Inverse-Probability Weighting for Modality Availability

Why Weight for Modality Availability?

An integrated association, such as the correlation between a transcriptomic feature and a metabolomic feature, requires both measurements on the same subject. If the subjects with both modalities differ systematically from the target population, the association among available pairs need not represent the target-population association. For example, pair availability may depend on age, clinical status, or site, and those characteristics may also relate to the omics measurements.

Inverse-probability weighting (IPW) addresses this selection by giving greater weight to observed subjects who had a lower probability of providing the required modalities. The method does not recover unavailable measurements one by one. Instead, under stated assumptions, it uses observed measurements and availability probabilities to estimate population averages, cross-products, and associations.

For instance, if an analysis needs modalities A and B but does not need C, a subject contributes whenever A and B are both observed, whether or not C is observed. Thus \(q_{\{A,B\}}(W)\) is the sum of the probabilities of every availability pattern that includes A and B. It is not necessarily the probability of observing exactly the pattern “A and B only.”

Assumptions and Estimating Availability Probabilities

The target is a population in which each subject is conceptually eligible to have the required measurements. Structural absence is different: if a modality is not defined or cannot exist for a subject, that subject is not simply a missing observation for that modality. Define the target population and required modalities accordingly.

For IPW to recover target-population moments, the key missingness assumption is that availability of the required modalities is conditionally independent of their values, given \(W\). In notation, for the measurements \(X_S\) needed by the analysis, \(R_S\perp X_S\mid W\). This is a missing-at-random assumption for this availability indicator. It is not enough that availability is unrelated to the measurements marginally, and it is not established merely by fitting a probability model.

• Observed predictors: \(W\) must be measured for observed and unavailable subjects, and should include relevant predictors of availability and of the target measurements.
• Correct availability probabilities: the fitted model must estimate \(q_S(W)\) adequately. If probabilities are estimated by a model, its specification and fit matter.
• Positive availability: every covariate profile represented in the target population must have a nonzero probability of providing the required modalities. Otherwise, its measurements cannot be represented by observed subjects with those covariates.
• Appropriate sampling: the enrolled subjects and analysis must correspond to the target population; additional sampling or dependence features may require their own adjustments.

A common approach is to treat each possible availability pattern as a category and model its probability conditional on \(W\), for example with a multinomial logistic model. Estimate the probability of each pattern, then sum probabilities over patterns that contain all modalities in \(S\). With a small number of well-populated covariate strata, empirical pattern proportions can instead estimate these probabilities. The fitted model should reflect the actual pattern definition and use predictors available for the full cohort.

Why the Weights Recover Population Moments

Let \(h(X_S,W)\) be a quantity that can be calculated when the required measurements are observed, such as \(X\), \(Z\), \(XZ\), or \(X^2\). Under the conditional independence assumption, the probability of observing the required set given the measurements and \(W\) is \(q_S(W)\). Conditioning on \(X_S,W\) gives the following justification:

Taking expectations over the target population shows that \(E[R_Sh/q_S(W)]=E[h]\). Thus, a weighted average of observed values can estimate the target-population average. In a sample, a common normalized, or Hájek, estimator is the sum of weighted observed values divided by the sum of their weights. Under suitable sampling and model conditions, it estimates the same target average. It is generally more stable in scale than dividing a weighted total by the full cohort size, although finite-sample results need not be identical.

To estimate an integrated association between two measurements \(X\) and \(Z\), use subjects with both observed and weights \(1/q_{\{X,Z\}}(W_i)\). Estimate the target means and cross-moments with the same weights:

Here \(R_i=1\) only when both measurements are available, and the sums include only such subjects. The weighted covariance is \(\widehat{E[XZ]}^w-\hat\mu_X^w\hat\mu_Z^w\). A weighted correlation divides this covariance by the square root of the product of the two weighted variances, each computed using the same normalized weights. These are estimates of marginal target-population association, not conditional association at fixed \(W\). Association alone does not establish causation.

Calculating a Weighted Integrated Association

Model Checks, Failure Modes, and Alternatives

Availability probabilities are estimated quantities, so inspect the pattern model before interpreting weighted results. Confirm that observed pattern counts and fitted pattern probabilities are sensible across important covariate groups, that probabilities for patterns sum to one, and that the inclusion probability for the required modalities is obtained by summing the correct patterns. Compare fitted availability with observed availability across covariate strata. A model that omits an important predictor can leave residual selection associated with the measurements, even if its overall predicted rate looks accurate.

Very small estimated inclusion probabilities produce large weights. A few such subjects can dominate weighted moments, making estimates unstable. Report the distribution of weights and assess whether results depend heavily on a small number of observations. Truncating or capping weights may reduce variability, but it changes the estimator and can introduce bias; it is not a repair for an unsupported missingness assumption. If a target subgroup has no observed subjects with the needed modalities, weighting cannot create information for that subgroup.

Complete-case analysis uses the observed pairs without adjustment. It is simple, but represents the target population only under stronger conditions, such as availability being independent of the relevant measurements and covariates, or under a correctly specified analysis that accounts for selection. IPW explicitly reweights for observed predictors of availability. Likelihood-based methods and multiple imputation instead use models for the measurements or missing data to make use of partial observations. Their assumptions and estimands should be checked in context; they are not automatically interchangeable with IPW. Weighting is attractive when an availability model is credible and the target is a population moment, but it may be imprecise when weights vary widely and does not itself supply values for unavailable blocks.

For inference, uncertainty in both the measured values and estimated availability probabilities matters. A variance procedure should account for propensity estimation and the study's sampling structure; treating estimated weights as fixed can understate uncertainty. Bootstrap procedures that refit the availability model in each resample are one possible approach when their sampling assumptions are appropriate. The weighted point estimate alone does not supply a valid standard error.

Putting the Procedure Together