Why Model Availability Using Unobserved Values?
A modality may be unavailable for reasons related to its own measurements. For example, a sequencing assay might fail more often when the underlying signal is weak, or a specimen might be selected for a specialized assay because an earlier measurement suggested an unusual profile. In either case, availability can depend on values that would have been measured had the modality been observed. A model that allows this dependence is called a selection model.
The central difficulty is that a missing value cannot be used to estimate its own association with availability from the observed records. A selection model makes that association explicit, but observed data alone generally do not determine its strength. The practical role of a sensitivity parameter is to state and vary that strength, rather than to claim that it has been learned from the data.
For clarity, suppose \(A\) and \(X\) are observed for every subject under study. In a multi-modality setting, this describes one target block \(B\) conditional on the other observed information. A more general model can assign probabilities to each availability pattern \(R\) conditional on the full omics vector and \(X\); the same identification issue arises whenever pattern probabilities depend on missing values.
Logistic Selection Model and Observed-Data Likelihood
A common specification for a binary availability indicator is
Here \(h(B)\) is a chosen summary or transformation of the target block. For a scalar \(B\), it could be \(B\) itself; for a high-dimensional block, it might include prespecified features such as a score or selected components. The coefficient \(\gamma\) governs how availability odds vary with those unobserved values. For example, if \(B\) is scalar and \(h(B)=B\), increasing \(B\) by one unit multiplies the odds of observation by \(\exp(\gamma)\), holding \(A\) and \(X\) fixed. The unit and scale of \(B\) therefore matter.
The model must also specify \(f(B\mid A,X)\), the distribution of the target modality given the observed information. For example, a scalar \(B\) might be modeled as normal with a mean depending on \(A,X\), while a binary feature could use a Bernoulli distribution. These are substantive assumptions about the full data, not consequences of the availability model.
For a subject with \(R=1\), both \(B\) and \(R\) are observed. The likelihood contribution conditional on \(A,X\) is \(f(B\mid A,X)\pi(A,B,X)\). For a subject with \(R=0\), the value of \(B\) is unobserved, so its possible values must be integrated out:
For discrete \(B\), the integral is a sum. This likelihood makes the selection assumption consequential: when \(B\) is missing, the probability of that missingness is averaged over possible values of \(B\), weighted by the assumed full-data distribution.
Worked Example: Likelihood Contributions for a Binary Omics Feature
Suppose \(B\) is binary, with \(P(B=1\mid A=1)=0.4\), and consider subjects with \(A=1\). Let the availability model be \(\operatorname{logit}\{\pi(A,B)\}=-1+0.5A+(\log 2)B\). Calculate the contributions for an observed \(B=1\), an observed \(B=0\), and a missing \(B\).
At \(A=1,B=0\), the log odds are \(-1+0.5=-0.5\), so \(\pi(1,0)=\operatorname{expit}(-0.5)=0.3775\). At \(A=1,B=1\), the log odds are \(-0.5+\log 2=0.1931\), so \(\pi(1,1)=\operatorname{expit}(0.1931)=0.5481\).
For an observed \(B=1\), the likelihood contribution is \(0.4(0.5481)=0.2192\). For an observed \(B=0\), it is \(0.6(0.3775)=0.2265\). For a missing \(B\), sum over both possible values, including the probability of missingness at each value:
The three contributions sum to approximately \(1.000\), as required for the probabilities of the mutually exclusive outcomes \((R=1,B=0)\), \((R=1,B=1)\), and \(R=0\), conditional on \(A=1\). This calculation depends on both the Bernoulli full-data model and the stated logistic availability model.
Why the Selection Dependence Is Not Identified
A parameter is identified when distinct parameter values imply distinct distributions for the observed data. In a nonignorable selection model, the observed records reveal the availability rate and the distribution of \(B\) among subjects with \(R=1\), but do not directly reveal the distribution of \(B\) among subjects with \(R=0\). Different combinations of the full-data distribution and the selection relationship can therefore yield the same observed distribution.
A binary example shows the issue exactly. Let \(p=P(B=1)\), \(s_1=P(R=1\mid B=1)\), and \(s_0=P(R=1\mid B=0)\). Suppose the observed availability rate is \(r=P(R=1)\), and the observed proportion with \(B=1\) among available subjects is \(q=P(B=1\mid R=1)\). The observed joint probabilities give only
There are three unknown quantities \(p,s_1,s_0\) but only two such constraints. Even with arbitrarily many subjects, the observed distribution does not generally determine all three.
A short derivation makes the role of a sensitivity parameter clear. Define \(\gamma\) as the log ratio of availability odds for \(B=1\) versus \(B=0\):
For a chosen \(\gamma\), the two observed-probability equations can be combined with this odds-ratio constraint to determine a compatible set of \(p,s_1,s_0\), if the resulting probabilities are valid. The data do not select \(\gamma\); it must be fixed or varied as an assumption. In a logistic model with binary \(B\), this \(\gamma\) is the coefficient of \(B\).
Worked Example: Varying a Binary Selection Sensitivity Parameter
Suppose \(r=0.60\) of subjects have the target modality observed, and \(q=0.70\) of those observed have \(B=1\). Compare the implied population prevalence \(p=P(B=1)\) under no selection dependence (\(\gamma=0\)) and under a selection odds ratio of \(2\) (\(\gamma=\log 2\)).
When \(\gamma=0\), availability odds are equal for \(B=1\) and \(B=0\), so \(s_1=s_0\). The observed distribution of \(B\) then matches its population distribution, giving \(p=q=0.700\).
For \(\gamma=\log 2\), \(s_1/(1-s_1)=2s_0/(1-s_0)\). Write \(t=\exp(\alpha)\) in the logistic availability model, so \(s_0=t/(1+t)\) and \(s_1=2t/(1+2t)\). The observed joint probabilities require \(p s_1=rq=0.42\) and \((1-p)s_0=r(1-q)=0.18\). Substituting \(p=0.42/s_1\) into the second equation gives
Thus \(s_0=0.975/1.975=0.4937\) and \(s_1=1.95/2.95=0.6610\). The implied population prevalence is
Both scenarios reproduce the same observed rate \(r=0.60\) and observed-case proportion \(q=0.70\), but imply different population prevalences. Under the positive selection dependence, \(B=1\) is more likely to be observed, so its population prevalence is lower than its observed-case proportion. This is a sensitivity comparison, not evidence that either value of \(\gamma\) is correct.
Choosing and Interpreting Sensitivity Parameters
A useful sensitivity parameterization has a scale that can be explained to subject-matter collaborators. For a continuous \(B\), standardizing \(B\) before modeling can make a coefficient correspond to a change per standard deviation rather than per assay unit. For a block, \(h(B)\) should be chosen deliberately: a coefficient on a prespecified score expresses a different assumption from coefficients on every feature. High-dimensional selection models can otherwise require many unidentifiable sensitivity parameters.
Worked Example: Interpreting a Continuous-Value Availability Effect
Suppose \(B\) is standardized, \(A=1\), and the availability model is \(\operatorname{logit}\{\pi(A,B)\}=-1+0.5A+0.4B\). At \(B=0\), the log odds are \(-0.5\), so availability is \(0.3775\). At \(B=1\), the log odds are \(-0.1\), so availability is \(\operatorname{expit}(-0.1)=0.4750\). The availability odds ratio for a one-standard-deviation increase in \(B\) is \(\exp(0.4)=1.492\).
The model therefore says that, at the same \(A\), a one-standard-deviation higher \(B\) is associated with 1.492 times the odds of the modality being observed. It does not say that availability probability rises by 0.4, nor that increasing \(B\) causes availability to rise. Because \(B\) is unobserved when \(R=0\), the value \(0.4\) is an assumption to examine in a sensitivity analysis, not a coefficient that can generally be estimated from those missing values.
A practical analysis should state a reference value, units, and direction for every sensitivity parameter. It should then recompute the target estimand across a scientifically defensible range. If an estimate or its uncertainty changes materially across that range, the conclusion is sensitive to the untestable selection assumption. If it changes little, that is useful robustness information, but it does not establish that the assumed range covers every plausible mechanism.
Diagnostics, Alternatives, and Limitations
No diagnostic based only on the observed data can confirm that missingness depends on unobserved values in the specified way. Diagnostics can still reveal incompatibilities and fragile choices. Check that the model reproduces observed availability rates and distributions of \(B\) among available subjects across levels of \(A\) and \(X\). Examine whether fitted probabilities approach zero or one, and whether the assumed distribution \(f(B\mid A,X)\) gives plausible values in regions contributing to the missing-block likelihood. Compare estimates across the prespecified sensitivity range, not only at the value that gives the preferred result.
The core assumptions include a correctly specified full-data model, a correctly specified availability model conditional on the modeled variables, and adequate support for the covariate combinations being analyzed. The selection dependence on missing \(B\) remains untestable from observed records alone. If several modalities can be missing, dependence among their availability indicators also needs an explicit model or sensitivity structure. A simple one-block logistic model should not be treated as an automatic description of every joint missingness pattern.
| Approach | How it represents missingness | Key distinction |
|---|---|---|
| Selection model | Models availability conditional on full-data values. | Its dependence on unobserved values is represented directly, typically through sensitivity parameters. |
| Pattern-mixture model | Models the data distribution separately by observed availability pattern. | It parameterizes differences between patterns rather than availability conditional on the full data. |
| MAR analysis | Assumes availability depends only on observed information after conditioning on modeled variables. | It is a special restriction, not something demonstrated by good fit to observed records. |
| Weighting under an availability model | Uses estimated observation probabilities to reweight observed subjects. | It requires the needed availability probabilities to be identified under its assumptions; it does not by itself resolve dependence on an unobserved value. |
A common mistake is to fit a logistic regression of \(R\) on observed covariates, find no strong associations, and conclude that availability is independent of the missing omics values. The regression cannot include the unobserved values for \(R=0\), so it cannot test their association with availability. The correction is to distinguish the observed-data model from the MNAR assumption: specify the unobserved-value dependence as a sensitivity parameter, explain its meaning, and report how conclusions vary under alternative values.
Check Your Understanding
For each question, first work out the answer, then compare it with the solution.
- In a logistic selection model with scalar \(B\), what does \(\exp(\gamma)\) mean when \(h(B)=B\)?
- Suppose \(P(R=1)=0.50\) and \(P(B=1\mid R=1)=0.80\). Under \(\gamma=0\) in the binary selection model, what is the implied population prevalence of \(B=1\)?
- Why can a large observed sample still fail to identify the selection dependence on a missing value?
- For a missing subject with discrete \(B\in\{0,1\}\), \(P(B=1\mid A,X)=0.3\), \(\pi(A,0,X)=0.4\), and \(\pi(A,1,X)=0.7\), calculate the likelihood contribution for \(R=0\).
Solutions. (1) Holding other variables fixed, it is the multiplicative change in the odds of availability for a one-unit increase in \(B\). (2) Under \(\gamma=0\), availability is independent of \(B\) in this model, so the population prevalence equals the observed-case proportion, \(0.80\). (3) More observations estimate the observed-data distribution more precisely, but do not reveal the distribution of \(B\) among subjects with \(R=0\); distinct full-data and selection parameter combinations can still produce the same observed distribution. (4) Sum over the unobserved values: \(0.7(1-0.4)+0.3(1-0.7)=0.7(0.6)+0.3(0.3)=0.51\).