Comparing MNAR Sensitivity Analyses Across Models

Why Compare Sensitivity Models on a Common Scale?

When an omics block is missing for some subjects, the observed data alone generally cannot reveal how the unobserved measurements differ from the observed ones if missingness may depend on those unobserved values. A selection model describes that dependence through the probability that a block is observed. A pattern-mixture model instead describes the distribution of the measurements separately by availability pattern. The introductory material on pattern-mixture sensitivity analysis and selection models for nonignorable modality missingness develops those parameterizations; here the focus is how their assumptions change the same scientific target.

A parameter such as a log-odds coefficient in a selection model and a mean shift in a pattern-mixture model do not have the same units or interpretation. Comparing their numerical values directly is therefore unhelpful. Instead, choose a target association, specify how each model completes the missing-data distribution, and calculate the resulting target under each assumption. A useful sensitivity comparison asks whether materially different assumptions lead to materially different scientific conclusions.

For the worked calculations, the target will be the population regression slope of \(Y\) on \(X\), with an intercept, in the full study population. It is a cross-omics association, not a causal effect. Because \(X\) is fully observed, the slope can be written as the covariance of \(X\) and \(Y\) divided by the variance of \(X\). If the scientific target were a correlation instead, each sensitivity scenario would also need to update the variance of \(Y\).

The calculations below treat the observed values and reference conditional means as fixed to make the mapping transparent. In an analysis, sampling uncertainty and uncertainty in fitted conditional models must also be considered. Neither parameterization makes the MNAR assumptions empirically identifiable from the observed data alone.

A Shared Numerical Setup

Consider a hypothetical study with six subjects. \(X\), measured in log2 abundance units, is recorded for everyone. \(Y\), also measured in log2 abundance units, is observed for four subjects and missing for two. A reference model for \(Y\) given \(X\) supplies baseline means of 4 and 6 for the two missing subjects. These are model-based reference values, not observed measurements.

The sensitivity assumption will shift each missing subject's conditional mean by a common amount \(\delta\), measured in log2 abundance units. This is a simplified pattern-mixture specification. It assumes the reference means adequately describe the missing values absent the sensitivity shift, and that a common shift is appropriate for these two \(X\) values. The observed \(X\) values have variation, but the assumed conditional model must still be credible at the missing subjects' \(X\) values.

Mapping a Selection Model to a Pattern-Mixture Shift

A simple selection model makes the connection algebraically explicit. Suppose, conditional on \(X=x\), the observed-outcome distribution is normal with mean \(m(x)\) and variance \(\sigma^2\). Suppose the odds of observing \(Y\) follow a logistic model whose \(Y\)-dependence is linear:

Here \(\gamma\) measures how the log odds of observing the modality change per one-unit increase in \(Y\), holding \(X\) fixed. The function \(\alpha(x)\) can vary with \(X\), allowing availability rates to differ across \(X\). A positive \(\gamma\) means larger \(Y\) values have greater odds of being observed at a fixed \(X\).

The result follows from Bayes' rule. Conditional on \(X=x\), the ratio of the two outcome densities is proportional to the inverse observation odds:

The terms that depend on \(x\), including the availability rate, are constants with respect to \(y\) and are absorbed into the normalizing constant. Substituting the normal density gives an exponent, up to constants,

Completing the square shows that the missing distribution has the stated mean and retains variance \(\sigma^2\). This is a distributional identity under the specified assumptions, not a general equivalence between all selection and pattern-mixture models.

When the Mapping Is Not One Number

The formula \(\delta=-\gamma\sigma^2\) depends on the normal distribution, a linear \(Y\)-term in the selection log odds, and a common conditional variance. If the variance changes with \(X\), the implied shift is \(-\gamma\sigma^2(x)\), so a single constant pattern-mixture shift cannot reproduce the selection model at every \(x\). If selection depends nonlinearly on \(Y\), the density tilt may change the shape as well as the mean of the missing distribution. In those cases, compare models by deriving or numerically calculating their implied missing distributions and then evaluating the same target association.

Comparing Assumptions, Diagnostics, and Alternatives

A practical comparison starts by specifying a plausible range for each model's sensitivity parameter, then translating those scenarios into the target association wherever possible. Plotting the target against each model's parameter can reveal whether conclusions change gradually or only under extreme assumptions. When parameters map exactly under stated conditions, show the mapping. When they do not, compare implied distributions and target values rather than suggesting that equal numerical parameter values mean equal assumptions.

Diagnostics cannot establish which MNAR model is true, but they can expose implausible implications. Check whether the assumed direction and size of selection are scientifically credible; whether conditional means and variances are extrapolated far beyond observed support; and whether the completed distributions produce impossible or implausible assay values. Inspect sensitivity curves for abrupt changes, and report how the target changes across a prespecified range rather than selecting a favorable parameter after seeing the result. If a normal tilt is used, verify that normality and the variance assumptions are reasonable approximations for the analysis scale.

A common mistake is to say, “\(\gamma=0.25\) and \(\delta=0.25\) are comparable because both equal 0.25.” They are not: \(\gamma\) is measured in inverse units of \(Y\), while \(\delta\) is measured in units of \(Y\), and their signs may have opposite implications. The correction is to derive a model-specific map such as \(\delta=-\gamma\sigma^2\), or compute each model's implied target separately. Another mistake is to treat agreement between two sensitivity analyses as evidence that MNAR assumptions are verified. Agreement only shows that the selected scenarios yield similar estimates; unobserved values remain unverified.

Relative to approaches that posit missingness independent of unobserved outcomes after conditioning on observed variables, selection and pattern-mixture sensitivity analyses explicitly explore departures from that condition. Their advantage is transparency about untestable assumptions. Their limitation is dependence on the chosen sensitivity scale, model form, and extrapolation. In particular, a target association can be sensitive to how missing blocks relate jointly to other modalities; simplifying that relationship to a single mean shift may miss changes in variance, tails, or dependence. A sensitivity analysis complements, rather than replaces, careful modeling of the observed data and transparent reporting of the assumptions.