Theory

Hernán MA, Robins JM (2018). Causal Inference. Boca Raton: Chapman & Hall/CRC, forthcoming.

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

Average Causal Effects

Causation vs. Association

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 follow-up
  - 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=1|A=1] and Pr[ Y^a=1|A=0]

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

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
   

Standardization

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 well-defined 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 ill-defined 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

IP Weighting and Marginal Structural Models

https://github.com/benhhu/R-Causal-Inference

• National Health and Nutrition Examination Survey Data I Epidemiologic Follow-up Study
   
• To estimate the average causal effect of smoking cessation A on weight gain Y
   
• 1,566 cigarette smokers aged 25-74 years
  - with a baseline visit and a follow-up visit 10 years later
  - A=1 if reported having quit smoking before the follow-up visit, and A=0 otherwise
  - weight gain Y was measured as the body weight at the follow-up 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 pseudo-population in which the arrow from the confounders L to the treatment A is removed
   
• The pseudo-population 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 pseudo-population created by the estimated IP weights

Estimating IP weights via modeling

Stabilized IP Weights

• The goal of IP weighting is to create a pseudo-population 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 pseudo-population 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 pseudo-population:

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(A|L) 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 non-missing 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 G-Formula

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

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

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