Sensitivity Analysis for Nonrandomly Missing Omics

Why Missingness Assumptions Matter

A protein or metabolite may be unmeasured because its concentration falls below an assay’s detection limit, because a sample fails quality control, or because sample availability depends on participant characteristics. If the chance of observing a measurement depends on the unobserved measurement itself, the data are missing not at random (MNAR). In that case, the observed molecular profiles may not represent the full set of profiles that would have been measured.

This matters when estimating a cross-layer association. For example, an association between gene expression and protein abundance calculated only among samples with both measurements can differ from the association in all eligible samples if protein availability depends on protein abundance or on its relationship with expression. The observed data generally cannot determine the extent of that dependence. Sensitivity analysis makes the uncertainty explicit by recalculating the association under a range of specified, scientifically plausible missingness assumptions.

For concreteness, let \(X\) be an expression measurement, \(G\) a genotype variable, and \(C\) a vector of measured covariates. Suppose the target is the coefficient \(\beta_X\) in a regression of \(Y\) on \(X\), \(G\), and \(C\). This coefficient describes a conditional association in the specified model; it does not, by itself, establish a causal effect. The analysis below assumes \(X\), \(G\), and \(C\) are available for the samples under study, while some values of \(Y\) are missing.

Two Ways to Represent Nonrandom Missingness

A selection model specifies how the probability of observing \(Y\) depends on its value. One common formulation uses a logistic model for the observation indicator and an outcome model for \(Y\). For example:

Here \(Y_i^*\) is \(Y_i\) on a prespecified scale, such as a standardized scale, and \(\gamma\) is the sensitivity parameter. If \(Y_i^*\) increases by one unit while the other terms stay fixed, the conditional odds of observing the measurement are multiplied by \(\exp(\gamma)\). A positive \(\gamma\) represents greater observation odds at higher values of \(Y\); a negative \(\gamma\) represents lower odds. The outcome distribution \(f_Y\) and the logistic form are modeling assumptions, not facts established by the sensitivity parameter.

For an observed measurement, the likelihood contribution includes both its outcome density and its probability of being observed. For a missing measurement, the unobserved \(Y_i\) must be integrated out:

The second approach, a pattern-mixture model, describes the distribution of \(Y\) separately for observed and missing cases. Its sensitivity parameter directly specifies how the missing-case distribution differs from the observed-case distribution, conditional on measured predictors. For example, one can specify a shift in the conditional mean among missing cases. This is often convenient when the scientific question can be expressed as “how much lower or higher might the unobserved values be?”

Worked Example: A Pattern-Mixture Shift Changes the Association

What the Pattern-Mixture Calculation Shows

For two expression groups, the calculation above has a simple general form. Let \(n_x\) be the total number in group \(x\), \(m_x\) the number with observed \(Y\), and \(r_x=n_x-m_x\) the number with missing \(Y\). Let \(\bar{y}_{1x}\) be the observed mean in that group. Under the mean-shift assumption, the implied mean for all samples in group \(x\) is:

The last equality follows by collecting the two terms involving \(\bar{y}_{1x}\) and using \(m_x+r_x=n_x\). If \(X\) is binary and the target is the difference between the \(X=1\) and \(X=0\) means, subtracting the two group means gives:

This expression makes an important feature visible: a common shift \(\delta_0\) can alter the association when the missing fractions differ between groups. A differential shift \(\delta_X\) can change it further because it allows the missing-versus-observed difference to vary with expression group. If both groups had the same missing fraction, a common \(\delta_0\) would cancel from the difference, but \(\delta_X\) would not necessarily cancel.

In an adjusted regression with continuous \(X\), genotype, and covariates, the same principle applies, but the coefficient is usually recalculated by fitting the prespecified analysis model to completed data or by fitting a likelihood under each fixed sensitivity setting. The simple group-mean formula is not a substitute for that model; it illustrates how assumptions about unobserved values can move an association estimate.

Worked Example: Interpreting a Selection Parameter

Choosing Scenarios and Reporting the Result

A sensitivity analysis is informative only if its parameter values are interpretable. For a selection model, report the scale of \(Y^*\), the meaning of \(\gamma\), and the corresponding odds multipliers \(\exp(\gamma)\). For a pattern-mixture model, state the units of the shifts and whether they represent a common difference or a difference that varies with \(X\), genotype, or another predictor. Choose a range that can be defended using assay knowledge, prior measurements, or explicit expert judgment. Since the parameter cannot generally be estimated from the observed data alone, do not present a single chosen value as data-determined.

For inferential analyses, report uncertainty as well as point estimates. Under each scenario, describe how standard errors or intervals were obtained and state whether they account for estimation of model parameters and imputation or integration uncertainty. An interval calculated under one sensitivity setting reflects uncertainty conditional on that setting; it does not account for uncertainty about which MNAR setting is appropriate. Presenting estimates across scenarios makes that second uncertainty visible rather than hiding it inside one interval.

Assumptions, Diagnostics, and a Common Pitfall

Sensitivity analysis does not remove the need for a defensible analysis model. State the outcome model, the observation model or pattern-mixture specification, and the adjustment variables. Check that the assumed outcome distribution and relationship with \(X\) are reasonable for the observed data, while recognizing that observed-data diagnostics cannot validate the distribution of missing \(Y\). In likelihood-based analyses, the likelihood contributions must correctly reflect whether each \(Y\) is observed or missing. In scenario-based imputation, the completed values must follow the stated conditional model, rather than an informal rule that ignores the covariates related to the association.

A common mistake is to say, “The missingness rate is explained by measured covariates, so missingness is random.” A model predicting \(R\) from observed variables does not rule out additional dependence on unobserved \(Y\). The correction is to distinguish the assumption that missingness is conditionally independent of \(Y\) given observed information from the MNAR alternatives considered here. The observed data may support a reference analysis under a stated assumption, but they generally cannot decide between that assumption and all possible MNAR mechanisms.