Constructing Weights for the Average Treatment Effect

Why ATE Weights Depend on Treatment Status

Inverse probability weighting assigns each unit a weight based on how likely that unit was to receive the treatment it actually received. For the average treatment effect (ATE), the weighting rule must account for both treatment groups: treated units receive the inverse of their probability of being treated, while untreated units receive the inverse of their probability of being untreated. The two denominators are different, so assigning the same formula to every unit is a consequential error.

Let \(D_i\) be unit \(i\)'s binary treatment indicator, with \(D_i=1\) for treated and \(D_i=0\) for untreated. Let \(X_i\) denote covariates measured before treatment, and define the propensity score as \(e(X_i)=P(D_i=1\mid X_i)\). In practice, the propensity score is usually estimated from a model; write the estimated score for unit \(i\) as \(\hat e_i\). It is a probability between 0 and 1.

The formula is constructed by taking the reciprocal of the relevant assignment probability. A treated unit's observed assignment probability is \(\hat e_i\), so its weight is \(1/\hat e_i\). An untreated unit's observed assignment probability is \(1-\hat e_i\), so its weight is \(1/(1-\hat e_i)\). This defines the ATE weights; it does not, by itself, establish that a weighted analysis identifies a causal effect. That interpretation also depends on causal assumptions and is the subject of the later treatment of why inverse probability weighting identifies the ATE.

This compact expression works because the treatment indicator selects one term. For a treated unit, \(D_i=1\), so the first term is \(1/\hat e_i\) and the second is zero. For an untreated unit, \(D_i=0\), so the first term is zero and the second is \(1/(1-\hat e_i)\). The notation is useful for computation, but the piecewise form is often clearer when checking a hand calculation.

Conditions and a Short Justification

The weights are defined for a binary treatment and an estimated propensity score that represents the probability of treatment conditional on baseline covariates. Covariates used to estimate the score should be measured before treatment; adjusting for variables affected by treatment can change the question and introduce bias. To interpret an ATE-weighted analysis causally, the standard requirements include consistency, no interference between units, conditional exchangeability given the covariates, and positivity. Positivity means that within covariate patterns relevant to the target population, the probability of either treatment status is greater than zero. These assumptions are not guaranteed simply because a propensity model has been fitted.

The reciprocal rule can be justified directly from the observed assignment mechanism. Conditional on \(X_i\), the chance of observing \(D_i=1\) is \(e(X_i)\), and the chance of observing \(D_i=0\) is \(1-e(X_i)\). Inverting the chance of the realized arm gives the two arm-specific weights. If a treated unit has a low probability of treatment, it receives a large weight; that unit represents treatment assignments that were relatively uncommon at its covariate values. The same logic applies to an untreated unit whose probability of remaining untreated is low.

This is a derivation of the weight construction, not a proof that the resulting weighted means identify the ATE. In a real analysis, the estimated score substitutes for the unknown assignment probability. If the score model is misspecified, the computed weights may not adequately account for differences in measured covariates. Even with a well-specified model, poor overlap can make some weights very large.

Calculating Weights Across a Dataset

The safest hand-calculation procedure is to record the treatment indicator, estimated score, relevant assignment probability, and reciprocal. That makes it easier to see whether the denominator matches the observed treatment arm. The following hypothetical dataset illustrates the calculation for multiple units.

In software, the same computation is often implemented with a vectorized expression. It is still important to inspect the treatment coding: if a dataset uses \(D_i=1\) for control rather than treatment, the interpretation of the formula must be changed to match the coding. A weight column without a clear treatment convention is difficult to audit.

Estimated Scores and Extreme Weights

Estimated propensity scores are rarely exact probabilities known in advance. They are predictions from a fitted model, and rounding can affect a displayed reciprocal. For example, a score close to zero creates a large treated-unit weight, while a score close to one creates a large untreated-unit weight. The problematic combination is not simply “a score near an endpoint”: it is a score near zero for a treated unit or near one for an untreated unit, because the probability of that unit's observed assignment is then small.

Useful diagnostics include plotting estimated scores separately by treatment group, inspecting the distribution and maximum of the weights, and checking whether a few units dominate the total weight. Covariate balance after weighting is also important: substantial residual imbalance can indicate that the fitted score model or its specification is inadequate. There is no single universal cutoff that makes a weight “too large”; the practical concern depends on sample size, the distribution of weights, the target population, and the resulting uncertainty.

Truncating or trimming extreme weights can reduce their influence, but it changes the procedure and can introduce bias; it should not be treated as an automatic repair. Likewise, removing units with extreme scores may change the population to which the results apply. Any such decision should be motivated, reported, and accompanied by a clear account of the target population.

How ATE Weights Differ From Other Approaches

ATE weights use both treatment arms to target the full study population, subject to the assumptions and overlap required for a causal interpretation. Other weighting schemes can target different populations, so weights that look similar algebraically are not interchangeable. For instance, the weights for the average treatment effect on the treated are designed around the treated population rather than the full population; those weights are treated separately in the later material.

Matching and subclassification offer different ways to use propensity scores. Matching selects or pairs units based on similarity, while subclassification groups units into score-based strata. ATE weighting instead assigns a weight to each unit, generally retaining all units unless the analyst makes an additional restriction. Weighting can be convenient for constructing an overall weighted analysis, but it can be unstable when observed treatment assignments are unlikely given the scores. Matching may discard units, and subclassification can leave residual differences within strata; no method automatically removes the need to check overlap and covariate balance.

A second common error is to treat a large weight as a computational nuisance that can be ignored. Large weights indicate that a unit's observed treatment status was estimated to be uncommon at its covariate values; they can make the result sensitive to a small number of observations. Check the arm-specific formula first, then inspect the score and weight distributions, balance, and the implications of any proposed restriction or truncation.