Selection Models For MNAR

Why Model Response Given Complete Values?

A missingness indicator records whether a value is observed, but it does not explain why that value is missing. A selection model makes that explanation explicit by modeling the probability of the response pattern as a function of the underlying complete-data values. Its defining feature is that the response model can depend on values that are themselves unobserved.

That feature matters because two people with the same recorded information may have different probabilities of responding if their unrecorded outcomes differ. For example, people with more severe unobserved symptoms might be less likely to complete a follow-up questionnaire. The recorded data alone cannot directly compare symptom levels among responders and nonresponders; the model must therefore specify how those unobserved levels relate to response.

The factorization is a probability identity when \(f\) and \(g\) are the corresponding conditional distributions; it does not assert that \(Y\) and \(R\) are independent. In a parametric analysis, \(f\) and \(g\) are given specific models indexed by parameters \(\theta\) and \(\psi\). Those models must define valid distributions and probabilities. If the likelihood is written as a product across individuals, that product additionally assumes independent sampling units, or a suitable model for their dependence.

From Complete Data to the Observed-Data Likelihood

A likelihood for observed data cannot use an unobserved value as though it had been recorded. Instead, it sums or integrates the complete-data joint distribution over every possible value of the missing components. Write \(Y_{\mathrm{obs}}\) and \(Y_{\mathrm{mis}}\) for the observed and missing components under the realized response pattern \(R=r\). The observed-data contribution is

For discrete missing values, the integral is replaced by a sum. This equation is the key operational consequence of the selection factorization: the response probability remains inside the integral whenever it depends on missing values. The observed-data likelihood is therefore generally not obtained by fitting \(f\) to observed values alone and ignoring the response process.

Response Models With Observed Predictors

In multivariate data, a response model can include both observed and missing variables. Consider a scalar outcome \(Y\) that may be missing and a fully observed predictor \(Z\). One possible selection model is logistic:

The model assumes that the log odds of observing \(Y\) are linear in \(Y\) and \(Z\), and that the resulting probability is given by the inverse-logit transformation. If \(\beta=0\), response depends on \(Z\) but not on the missing \(Y\), which is compatible with MAR for this outcome when \(Z\) is observed. If \(\beta\ne 0\), the response probability varies with \(Y\) even at fixed \(Z\), so the model is MNAR for records in which \(Y\) is missing. This is a statement about the specified model, not a conclusion established by observing a nonzero estimate from data that do not reveal the missing \(Y\) values.

A Likelihood Contribution Under a Continuous Model

For a continuous outcome, a selection model combines a density for the complete outcome with a response probability. The density contribution for a respondent includes both factors. For a nonrespondent, the outcome is unknown, so its contribution requires integration across its possible values.

Justification, Assumptions, and What Can Be Checked

The observed-data likelihood follows directly from the complete-data factorization. For each possible value \(y_{\mathrm{mis}}\), the joint probability of the complete values and realized response pattern is \(f(y_{\mathrm{obs}},y_{\mathrm{mis}};\theta)g(r\mid y_{\mathrm{obs}},y_{\mathrm{mis}};\psi)\). Since the missing value is not observed, the observed event includes every possible \(y_{\mathrm{mis}}\). Summing or integrating those mutually exclusive possibilities gives the observed-data likelihood contribution. In particular, if \(g\) changes across possible missing values, it cannot be taken outside that sum or integral.

For continuous variables, the same reasoning uses an integral in place of the sum. A sound analysis also requires that the complete-data distribution and response model are appropriate for the scientific setting, the response indicators are correctly defined, and the probability model assigns valid probabilities to possible response patterns. For likelihood-based estimation, relevant parameter values must also be informed by the observed-data likelihood; a mathematically specified model is not automatically estimable or identified.

Observed-data checks can reveal conflicts between a proposed model and recorded information. For example, the model may imply response rates or distributions among respondents that are inconsistent with observed summaries. But the dependence of response on the missing value cannot generally be checked by comparing response rates across the unobserved values: those values are unavailable precisely for nonrespondents. In particular, observed data alone do not establish that an MNAR coefficient such as \(\beta\) is zero or nonzero. This is the nonidentifiability problem already encountered in the earlier material on nonidentifiability of MNAR mechanisms.

A practical response is to treat the unverified part of the selection model as an explicit assumption and examine how conclusions change over scientifically plausible values. If estimated response probabilities are near zero for parts of the outcome distribution, the model may also rely on extrapolation with little data support. Such weak support, implausible probabilities, or severe sensitivity of conclusions to small changes in the response model are warning signs, not evidence that one chosen MNAR specification is correct.

How Selection Models Fit Among Alternatives

A selection model specifies the complete-data distribution first and then models response conditional on complete values. A MAR analysis instead assumes that, conditional on observed information, response does not additionally depend on the missing values; this can allow the missingness process to be ignored for some likelihood analyses under the usual parameter distinctness and model conditions. The selection formulation does not make that assumption when its response model includes missing values. Pattern-mixture models express the joint distribution using distributions conditional on response patterns, a different factorization rather than a resolution of the same identification problem.

These formulations can describe the same observed-data distribution while making different assumptions about unobserved quantities. A selection model is useful when a response mechanism can be described meaningfully in terms of complete values, but it does not make MNAR dependence empirically verifiable. Nor does merely writing a response regression solve the problem: estimation and interpretation depend on the specified complete-data model, the response model, and the assumptions that connect them. The introductory material on observed-data equivalence of MAR and MNAR motivates why observed fit by itself cannot choose between these assumptions.

Common Pitfall

A common mistake is to argue, “The response model contains \(Y\), so the data prove response depends on the unobserved \(Y\).” That reverses assumption and evidence. The term involving \(Y\) is part of the model specification; for nonrespondents, the relevant \(Y\) values are not available to verify it. The correction is to report the dependence as an MNAR assumption, state how its parameters are chosen or varied, and distinguish model-based conclusions from directly observed findings.