Notes from book: What if (Part 2)

IP weighting, standardization (g-computation)

Validatidy of causal inference requires exchangeability, positivity, consistency, no measurement error and no model misspecification.

Code: What If. R code

Causal question: estimate the average causal effect of smoking cessation (treatment, A) on weight gain (outcome, Y)

Dataset: NHEFS. 1566 smokers, age 25-74. Baseline visit and follow-up 10 years later. A=1 means they quit smoking, 0 otherwise. Outcome weight gain measured in kg.

Adjust for 9 variables measured at baseline:

• sex (0M 1F)
• age
• race (0 white 1 other)
• education (5 categories)
• intensity of smoking (n cigarettes per day)
• duration of smoking (years)
• physical activity in daily life (3 categories)
• recreational exercise (3 categories)
• weight (kg)

R code for the two methods: link

IP weighting

IP weighting creates pseudo-population to remove covariates L to the treatment A. Properties of the pseudo-population:

• A and L are statistically independent
• \(E_{ps}[Y|A=a]\) equals the standardized mean in the actual population, \(\sum_{l}E[Y|A=a,L=l]P[L=l]\).

Individual-specific IP weights for treatment A is \(W = 1/f(A|L)\). For the quitters (A=1), \(f(A|L) = P[A=1|L]\). This is also the propensity score.

The weights can be estimated non-parametrically when the problem is simple: count the number of people treated (A=1) in each stratum (L=1, L=0) and then divide by the number in the stratum. When the problem has more variables (confounders), fit a logistic regression to estimate the probability.

(more details to be filled in)

g-formula, standardization

IP weighting and standardization are estimators of g-formula (1986). g-formula is a synonym of g-computation. (g-estimation is a different method)

flowchart LR
  L(L) --> A[A]
  A[A] --> Y{Y}
  L(L) --> Y{Y}

Procedures:

1. fit a regression with Y as outcome, A as treatment, L as control. It is possible to make polynomials and/or interactions.
2. create a dataset identical to the original data, but \(A = 1\) in every row
3. create a dataset identical to the original data, but \(A = 0\) in every row
4. use model from step 1 to compute adjusted predictions in the two counterfactual datasets.

The quantity of interest is the difference between the predictions.

(Note that in Hernan & Robins, three blocks of data are binded into one. The counterfactual datasets (2nd and 3rd) have NA in the outcome. So even though in the regression model it’s using the large dataset with duplicates, it is effectively only using the original data to estimate the parameters.)

Robust methods

Doubly robust methods combine models for treatment (ipw) and for outcome (standardization) in the same estimator.

• Augmented IP weighted estimator
• doubly robust plug-in estimator