Why combine weighting and outcome regression?
A trial may enroll patients who receive the experimental treatment, while a registry supplies patients who received usual care. Even after careful alignment, the two groups may differ in measured prognosis. Outcome regression addresses those differences by predicting registry control outcomes at the trial participants’ covariate values. Inverse probability weighting instead reweights registry controls to resemble the trial group. Doubly robust estimation combines these two adjustments in one estimator.
Here the target is the average treatment effect among the trial participants who received treatment. This choice is natural when the treatment outcomes are observed for the trial group and the external controls provide information about what those same participants might have experienced under control. It is not automatically the effect for every eligible patient or for the registry population.
For participants with \(A=1\), consistency means the observed outcome is \(Y(1)\); for those with \(A=0\), it is \(Y(0)\). The treatment-effect estimate compares the treated group’s observed mean outcome with an estimate of its mean untreated potential outcome. The latter is the part that must be recovered from the registry and the measured covariates.
Estimator and assumptions
Fit a treatment-assignment model to estimate \(\hat e(X_i)\), and an outcome model among external controls to estimate \(\hat m_0(X_i)\). The assignment model describes the probability of belonging to the trial-treated group rather than the external-control group, conditional on measured covariates. The outcome model predicts the control outcome conditional on those covariates. The augmented estimate of the trial group’s mean outcome under control is:
The first sum predicts each treated participant’s control outcome. The second sum adds weighted prediction errors observed among controls. The weight \(\hat e(X_i)/(1-\hat e(X_i))\) is the estimated odds of being in the treated trial group rather than the external-control group at that covariate value. Subtracting \(\hat\mu_{0,1}\) from the treated group’s observed mean gives the estimate:
This estimate is on the outcome scale: for a binary endpoint coded 0 or 1, it is a risk difference; for a continuous endpoint, it has the endpoint’s measurement units. The formula does not itself make a registry comparison causal. Its interpretation depends on assumptions about how the groups and outcomes relate.
Conditional exchangeability is a substantive assumption, not something established by fitting both models. It can fail if an important prognostic factor is unmeasured, measured differently, or inadequately aligned between the trial and registry. The earlier material on confounding diagrams and confounder selection helps clarify which covariates need to be considered. The earlier material on propensity scores and outcome regression supplies the two component models; the key addition here is their augmentation.
Why the estimator is doubly robust
The justification can be seen by considering the population version of the estimated control mean. Let \(\pi=P(A=1)\), and temporarily use candidate functions \(e(X)\) and \(m(X)\). Its numerator has two terms: \(E[A m(X)]\) and \(E[(1-A)e(X)\{Y-m(X)\}/\{1-e(X)\}]\). The target numerator is \(E[A Y(0)]\). Conditional exchangeability and consistency imply \(E[Y\mid A=0,X]=E[Y(0)\mid X]\), which we denote by \(m_0(X)\).
The final expression equals \(E[A Y(0)]\), because \(E[A Y(0)\mid X]=e(X)E[Y(0)\mid X]\). Thus, if the assignment model is correct, the augmented expression targets the right control mean even if \(m(X)\) is misspecified. Alternatively, if \(m(X)=m_0(X)\), the conditional mean of the control residual \(Y-m(X)\) is zero. The weighted residual term then has expectation zero, and \(E[A m_0(X)]\) is the target numerator. Dividing by \(\pi\) yields the target mean among treated participants in either case.
This is the double-robust property: under the causal and support assumptions, consistency can hold when either one of the two nuisance models is correctly specified. It does not mean that both models can be wrong, nor does it mean that the estimate is unbiased in every finite sample. The argument also relies on the models being estimated well enough and on the data providing usable support.
Worked Example: A trial group and registry controls
Hypothetical data. Suppose a trial has 100 treated participants and a registry has 100 controls. Baseline prognosis is summarized by a binary covariate, high risk or low risk. There are 40 high-risk and 60 low-risk treated participants; among controls there are 20 high-risk and 80 low-risk participants. The outcome is a hypothetical symptom score at follow-up, in points, with lower scores preferred. The control-group mean is 4 points for high-risk participants and 2 points for low-risk participants. The treated-group mean is 8 points in each risk category.
For illustration, use the risk category to fit the models. The outcome predictions are \(\hat m_0(\text{high})=4\) and \(\hat m_0(\text{low})=2\). In the pooled sample, the estimated assignment probabilities are \(\hat e(\text{high})=40/(40+20)=2/3\) and \(\hat e(\text{low})=60/(60+80)=3/7\). Thus the control weights are \(2\) for high-risk controls and \(3/4\) for low-risk controls. Both risk categories occur in each group, so these simple data do not have a structural absence of controls in a treated category.
The observed treated mean is \((40\times8+60\times8)/100=8\) points. The outcome-model prediction averaged over treated participants is \((40\times4+60\times2)/100=2.8\) points. The control residuals average zero within each risk category, because the stated control means are the model predictions. Therefore the weighted residual contribution is zero:
It follows that \(\hat\mu_{0,1}=2.8+0=2.8\) points and \(\hat\tau_{\mathrm{DR}}=8-2.8=5.2\) points. Under the assumptions, the estimate says that the treated trial group’s mean symptom score is estimated to be 5.2 points higher under treatment than their mean score would have been under control. Since lower scores are preferred, the direction is unfavorable for treatment. This calculation demonstrates the estimator, not evidence about any actual therapy.
Worked Example: Correct assignment model, misspecified outcome model
Keep the hypothetical counts and outcomes above, but suppose the outcome model is misspecified and predicts 3 points for everyone. The assignment model remains correct for the two risk categories. The treated participants’ average prediction is \(3\) points. High-risk controls have residual \(4-3=1\) point and weight \(2\); low-risk controls have residual \(2-3=-1\) point and weight \(3/4\). The weighted correction per treated participant is:
Thus \(\hat\mu_{0,1}=3-0.2=2.8\) points, and \(\hat\tau_{\mathrm{DR}}=8-2.8=5.2\) points. In these data the correct assignment model allows the weighted residuals to correct the misspecified constant outcome prediction. In practice, the correction may be unstable if the weights are very large.
Worked Example: Correct outcome model, misspecified assignment model
Again use the same hypothetical data, but now suppose the assignment model wrongly assigns every participant \(\hat e(X)=1/2\), giving every control a weight of \(1\). Retain the correct risk-specific outcome predictions of 4 and 2 points. The weighted residual contribution is still zero because each group’s residual sum is zero within each risk category:
The treated average prediction remains \(2.8\) points, so the estimated control mean is \(2.8\) points and the estimated treatment effect is \(8-2.8=5.2\) points. This example illustrates the other route to consistency. With real data, conditional residual means will not generally be exactly zero in every observed sample; a correctly specified outcome model supports consistency as sample information grows, not a guarantee of exact equality in a particular data set.
Diagnostics, alternatives, and limits
Before interpreting a doubly robust estimate, examine whether the data and fitted models support the calculation. Check that covariates are measured before treatment and represented comparably across sources. Inspect estimated assignment probabilities and the distribution of control weights, especially whether a few controls carry a large share of the weighted contribution. Confirm that the outcome model produces clinically plausible predictions across the treated group’s covariate patterns. Compare the covariate distributions after weighting as a diagnostic, while remembering that balance on measured covariates does not demonstrate absence of unmeasured confounding.
A large change in the estimate when using outcome regression alone, weighting alone, or the augmented estimator is a warning to investigate model specification, overlap, influential observations, and data harmonization. It is not a reason to select whichever answer is most favorable. Uncertainty calculations should account for estimation of both nuisance models; a resampling procedure, when appropriate to the design, should refit both models in each resample. Very small standard errors do not compensate for a biased external comparison.
Outcome regression alone depends on the outcome model being adequate for the target group. Weighting alone depends on an adequate assignment model and can be sensitive to extreme weights. The augmented estimator includes both approaches and has the stated protection against misspecification of one model, but it may still be unstable when overlap is poor. Truncating weights can reduce sensitivity to extreme values, but changes the estimator and may introduce bias; it does not restore missing support.
Most importantly, double robustness does not protect against unmeasured confounding, incompatible endpoint definitions, selection bias, or systematic differences in follow-up and calendar time that remain after adjustment. It does not turn observational external controls into randomized controls. The estimate’s causal meaning therefore rests on design and subject-matter arguments as well as model fitting. For a Bayesian treatment-effect analysis after adjustment, the treatment of uncertainty and inferential target requires additional choices beyond this estimator.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- What treatment effect does the estimator above target?
- In the first worked example, why is the weighted residual contribution zero?
- If the outcome model is wrong but the assignment model is correct, what role does the weighted residual term play?
- Does double robustness address an important confounder that was never measured? Explain.
- What does a very large control weight suggest checking?
Solutions. (1) The average treatment effect among the treated trial participants. (2) Within each risk category the observed control mean equals its outcome-model prediction, so each category’s mean residual is zero. (3) It corrects the outcome-model prediction; with a correct assignment model, the population augmented control mean targets the treated group’s mean untreated potential outcome. (4) No. Conditional exchangeability may fail, and fitting two models cannot remove unmeasured confounding. (5) Check estimated assignment probabilities, covariate overlap, influential controls, and model or data problems that may create unstable weights.