Outcome Regression With Missing Outcomes

Why Model Outcomes When Some Are Missing?

When an outcome is unobserved for some people, an analysis limited to observed outcomes can change the composition of the groups being compared. Outcome regression offers a different route: model the outcome using treatment and covariates among people with observed outcomes, then use the fitted model to predict outcomes under each treatment for everyone in the target population. The predictions are averaged over that population’s covariate distribution, not just over the people whose outcomes happened to be recorded.

This procedure can identify a causal effect only under assumptions about treatment assignment and missingness, as well as a suitable outcome model. In particular, the missingness mechanism must be ignorable conditional on the variables used in the model, and there must be observed outcomes across the relevant treatment and covariate groups. A fitted regression does not, by itself, make those assumptions true.

Identification Before Estimation

A target causal effect is identified from the observed data only if the assumptions connect potential outcomes to the available outcomes. For the ATE, the main assumptions are:

• Consistency: a person who receives treatment \(A=a\) has observed outcome \(Y=Y(a)\) if that outcome is recorded.
• Conditional exchangeability of treatment: \(Y(a)\perp A\mid X\), for \(a=0,1\). Given the measured covariates, treatment groups are comparable with respect to their potential outcomes.
• Missing at random (MAR) for the outcome: \(Y\perp R\mid A,X\). Within treatment and covariate levels, the probability that the outcome is recorded does not depend on its value.
• Treatment positivity: each treatment value has positive probability at covariate values represented in the target population.
• Response positivity: the outcome has positive probability of being observed in each relevant treatment and covariate group.

The covariates \(X\) must be measured for people in the target population, including those whose outcomes are missing. The target population must also be clearly specified. If the target is the study population, its observed covariates can define the averaging distribution; targeting another population requires that population’s covariate distribution and additional justification for applying the fitted outcome model there.

Define the observed-outcome regression function as \(m(a,x)=E[Y\mid A=a,X=x,R=1]\). Under MAR, \(m(a,x)=E[Y\mid A=a,X=x]\). Consistency and conditional exchangeability then connect this conditional mean to the potential outcome mean. In combination, these assumptions give the identification result below.

The expectation over \(X\) is taken over the covariate distribution in the target population. Importantly, a conditional regression contrast need not itself equal the ATE: averaging the two treatment-specific conditional means over the target covariates is what produces the marginal effect.

Why the Identification Formula Holds

For a fixed treatment value \(a\), conditional exchangeability says the potential outcome mean among people with \(A=a\) equals the potential outcome mean at that covariate value in the whole population. Consistency equates that potential outcome with the observed outcome for people who received \(a\). MAR then allows the observed outcomes to represent the conditional outcome distribution even though some outcomes are missing. In sequence:

The first equality uses conditional exchangeability, the second uses consistency, and the third uses MAR together with response positivity. Averaging the resulting equality over the target distribution of \(X\) gives \(E[Y(a)]=E_X\{m(a,X)\}\). Subtracting the expressions for \(a=0\) and \(a=1\) gives the ATE formula. This derivation shows why each assumption matters: if one link fails, the observed-data regression need not recover the relevant potential-outcome mean.

Estimation by Prediction and Averaging

In a sample, fit a model for \(E[Y\mid A,X,R=1]\), using only records with observed outcomes. The model may be linear, logistic, or another appropriate regression model, depending on the outcome and the intended conditional mean. Then, for every member \(i\) of the target sample, predict the outcome twice: once setting treatment to 1 and once setting it to 0, while retaining that person’s observed covariates \(X_i\).

Here \(n\) is the number of people in the target sample, including those with missing outcomes, and \(\widehat m\) is the fitted conditional mean. For a target population represented by unequal sampling weights, the averaging step must instead use the corresponding normalized target weights. The outcome model is estimated among observed outcomes; the standardization step uses the covariates of the target population.

Model Form and the Scale of the Effect

The regression model must estimate the conditional mean well enough over the covariate values being averaged. A model that omits important treatment-by-covariate variation can give poor predictions even if its overall fit appears acceptable. When the effect may differ by \(X\), the model needs a structure that can represent that difference, such as an interaction or separate treatment-specific models. The right choice depends on subject-matter knowledge and the observed data; a simple formula should not be mistaken for evidence that effects are truly constant.

For a continuous outcome, a linear model can directly predict conditional means in outcome units. For a binary outcome, a logistic model predicts conditional probabilities after applying the inverse-logit transformation. Average those predicted probabilities to estimate each marginal risk, then subtract them for a risk difference. A logistic regression coefficient for treatment is a conditional log odds ratio, not the standardized marginal risk difference; in general, it is not the ATE on the risk-difference scale.

Diagnostics, Uncertainty, and Common Pitfalls

MAR is an assumption about the relationship between the missingness indicator and the unobserved outcome after conditioning on \(A\) and \(X\). It cannot generally be verified from the observed data alone, because the outcomes needed to assess dependence among people with \(R=0\) are precisely the outcomes that are unavailable. Investigators should use substantive knowledge to assess whether the conditioning variables make MAR plausible and should be explicit about this limitation.

Response positivity can also be fragile. If a treatment-covariate group has very few observed outcomes, its fitted mean may depend heavily on model extrapolation. Inspect the observed response patterns across treatment and important covariates, check whether predictions are being made far outside the support of the observed outcomes, and report limitations when overlap is weak. A mathematically computable prediction is not necessarily a well-supported one.

The estimator has sampling uncertainty from both fitting the outcome model and averaging predictions over the target sample. Confidence intervals should account for the complete estimation procedure; for example, a bootstrap can resample the relevant sampling units and repeat both model fitting and standardization in each resample. The interval’s validity still relies on the identification assumptions and a suitable estimation procedure. It does not measure uncertainty about whether MAR or no unmeasured confounding is true.

• Pitfall: “MAR means the observed cases are representative without adjustment.” This is wrong if response depends on treatment or covariates. MAR permits dependence on \(A,X\); the conditional model and target-population averaging are what address those differences.
• Pitfall: “A regression coefficient is automatically the causal effect.” The fitted outcome regression is used to predict potential-outcome means. The causal contrast comes from averaging both predictions over the target covariate distribution, and its scale depends on the chosen outcome summary.
• Pitfall: “MAR guarantees a correct estimate.” MAR does not correct a misspecified conditional mean model, lack of response positivity, or unmeasured confounding of treatment and outcome. These are distinct requirements.