Doubly Robust Estimation With Missing Modalities

Why Combine Two Models?

When an omics modality is unavailable for some subjects, its measurements cannot simply be averaged over the observed subjects and assumed to represent the entire cohort. The observed subjects may differ systematically from those whose modality is missing. Inverse-probability weighting addresses this by giving more influence to subjects who resemble those less likely to have the modality observed. An outcome model instead predicts the missing measurements from information available for everyone. Doubly robust estimation combines these strategies: under stated conditions, it can remain consistent if either one of the two models is correctly specified.

We focus on a scalar target, such as the cohort mean of a prespecified molecular feature or a score derived from a modality. The same construction can be applied to other defined full-data means, but it does not make an unidentified target identifiable. In particular, the availability and modeling assumptions must support inference about the intended cohort quantity.

Here \(\hat{\pi}(Z_i)\) and \(\hat{m}(Z_i)\) are estimates of the availability probability and conditional outcome mean. Because \(Z_i\) is observed for every subject, the fitted outcome model supplies a prediction for each subject, including those with \(R_i=0\). The residual correction then adjusts predictions using observed outcomes, with larger corrections for observations that represent subjects less likely to have their modality observed.

Assumptions for the Doubly Robust Result

The central missingness condition is that, conditional on \(Z\), availability carries no further information about the unobserved measurement: \(R\) is independent of \(Y\) given \(Z\). This is a missing-at-random condition for this target and these predictors. It is an assumption, not a fact established merely because the availability model fits well. If availability depends on the unseen value even after conditioning on \(Z\), the observed data generally do not identify the full-cohort mean without additional assumptions.

The other key condition is positivity: for covariate values represented in the target population, \(\pi(Z)>0\). If a subgroup has zero chance of having the modality observed, its outcome distribution cannot be learned from observed outcomes under this setup. Very small probabilities are also problematic in practice because they can produce unstable, highly influential corrections. The earlier material on positivity and extreme weights discusses this issue in more detail.

For the usual consistency result, subjects are sampled independently from the target population (or an appropriate sampling design is accounted for), the stated conditional mean and availability probabilities are well-defined, and the nuisance estimates converge to suitable limits. The result concerns large-sample consistency. It does not guarantee that every finite-sample estimate is stable or unbiased, nor does it protect against misspecification of both models.

Why Either Model Can Be Enough

The argument is easiest to see by considering the population version of the estimator, with fitted nuisance functions replaced by limiting functions \(\pi^*(Z)\) and \(m^*(Z)\). Conditional on \(Z\), its expected contribution is \(m^*(Z)+E(R\mid Z)E\{Y-m^*(Z)\mid R=1,Z\}/\pi^*(Z)\). Under missing at random, \(E(Y\mid R=1,Z)=E(Y\mid Z)=m(Z)\), and \(E(R\mid Z)=\pi(Z)\). Therefore the conditional expectation becomes \(m^*(Z)+\{\pi(Z)/\pi^*(Z)\}\{m(Z)-m^*(Z)\}\).

If the availability model is correct, \(\pi^*(Z)=\pi(Z)\), and this expression simplifies to \(m(Z)\), even when \(m^*(Z)\) is not the true outcome regression. If the outcome model is correct, \(m^*(Z)=m(Z)\), the second term is zero, even when \(\pi^*(Z)\) is not the true availability probability, provided the fitted probabilities remain valid and bounded away from zero as needed. Averaging over \(Z\) gives \(E\{m(Z)\}=E(Y)=\theta\). This establishes the two population routes to consistency.

The calculation also shows what the label “doubly robust” does not mean. It does not mean robust to any two errors, to missing-not-at-random mechanisms, or to a lack of overlap. If both limiting models are wrong, the conditional expectation generally differs from \(m(Z)\). Specifically, its discrepancy from \(m(Z)\) is \(\{1-\pi(Z)/\pi^*(Z)\}\{m^*(Z)-m(Z)\}\), which need not average to zero.

Two Routes Illustrated With Population Values

The following hypothetical two-group example makes the consistency argument concrete. Let \(Z\) identify group A or group B, with each group comprising half of the target population. Suppose the true mean \(Y\) is 10 in group A and 20 in group B. Under missing at random given group, the availability probabilities are 0.50 and 1.00, respectively. The target mean is therefore \(0.5(10)+0.5(20)=15\).

Diagnostics, Inference, and Practical Choices

Doubly robust estimation should be accompanied by checks of both fitted models and of the data support. Examine the distribution of \(\hat{\pi}(Z_i)\), particularly its minimum and lower tail, and identify covariate regions with few or no observed modality measurements. Very small fitted probabilities can make \(R_i\{Y_i-\hat{m}(Z_i)\}/\hat{\pi}(Z_i)\) large. Inspect influential contributions and compare results under defensible model specifications. Truncating or stabilizing weights may reduce variance, but it changes the estimator and can introduce bias; it does not restore information in a region with no observed data.

Assess whether the availability model includes predictors associated with modality observation and the outcome, and whether the outcome model captures relevant patterns in \(Y\) across the fully observed \(Z\). Flexible models may reduce functional-form error, but do not automatically solve positivity failures or leakage from information unavailable for subjects with \(R=0\). If nuisance models are estimated using the same observations, overfitting can distort residual corrections. Cross-fitting, which estimates nuisance functions on other folds and evaluates each subject's correction on a fold not used to fit those functions, can help control this problem for flexible estimators.

For uncertainty quantification, a useful first-order influence function when the nuisance functions are correct is \(\phi(O)=m(Z)-\theta+R\{Y-m(Z)\}/\pi(Z)\), where \(O\) denotes the observed data. Under suitable regularity conditions, the sampling variance of the estimator is approximately \(\operatorname{Var}\{\phi(O)\}/n\). Estimating this variance requires accounting for the fitted nuisance models and the estimator's sampling scheme. In flexible settings, cross-fitting and appropriate nuisance-estimation rates are often used to justify standard errors; consistency alone is not a guarantee of valid confidence intervals. A common sufficient type of rate condition requires the product of the two nuisance estimation errors to be smaller than order \(n^{-1/2}\), alongside boundedness and other regularity conditions.

The principal alternatives answer related but not identical practical needs. Complete-case analysis averages outcomes among subjects with \(R=1\); it is generally representative of the full cohort only under additional conditions, such as availability being independent of the outcome in the relevant population. Inverse-probability weighting uses observed outcomes weighted by \(1/\hat{\pi}(Z_i)\); it relies heavily on the availability model and can be unstable with extreme weights. An outcome-model approach averages predicted outcomes over all subjects and depends on the outcome regression. The augmented estimator combines prediction and residual correction, offering the two-model consistency property under its assumptions. Likelihood and imputation approaches can use richer distributional or joint-data structures, but they likewise depend on their modeling and missingness assumptions. No method can use the observed data alone to identify a full-cohort mean when the assumptions fail in an unrestricted way.

Common Mistakes and Limitations

• Calling the estimator robust to missing-not-at-random data. The derivation uses \(R\) independent of \(Y\) given \(Z\). If unobserved values still affect availability conditional on \(Z\), the two-model argument does not establish consistency.
• Assuming one correct model removes every other requirement. Either nuisance model can be correct, but positivity, adequate observed-data information, and the missingness condition remain necessary.
• Treating a good model fit as proof of correctness. The unseen outcomes for \(R=0\) cannot directly validate their predicted distribution. Diagnostics can reveal problems, but cannot verify assumptions about unobserved values.
• Ignoring unstable contributions. A small \(\hat{\pi}\) can amplify a residual substantially. Report and investigate influential corrections rather than relying only on an overall fit statistic.
• Interpreting the result as causal. The estimator targets a descriptive full-cohort mean unless a separate causal estimand and its identification assumptions have been specified.

In practice, the estimator is most useful when predictors are available for everyone, there is meaningful overlap in modality availability, and at least one nuisance model is credible. Reporting the target, predictors, missingness assumption, probability range, model checks, and sensitivity to reasonable specifications makes the result easier to assess. Doubly robust construction is a protection against one particular form of model error, not a substitute for careful design or transparent assumptions.