Inverse Probability Weighting

Inverse probability weighting (IPW) removes confounding by reweighting. Each unit is weighted by the inverse of the probability of the treatment it actually received, which up-weights the under-represented combinations of treatment and confounders and down-weights the over-represented ones — building a pseudo-population in which treatment is no longer associated with the measured confounders. It is the same idea as reweighting a biased survey to represent the target population, applied to a treatment instead of a sampling design, and it reuses the propensity score as its engine.

Figure 1. Unstabilized weights (1/e for the treated, 1/(1−e) for controls) have a heavy right tail — units with extreme propensity get enormous weight — while stabilized weights cluster near 1, preserving the same balance with far less variance.

The pseudo-population and the weights#

With propensity $e(L) = \Pr(A=1\mid L)$ , assign each unit the inverse-probability weight

$w_i = \frac{A_i}{e(L_i)} + \frac{1 - A_i}{1 - e(L_i)}.$

In the weighted sample, treatment and confounders are independent, so a simple weighted difference in outcomes estimates the causal effect:

$\widehat{\text{ATE}} = \frac{\sum_i w_i A_i Y_i}{\sum_i w_i A_i} - \frac{\sum_i w_i (1-A_i) Y_i}{\sum_i w_i (1-A_i)},$

the Hájek (self-normalized) estimator, generally preferred to the plain Horvitz–Thompson version because it is less sensitive to a few large weights. The same assumptions as any propensity method apply — no unmeasured confounding and positivity — and positivity bites harder here, because a propensity near $0$ or $1$ produces a colossal weight.

Stabilized weights#

Raw weights can be wildly variable, and a handful of huge ones wreck the variance. Stabilized weights replace the numerator $1$ with the marginal probability of the treatment received,

$sw_i = A_i\,\frac{\Pr(A=1)}{e(L_i)} + (1 - A_i)\,\frac{\Pr(A=0)}{1 - e(L_i)},$

which leaves the balancing property intact but pulls the weights toward $1$ , sharply cutting variance (Figure 1). When extreme weights remain, truncate them at, say, the 1st and 99th percentiles — trading a little bias for a lot of stability.

Marginal structural models#

Weighting generalizes past a single binary contrast. Fit an outcome model — a marginal structural model (MSM) — in the weighted pseudo-population, e.g. $\mathbb{E}[Y^a] = \beta_0 + \beta_1 a$ , and its coefficients are marginal (population-level) causal effects, not the conditional-on- $L$ effects a plain outcome regression would give. The MSM can carry continuous exposures, interactions, and — crucially — treatment histories.

Time-varying treatments: the reason IPW exists#

The setting IPW was really built for is a treatment given repeatedly over time, with a time-varying confounder that is itself affected by earlier treatment (Robins, Hernán & Brumback 2000). The canonical example: HIV treatment $A$ decided by CD4 count $L$ , while treatment also changes future CD4 (Figure 2).

Figure 2. The time-varying covariate L₁ is simultaneously a confounder of the later treatment A₁ and a mediator of the earlier treatment A₀. Adjusting for it to fix the confounding blocks part of A₀’s effect (and opens collider bias); not adjusting leaves A₁ confounded.

Here standard regression is stuck: you must adjust for $L_1$ to de-confound $A_1$ , but adjusting for $L_1$ blocks the effect of $A_0$ that runs through $L_1$ and conditions on a collider — biasing the estimate either way. Weighting sidesteps the trap: weight by the inverse probability of the whole treatment history and fit an MSM, and the time-varying confounders never enter the outcome model at all. G-estimation is the other principled solution to the same problem.

A worked example#

The point-treatment version, on the running confounded example (true effect $-2$ ).

Python

import numpy as np
import statsmodels.api as sm

rng = np.random.default_rng(2)
n = 4000
L = rng.normal(0, 1, n)
A = rng.binomial(1, 1 / (1 + np.exp(-0.8 * L)))
Y = -2.0 * A + 1.5 * L + rng.normal(0, 1, n)     # true ATE = -2

ehat = sm.Logit(A, sm.add_constant(L)).fit(disp=0).predict()
w = A / ehat + (1 - A) / (1 - ehat)              # inverse-probability weights
pA = A.mean()
sw = A * pA / ehat + (1 - A) * (1 - pA) / (1 - ehat)   # stabilized

def hajek(weight):
    return (np.sum(weight * A * Y) / np.sum(weight * A)
            - np.sum(weight * (1 - A) * Y) / np.sum(weight * (1 - A)))

print(f"naive              {Y[A==1].mean() - Y[A==0].mean():+.2f}")
print(f"IPW (unstabilized) {hajek(w):+.2f}   max weight {w.max():.1f}")
print(f"IPW (stabilized)   {hajek(sw):+.2f}   max weight {sw.max():.1f}   (true -2.00)")

naive              -0.93
IPW (unstabilized) -1.98   max weight 13.9
IPW (stabilized)   -1.98   max weight 6.9   (true -2.00)

Both weightings recover the true $-2$ ; stabilizing barely moves the point estimate but shrinks the largest weight, which in a bigger or more extreme problem is the difference between a usable estimate and a hopelessly noisy one.

In code#

R#

library(WeightIt)         # weights + balance diagnostics; survey/ipw also common
W <- weightit(A ~ L, data = d, method = "glm", estimand = "ATE", stabilize = TRUE)
library(survey)
des <- svydesign(~1, weights = W$weights, data = d)
svyglm(Y ~ A, design = des)     # the marginal structural model

Julia#

Julia

using GLM, DataFrames
e = predict(glm(@formula(A ~ L), d, Binomial()))
d.w = @. d.A / e + (1 - d.A) / (1 - e)          # inverse-probability weights
lm(@formula(Y ~ A), d, wts = d.w)               # weighted outcome model (the MSM)

Why it matters#

Inverse probability weighting is the standard tool for causal questions with treatment sustained over time — antiretroviral therapy, statins, occupational exposures — where the thing that decides today’s treatment is itself moved by yesterday’s, and ordinary regression cannot give an unbiased answer. Even for a single time point it has a clean interpretation (a reweighted pseudo-experiment) and pairs naturally with an outcome model to become doubly robust. Its Achilles heel is positivity: when some units almost never receive a given treatment, their inverse weights explode, and no amount of stabilizing rescues an estimate that is really being carried by three people.