Shared-Parameter Models For Informative Missingness

Why Share a Latent Variable?

A response indicator may be associated with an outcome even when the model gives no direct response effect to the outcome. For example, a latent characteristic such as underlying illness severity could influence both a patient’s measured health score and the likelihood of completing a follow-up assessment. If severity is not fully observed, the outcome and its response indicator can remain associated after that latent characteristic is averaged out. When response then depends on the outcome itself, the missingness mechanism is MNAR.

A shared-parameter model represents this idea by placing the same latent variable in both an outcome model and a response model. It is a substantive model for the joint relationship among the outcome, response, observed predictors, and latent variable—not a guarantee that the missing values can be recovered uniquely from observed data.

Here \(Y \perp R \mid X,Z\) means that, conditional on \(X\) and \(Z\), knowing the outcome gives no additional information about response. It does not mean that \(Y\) and \(R\) are independent given only \(X\). The distinction is central: the latent variable is shared, but it is not observed for conditioning in an ordinary observed-data analysis.

Model and Assumptions

A basic specification has three parts: a distribution for \(Z\) given \(X\), an outcome model, and a response model. The shared-parameter factorization is

This factorization assumes conditional independence of \(Y\) and \(R\) given \(X,Z\). It also assumes that the chosen distributions and their parameters adequately describe the latent-variable distribution, the outcome, and response. These are substantive modeling assumptions; observed response rates alone do not establish that they are correct.

For \(R=1\), the observed-data contribution at \(Y=y\) is

For \(R=0\), the value \(y\) is not observed, so the contribution is

The response probability conditional on a particular outcome value is instead

The distribution \(f(z\mid y,x)\) generally differs across values of \(y\). Thus, even though response and outcome are conditionally independent given \(X,Z\), their response probability can vary with \(y\) given \(X\). If it does, the mechanism is MNAR. In this scalar setting, \(Y\) is unavailable precisely when \(R=0\), so dependence of response on \(Y\) cannot generally be checked directly from the observed data.

A Binary Latent-Variable Derivation

A two-category latent variable makes the induced dependence transparent. Suppose \(Z\) is either 0 or 1, and the fully observed predictors \(X\) are held fixed. Write \(\pi=P(Z=1\mid X)\), \(a_z=P(Y=1\mid Z=z,X)\), and \(q_z=P(R=1\mid Z=z,X)\). Under conditional independence, Bayes’ rule gives

If \(Y\) is binary, then

When \(Y\) informs us about \(Z\) and \(Z\) informs us about response, these posterior weights can differ by outcome value. The resulting response probabilities can differ too. In this model, if \(a_1=a_0\), \(Y\) provides no information about \(Z\); if \(q_1=q_0\), response does not vary by \(Z\). Either condition removes this particular path from \(Y\) to \(R\), though other direct dependencies could still make missingness MNAR.

Implications for the Observed Sample

When response depends on the outcome through \(Z\), respondents and nonrespondents can have different outcome distributions. Complete cases then need not represent the full outcome distribution, even if the outcome and response are independent within latent-variable strata. The following calculation uses the same hypothetical probabilities as above.

A Continuous Shared-Parameter Model

A useful continuous example separates latent variation in the outcome from measurement noise. Suppose \(Z\sim N(0,1)\), \(Y=\mu+\beta Z+\varepsilon\), and response is determined by a latent threshold: \(R=1\) when \(\delta+\gamma Z+u>0\), and \(R=0\) otherwise. Assume \(\varepsilon\sim N(0,\sigma^2)\), \(u\sim N(0,1)\), and that \(Z,\varepsilon,u\) are mutually independent. The response model is a probit model conditional on \(Z\): \(P(R=1\mid Z)=\Phi(\delta+\gamma Z)\), where \(\Phi\) is the standard normal cumulative distribution function.

These assumptions matter. The shared variable \(Z\) affects the mean of \(Y\) when \(\beta\ne0\), and affects response when \(\gamma\ne0\). Conditional independence of \(Y\) and \(R\) given \(Z\) follows from the independent disturbances \(\varepsilon\) and \(u\). The following derivation shows the induced response probability given \(Y=y\).

To see the posterior parameters, the conditional density is proportional to the prior density times the outcome density. The terms involving \(z\) in the exponent are

Collecting the quadratic and linear terms in \(z\) gives precision \(1+\beta^2/\sigma^2\) and linear coefficient \(\beta(y-\mu)/\sigma^2\). Completing the square therefore gives posterior variance \(1/(1+\beta^2/\sigma^2)=\sigma^2/(\beta^2+\sigma^2)\) and posterior mean \(\beta(y-\mu)/(\beta^2+\sigma^2)\), as stated.

Given \(Y=y\), the threshold quantity \(\delta+\gamma Z+u\) is normal with mean \(\delta+\gamma E(Z\mid y)\) and variance \(1+\gamma^2\operatorname{Var}(Z\mid y)\). Hence

When \(\beta\gamma\ne0\), this probability varies with \(y\), so the model is MNAR. If either coefficient is zero, this shared-\(Z\) pathway does not produce dependence of \(R\) on \(Y\).

Model Checks, Alternatives, and Limitations

A shared-parameter model is useful when a plausible latent characteristic would affect both the outcome and response. Its assumptions should be made explicit, because the latent structure is not directly observed. In practice, investigate whether the fitted model reproduces observed outcome distributions among respondents, response rates across observed predictors, and any available repeated-measure or auxiliary-variable patterns. Poor fit to these observed features is evidence against the specification. Good fit, however, cannot confirm the assumed dependence on unobserved outcomes.

Several warning signs deserve attention:

• Little variation in response by modeled latent predictors: if the fitted response model effectively has \(\gamma=0\), it does not capture the proposed shared pathway.
• Weak outcome information about the latent variable: if \(\beta\) is near zero, outcome values provide little information about \(Z\), weakening the induced response-outcome association.
• Extrapolation or near-zero response probabilities: estimates for poorly represented latent strata can be highly sensitive to distributional assumptions.
• Unstable estimates: a complex latent model can be weakly identified, especially when there are few observed outcomes or limited predictors. Convergence alone does not establish identification or model adequacy.

Shared-parameter models are one way to formulate MNAR. Selection models express response conditional on the outcome and observed predictors; pattern-mixture models specify outcome distributions within response patterns. These are different factorizations of the joint data structure, not interchangeable assumptions. The earlier material on selection models for MNAR and pattern-mixture models for MNAR introduces those formulations; the shared-parameter approach instead makes the latent common cause explicit.

The main limitation is that observed data generally cannot determine how outcomes among nonrespondents relate to their unobserved values. Different MNAR assumptions may be compatible with the same observed-data distribution. This is the nonidentifiability of MNAR mechanisms: a shared-parameter model imposes a structure that can support analysis, but it does not make that structure empirically verifiable from the observed outcomes and response indicators alone. Consequently, its conclusions depend on the latent-variable distribution, link functions, and parameter restrictions. Standard MICE conditional models do not automatically account for such a shared latent response mechanism; an MNAR analysis needs an explicit model or sensitivity assumptions that represent it.

A common mistake is to say, “The outcome and response are independent given the latent variable, so missingness is MAR.” That conclusion conditions on \(Z\), which is unobserved. The relevant question is whether \(P(R=1\mid Y,X)\) depends on \(Y\) after \(Z\) is averaged out. In the examples, it does; therefore, the mechanism is MNAR. Conditional independence given an unobserved variable is not the same as conditional independence given observed data.