Matching and weighting

Causal inference

Propensity score

Author

Chi Zhang

Published

September 3, 2023

Propensity score

(Rosenbaum and Rubin, 1983)

  • in observational studies, conditioning on propensity scores can lead to unbiased estimates of the exposure effect
  • given that there are no unmeasured confounders
  • every subject has a non-zero probability of receiving exposure

Fit a logistic regression:

  • binary outcome: exposure (1,0)
  • covariates: all but exposure

Predict the values (probability), they are the propensity scores.

PS can also be estimated using other methods that produce probabilities, not just logistic regression: random forest, lasso logistic regression etc.

Matching

This overview summary is based on the review paper Stuart 2010: Matching methods for causal inference: a review and a look forward

Goal of matching: choosing well-matched samples of the original groups to reduce confounding - acquire treatment and control groups with similar covariate distributions.

Alternatives to matching: adjust for covariates in a regression model, instrumental variables, structural equation modeling etc.

Benefits of matching:

  • complementary to regression adjustment, can and should be used together
  • have straightforward diagnostics, hence performance can be assessed

Two settings to use matching:

  • when outcome values are NOT available, and matching is used to select subjects to follow up
  • outcome values are available, matching is to reduce bias in treatment effect estimation

History of matching methods

  • they’ve been used in 1940s, but a theoretical basis was not developed until 1970s (Rubin et al)
  • with multiple covariates it is difficult to find exact matches on even small number of covariates (Chapin 1947). The introduction of propensity score (1983, Rosenbaum and Rubin) helped.

Comments

  • The outcomes are usually not used even when they are available
  • PS matching is ONE of the many matching techniques that uses PS as the difference
  • Matching would reduce number of observations, hence loss of power
  • matching gives estimates for ATT

Steps to implement matching methods

  1. Define closeness: distance measure used to determine whether an individual is a good match for another
  2. implement a matching method
  3. assess the quality of a matching method. Might require iteration of step 1 and 2
  4. analyse the outcome given the matched data

Vignette: Estimating effects after matching by Noah Greifer

Exact matching:

  • perfect covariate balance; \(F(X_i|T_i = 1) = F(X_i|T_i=0)\)
  • infeasible when covariate is continuous, and when there are many covariates.

Probability of receiving treatment, \(\pi(X_i) = P(T_i = 1 | X_i)\)

Matching based on distance measures

  • Mahalanobis distance
  • Estimated propensity score, \(D(X_i, X_j) = |P(T_i = 1|X_i) - P(T_j=1 | X_j)|\)

Check covariate balance

  • ideally compare joint distribution of all covariates
  • practically check lower-dimensional summaries (standardized mean difference, variance ratio, empirical CDF difference)

Balance test

Weighting

Weighting can be viewed as a generalization of matching. Weighting is commonly used for estimating ATE.

Weights used in weighting:

  • ATE (population) \(w_{ATE} = \frac{Z_i}{p_i} + \frac{1-Z_i}{1-p_i}\)

  • ATT \(w_{ATT} = \frac{p_i Z_i}{p_i} + \frac{p_i(1-Z_i)}{1-p_i}\)

  • ATC \(w_{ATC} = \frac{(1 - p_i) Z_i}{p_i} + \frac{(1 - p_i)(1-Z_i)}{1-p_i}\)