Pattern-Mixture Sensitivity Analysis for Missing Blocks

Why Model Missingness Patterns Explicitly?

When an omics block is absent, the observed data do not by themselves reveal its distribution among subjects with that availability pattern. An analysis can fill the gap using a reference model, often an assumption that missing values follow the same conditional distribution as values in subjects with the block observed. Pattern-mixture sensitivity analysis asks a further question: how would the integrated result change if that reference distribution were systematically wrong for a particular missingness pattern?

This is useful because different subjects may be missing different modalities, and the unobserved values may differ from predictions based on observed subjects. The sensitivity analysis does not recover those values from the data. Instead, it makes the untestable extrapolation assumption explicit, varies it in scientifically interpretable units, and reports the resulting range of integrated estimates.

Reference Distributions and Sensitivity Parameters

Write \(O_r\) for the blocks observed and \(M_r\) for the blocks missing under pattern \(r\). The observed-data distribution within a pattern, \(f_r(Y_{O_r})\), can be estimated from that pattern's observed records. The conditional distribution \(f_r(Y_{M_r}\mid Y_{O_r})\), however, is not identified from those records: its values concern blocks that are absent. A reference model supplies an extrapolation for this conditional distribution. One common reference is the conditional distribution implied by a missing-at-random model, in which the missingness pattern adds no further information about the missing block after conditioning on observed information and modeled covariates.

A sensitivity model modifies that reference distribution. For a continuous block, a transparent option is a location shift: preserve the reference conditional distribution's shape and shift its conditional mean by a specified amount. In a multivariate Gaussian reference model, for example, the modification is:

Here \(m_{0,r}\) and \(V_{0,r}\) are the reference conditional mean and covariance, and \(\Delta_r\) is a sensitivity parameter vector for the missing blocks under pattern \(r\). A positive component says that the corresponding missing-block value is assumed, on average, higher than its reference prediction. The shift is on the original measurement scale unless the model is explicitly defined on a transformed scale. This location-shift model is a choice, not a fact established by the observed data.

For a target \(\theta=E\{h(Y)\}\), where \(h\) is the integrated quantity of interest, the pattern-mixture expression is:

where \(\pi_r=P(R=r)\) is the population proportion in pattern \(r\). The expression averages the pattern-specific expected target over the observed values in each pattern, then weights by the pattern's prevalence. In practice, estimates use estimated pattern proportions and observed-data distributions. For a linear target and constant shifts, the change from the reference estimate is especially simple:

The components of \(c\) determine how each modality contributes to the integrated quantity. Thus a shift in a modality with a large target weight may matter more than an equal-sized shift in a lightly weighted modality.

How the Shift Formula Follows

The linear-target result comes directly from the pattern-specific conditional mean. Under a constant shift, the missing-block conditional expectation changes from \(m_{0,r}(Y_{O_r})\) to \(m_{0,r}(Y_{O_r})+\Delta_r\). The observed blocks are unchanged. Therefore, for \(h(Y)=c^TY\), the conditional expectation of the target changes by:

This difference is constant within pattern \(r\), so averaging over that pattern's observed values leaves it unchanged. Weighting the resulting difference by \(\pi_r\) and summing over patterns gives \(\theta(\Delta)-\theta(0)=\sum_r\pi_r c_{M_r}^{T}\Delta_r\). This derivation depends on a linear target and a constant location shift. For nonlinear targets, the change generally cannot be obtained by this simple formula; the modified distribution must be integrated or simulated through the target calculation.

Choosing Scenarios and Finding a Tipping Point

Sensitivity values should be selected and reported in units that collaborators can interpret. Possible anchors include a fraction of the block's observed standard deviation, a clinically meaningful assay difference, or a range elicited from subject-matter experts. If a modality has been standardized, shifts are in standard-deviation units; if it is measured in concentration units, shifts must be stated in those units. A grid should include the reference value of zero and plausible departures in both directions when either direction is credible.

Assumptions, Diagnostics, and Alternatives

A useful sensitivity analysis makes its assumptions visible. Pattern membership and the observed blocks must be correctly recorded; the reference model must reasonably represent the observed-data relationships used for extrapolation; and the sensitivity parameters must describe a plausible departure for the specific pattern. The model also assumes the chosen form of departure, such as a constant mean shift with unchanged covariance. If a Gaussian conditional reference model is used, its conditional distribution and covariance structure should be appropriate for the variables and patterns being modeled. Adequate overlap and information are needed to estimate reference relationships; a large pattern with little relevant observed data can make those relationships unstable.

Inspect whether the reference model fits observed relationships and whether estimates are driven by patterns with few subjects. Compare pattern sizes, the spread of observed blocks, and the ranges of model predictions with the observed data. For each scenario, check that the implied values are scientifically possible. A constant shift can create impossible values for bounded measurements or counts; a transformation, truncated distribution, or alternative distributional modification may then be more appropriate. Shifting a mean while holding covariance fixed may also be inadequate if the plausible departure concerns variability or cross-modality dependence.

Pattern-mixture sensitivity analysis complements, rather than replaces, the methods already used for incomplete blocks. Complete-case analysis targets subjects with fully observed data and may not represent the whole study. Inverse-probability weighting and doubly robust estimation address missingness through specified availability or outcome models under their respective assumptions; they do not make unidentified missing-block distributions observable. Multiple imputation under a reference model can produce uncertainty estimates conditional on that model, while pattern-mixture shifts explicitly vary its extrapolation. A selection-model approach parameterizes missingness conditional on unobserved values; it is a different sensitivity parameterization, not a validation of a chosen pattern-mixture scenario.

For a fixed sensitivity value, uncertainty in the estimate still matters. It can be assessed using an appropriate variance procedure, bootstrap, or imputation-based calculation that repeats the estimation under that fixed scenario. A range of estimates across deliberately chosen sensitivity values is not a confidence interval and should not be described as one. Likewise, averaging over sensitivity scenarios as though the scenarios were random requires an explicit probability model for those assumptions.