refer to it as an unconditional or marginal structural mean model  The parameters for treatment in structural models correspond to average causal effects
 The above model is saturated because smoking cessation A is a dichotomous treatment
 the model has 2 unknowns on both sides of the equation: and vs and
Marginal Structural Models
 Treatments are often polytomous or continuous
 Change in smoking intensity
 number of cigarettes smoked per day in 1982 minus number of cigarettes smoked per day at baseline
 Nonsaturated model
 For a continuous treatment A, f(AL) is a probability density function (PDF)
 PDFs are generally hard to estimate correctly
 Using IP weighting for continuous treatments will often be dangerous
Censoring and Missing Data
 We restricted our analyses on 1566 individuals with nonmissing data
 may introduce selection bias 
C: indicator for censoring
 C=1 if censored, C=0 if uncensored
 If no selection bias (no arrow from either L or A into C):
Standardization and The Parametric GFormula
Standardization as an alternative to IP weighting
 The standardized mean outcome in the uncensored treated is a consistent estimator of the mean outcome if everyone had been treated and had remained uncensored
 The standardized mean outcome in the uncensored untreated is a consistent estimator of the mean outcome if everyone had been untreated and had remained uncensored
Under exchangeability and positivity conditional on the variables in L,
We first need to compute the mean outcomes in the uncensored treated in each stratum l of the confounders L
Estimating the Mean Outcome via Modeling
 We fit a linear regression model for the mean weight gain with treatment A and all 9 confounders in L included as covariates
 We can then obtain an estimate
for each combination of values of A and L.
Standardizing the Mean Outcome to the Confounder Distribution
 When all variables in L are discrete, each mean receives a weight equal to the proportion of subjects with values L=l
 The proportions Pr[L=l] could be calculated nonparametrically from the data, but this method becomes tedious for high dimensional data with many confounders
 There is another faster, but mathematically equivalent, method to standardize means
Standardizing the Mean Outcome to the Confounder Distribution
 Create a dataset in which the original data is copied 3 times
 In the second block, set the value of A to 0, and delete the data on the outcome
 In the third block, set the value of A to 1, and delete the data on the outcome
 Fit a regression model for the mean outcome given A and L
 Use the parameter estimates from the model to predict the outcome values for all rows in the second and third blocks
 Compute the average of all predicted values in the second block, which is the standardized mean outcome in the untreated
 The standardized mean outcome in the treated is the average of all predicted values in the third block
Use bootstrap to obtain a 95% confidence interval
IP Weighting vs. Standardization
 IP weighting relies on the correct specification of the treatment model
 Standardization relies on the correct specification of the outcome model
 We should not choose between IP weighting and standardization when both methods can be used to answer a causal question
 Use both methods whenever possible  We can also use doubly robust methods that combine models for treatment and for outcome in the same approach
Summary
 Exchangeability
 Positivity
 Consistency
 No measurement error
 No model misspecification
The validity of causal inferences based on observational data requires the following conditions:
Causal Inference  PHC6016
By Hui Hu
Causal Inference  PHC6016
Slides for the Social Epidemiology guest lecture, Fall 2017
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