Nonidentifiability Of MNAR Mechanisms

Why the Response Mechanism May Not Be Identifiable

When a value is missing, the data record whether it was observed, but not what the value would have been. This creates a specific inferential problem: the distribution of observed records may be compatible with more than one relationship between the missing value and the probability of response. The issue is not merely that a particular model is difficult to estimate. Without additional information or restrictions, the observed distribution may not determine the response mechanism at all.

Consider one outcome \(Y\) and an indicator \(R\), where \(R=1\) means \(Y\) is observed and \(R=0\) means it is missing. For clarity, begin with a population of independent units and no other variables. Let \(f(y)\) denote the population distribution of \(Y\), and let \(\pi(y)=P(R=1\mid Y=y)\) denote the response probability at value \(y\).

In this one-variable setting, an MNAR mechanism has \(\pi(y)\) varying with \(y\). The observed records reveal the overall response rate and the distribution of \(Y\) among responders. They do not directly reveal the distribution of \(Y\) among nonresponders, which is where the competing explanations can differ.

The Observable Part Is a Product

For a unit with \(R=1\) and observed value \(Y=y\), the joint density or probability mass is the product of the probability of that value and the probability of responding at that value. Write this observable response subdistribution as \(h(y)\).

For continuous \(Y\), the equality is an equality of densities; for discrete \(Y\), it is an equality of probability masses. The response rate is the total mass of this subdistribution, and the distribution of values among responders is the normalized subdistribution:

The observed data identify \(h(y)\), or equivalently the response rate \(\rho\) and the respondent distribution \(g(y)\), in the population. But the equation \(h(y)=f(y)\pi(y)\) gives only a product. It does not, by itself, determine which factor is the full-data distribution and which is the response probability. For nonresponders, the observed probability is only the total missing mass:

This quantity does not report how that mass is divided across possible values of \(Y\). The same missing fraction can therefore arise from very different distributions of values among nonresponders.

A General Construction and Its Limits

The binary example is not a special trick. The same factorization applies to any outcome distribution. Let \(h(y)\) be the observable subdensity or sub-probability mass for responders. A candidate full-data distribution \(f(y)\) must satisfy \(f(y)\geq h(y)\) pointwise, because \(h(y)=f(y)\pi(y)\) and a response probability cannot exceed one. If \(f\) is a valid distribution, define

Then \(0\leq\pi(y)\leq1\), and \(f(y)\pi(y)=h(y)\), so this candidate reproduces the observed responder data. The remaining probability mass has total \(1-\rho\), since

There can be many candidate distributions \(f\) satisfying these conditions. Each can imply a different response mechanism and a different distribution of missing values, while retaining exactly the same observed subdistribution. The construction does not say that every imaginable response function is compatible with the data: the implied \(f\) must be a valid distribution, and the implied \(\pi\) must remain between zero and one. Nonidentifiability means that multiple valid possibilities remain, not that constraints disappear.

Conditioning on Observed Covariates Does Not Automatically Resolve It

A fully observed covariate can help describe differences in response rates and respondent distributions across groups. But observing and conditioning on such a covariate does not, on its own, reveal the missing outcome values within each group. The same factorization applies separately at each covariate value: observed responder information constrains the product of the outcome distribution and response probability, not necessarily either factor individually.

Justification: What the Observed Likelihood Can Contain

The identification problem can also be seen from the contribution each unit makes to the observed-data distribution. A responder with value \(y\) contributes \(f(y)\pi(y)=h(y)\). A nonresponder contributes the probability of \(R=0\), with the unobserved value summed or integrated out:

Thus the observed-data distribution contains the responder subdistribution \(h\) and the total nonresponse probability \(1-\rho\). The latter is already determined by the total mass of \(h\). Neither contribution separately reveals \(f\) and \(\pi\). This argument is a direct justification for the nonidentifiability claim: if two valid pairs \((f,\pi)\) have the same product \(h\), their responder contributions agree; if they also have the same \(\rho=\int h\), their nonresponder contributions agree as well. Their full observed-data distributions are therefore identical.

The conclusion concerns identification in the population, not just small-sample precision. Collecting more observations from the same observed-data distribution can estimate that distribution more accurately, but it cannot distinguish models that imply exactly the same distribution of observed records.

Diagnostics, Alternatives, and Practical Limits

Observable checks remain useful, but their scope must be stated correctly. One can examine whether response rates vary with observed predictors, whether a model reproduces observed response patterns, or whether the distribution of observed outcomes is represented adequately. Such checks can reveal a model that fails to describe observable features. They cannot establish how the response probability varies with values that were not recorded.

For example, finding that response rates vary with an observed covariate is evidence against MCAR in the relevant setting. Finding no such association does not establish that response is independent of the missing outcome. A hidden dependence on that outcome can remain compatible with the observed response rates and respondent values. This is the central limitation of observable diagnostics for MNAR mechanisms.

The principal alternative is not a data-only test that selects the correct MNAR explanation. It is to add information or restrictions and make clear what they contribute. Possibilities include a validation or follow-up sample that obtains values for some initial nonresponders, external data that constrain the full outcome distribution, or explicit assumptions narrowing the allowed response mechanisms. In the absence of such information, analysts can examine conclusions under multiple plausible assumptions rather than treating one mechanism as established by the observed data.

A common mistake is to reason, “The response rate is known, and the respondent distribution is known, so the response mechanism is known.” The correction is that these determine \(h(y)=f(y)\pi(y)\), not its two factors. A second mistake is to interpret a well-fitting model for response given observed predictors as proof that the response mechanism is not MNAR. Good fit to observed quantities checks observable implications; it does not reveal the unseen values among nonresponders. Nonidentifiability is a limitation on what can be learned from the observed distribution alone, not a claim that every model is equally plausible for substantive reasons.