Hájek Normalization for Inverse-Probability Weights

Why Normalize Inverse-Probability Weights?

Inverse-probability weighting uses treatment probabilities to reweight observed outcomes. For the average treatment effect, treated units receive weights proportional to the inverse of their treatment probabilities, while control units receive weights proportional to the inverse of their control probabilities. The Horvitz–Thompson estimator uses these weights to form weighted totals. The Hájek estimator divides each weighted total by the corresponding sum of weights, producing a normalized weighted mean.

That division matters in a finite sample. The sum of the weights in an arm need not equal the sample size, or the population quantity it is intended to represent. Consequently, the Horvitz–Thompson and Hájek estimates can differ even when calculated from the same observations and propensity scores. Normalization can make a weighted mean more stable in some samples, but it does not guarantee lower bias, lower variance, or better causal identification.

Setup and Identification

For unit \(i\), let \(A_i\in\{0,1\}\) indicate treatment, let \(Y_i\) be the observed outcome, and let \(X_i\) denote observed pre-treatment covariates. Write \(Y_i(1)\) and \(Y_i(0)\) for the potential outcomes under treatment and control, respectively. The observed outcome satisfies \(Y_i=A_iY_i(1)+(1-A_i)Y_i(0)\). The propensity score is \(e(X_i)=P(A_i=1\mid X_i)\).

The weighting formulas require a causal identification framework. Assume consistency, so the observed outcome under the treatment actually received equals the corresponding potential outcome. Assume conditional exchangeability, \((Y(1),Y(0))\mathbin{\perp\!\!\!\perp} A\mid X\), and positivity, \(0<e(X)<1\) for covariate values in the target population. These assumptions let observed outcomes, after weighting, represent the potential-outcome means of interest. In practice, the propensity score is usually estimated; the formulas below use \(e(X_i)\) to describe the weighting logic.

For the average treatment effect (ATE), the Horvitz–Thompson estimator is

Its first weighted total estimates the population mean of \(Y(1)\); its second estimates the population mean of \(Y(0)\). The Hájek estimator instead calculates a normalized mean within each weighted arm:

Each Hájek arm estimate is a weighted average of observed outcomes in that arm, so it lies between the smallest and largest outcome with positive weight in that arm. By contrast, each Horvitz–Thompson component is a weighted sum divided by the full sample size \(n\); it is not generally a weighted average of outcomes in the observed arm.

Why the Normalization Is a Ratio

The normalized form follows from expressing each target mean as a ratio of expectations. For example, conditional exchangeability and positivity imply

To see the first equality, condition on \(X\). Since \(Y=Y(1)\) when \(A=1\), exchangeability gives

Averaging over \(X\) yields \(E[Y(1)]\). The denominator identity follows in the same way: \(E[A/e(X)\mid X]=e(X)/e(X)=1\). Therefore the treated Hájek mean estimates the ratio \(E[AY/e(X)]/E[A/e(X)]=E[Y(1)]\). Replacing these expectations by their sample averages produces the observed normalized weighted mean. The common factor \(1/n\) cancels from its numerator and denominator.

The same argument applies to controls: \(E[(1-A)Y/\{1-e(X)\}]=E[Y(0)]\) and \(E[(1-A)/\{1-e(X)\}]=1\). These population identities explain why both approaches target the same arm means under the assumptions. They do not imply that their finite-sample estimates are equal.

Normalization for the Treated-Group Effect

The average treatment effect for the treated (ATT) is \(E[Y(1)-Y(0)\mid A=1]\). The observed treated mean estimates \(E[Y(1)\mid A=1]\). To represent the treated group’s covariate distribution among controls, a control unit receives the inverse-odds weight \(e(X_i)/\{1-e(X_i)\}\). Under conditional exchangeability, weighting control outcomes this way targets \(E[Y(0)\mid A=1]\).

Let \(n_1=\sum_i A_i\), the number of treated units in the sample. A Horvitz–Thompson-style ATT estimator uses the treated sample mean and scales the weighted control total by \(n_1\). The Hájek form normalizes the control weights by their observed sum:

Here the treated mean is already normalized by the number of treated units; the key difference is the denominator for the reweighted control mean. The Horvitz–Thompson scaling uses \(n_1\), while the Hájek scaling uses the realized sum of inverse-odds weights.

Finite-Sample Behavior: What Normalization Does and Does Not Do

For an ATE arm, the Hájek estimator can be written as a ratio of a weighted outcome total to a weight total. If a sample average in the numerator converges to its expectation and the corresponding denominator converges to a positive expectation, the ratio converges to the ratio of those expectations. This is the usual ratio argument: if \(U_n\) converges to \(u\) and \(V_n\) converges to \(v\ne0\), then \(U_n/V_n\) converges to \(u/v\). Applied to the treated arm, \(u=E[Y(1)]\) and \(v=1\); the control arm follows analogously. This explains consistency under the identification assumptions and appropriate sampling conditions, not exact finite-sample equality.

In finite samples, the denominator is random. A ratio involving random quantities is generally not equal to the ratio of their expectations, so Hájek estimates are not generally exactly unbiased just because the weighted totals have useful expectation identities. Conversely, an unusually large inverse-probability weight can make a Horvitz–Thompson total highly variable. Dividing by the realized weight sum can dampen the impact of a mismatch between the weight sum and its target, but can also introduce ratio bias. Which estimate performs better depends on the data-generating process and sample.

For the normalized weighted mean, positive weights also give a useful arithmetic check: the estimate must lie between the minimum and maximum observed outcomes among units with positive weight in that arm. If it does not, there is likely a calculation or denominator error. That check does not establish that the estimate is causally valid or precise.

Common Mistakes and Practical Interpretation

• Calling the Hájek estimator unbiased by definition. Normalizing creates a ratio with a random denominator. It may have finite-sample bias; its attraction is not a universal guarantee of unbiasedness or lower error.
• Dividing by the wrong quantity. For the ATE Hájek estimator, divide each arm’s weighted outcome total by that arm’s sum of weights. For the ATT form shown here, divide the control weighted total by the control inverse-odds weight sum. Do not substitute the total sample size or the treated count for these Hájek denominators.
• Interpreting the weight sum as a sample size. A sum of weights is a normalization quantity, not the number of observed people. It can be fractional and can vary substantially across samples.
• Taking a difference as proof of a causal effect. Weighting identifies a causal target only when the relevant identification assumptions hold and the weights are constructed appropriately. Normalization itself does not make confounding disappear.