Identified Functionals and Unidentified Functionals

What Does It Mean for a Quantity to Be Identified?

When an outcome is missing, the observed records do not reveal the complete-data distribution. The central question here is narrower than whether a model fits or an estimate can be computed: does the observed-data distribution determine the target quantity uniquely? This distinction lets us classify response rates and observed-case means separately from full-population means.

Consider a population with outcome \(Y\) and response indicator \(R\), where \(R=1\) means \(Y\) is observed and \(R=0\) means it is missing. The observed record consists of \(R\) and, when \(R=1\), the value of \(Y\). An observed-data distribution specifies the probabilities of these records. A complete-data law specifies the joint distribution of \(Y\) and \(R\), including outcomes for individuals with \(R=0\).

Identification is therefore relative to the model being considered. With no restrictions on how outcomes relate to response, the model includes all complete-data laws compatible with the observations. An assumption such as missingness being independent of the outcome can restrict that set and make a target identifiable under the restricted model. That conclusion depends on the assumption; it does not mean the assumption is established by the observed data.

Identification is also distinct from estimation. Identification asks whether the population target is uniquely determined in principle by the observed-data distribution. Estimation asks how to use a finite sample to estimate that target, with uncertainty. A sample can yield a numerical observed-case mean even when the full-population mean is not identified.

Quantities Determined Directly by Observed Data

The response rate is \(P(R=1)\). Since response status is recorded for everyone, the observed-data distribution determines it. When \(P(R=1)>0\), the observed-case mean is \(E(Y\mid R=1)\). It too is determined by the observed-data distribution: it uses only outcomes among respondents and their response status.

These statements are about the population quantities represented by the observed-data distribution. In a finite sample, the response proportion and respondent average are estimates of those quantities. Neither statement says that the observed-case mean equals the mean among everyone.

Why the Full-Population Mean Can Be Unidentified

Write \(r=P(R=1)\), let \(\mu_1=E(Y\mid R=1)\), and let \(\mu_0=E(Y\mid R=0)\), when these conditional means exist. Splitting the population according to response status gives the mixture identity

The observed-data distribution determines \(r\) and \(\mu_1\), but it does not generally determine \(\mu_0\): outcomes are unobserved when \(R=0\). If \(0<r<1\), changing the distribution of missing outcomes while keeping the respondent outcomes and response statuses unchanged preserves the observed-data distribution. If that change alters \(\mu_0\), it alters \(E(Y)\). This gives a direct way to establish nonidentification.

The argument depends on the target and the assumptions. If everyone responds, then \(r=1\) and the full mean equals the observed mean. If a substantive model imposes a restriction on missing outcomes, the full mean might be identified under that model. But without such a restriction, the observed-data distribution alone does not generally pin down the full-population mean.

Identification Under an Explicit Assumption

An additional assumption can make a functional identifiable relative to a smaller model. For example, suppose \(X\) is a fully observed categorical covariate and \(Y\) is observed only when \(R=1\). Consider the conditional independence assumption \(Y\perp R\mid X\): within each covariate group, the outcome distribution is the same for respondents and nonrespondents. If each group has a positive response probability and the relevant means exist, then

Averaging over the observed population distribution of \(X\) gives

Every term on the right is determined from the observed-data distribution when the group has respondents. Thus the mean is identified under the stated conditional independence and response-support assumptions. Without the conditional independence restriction, the respondent mean within a group need not equal the mean among all people in that group.

How to Classify a Proposed Functional

A practical classification begins by stating the target precisely and listing what the observed-data distribution reveals. Then ask whether two compatible complete-data laws can preserve all those observed quantities while changing the target. If they can, the target is not identified without additional restrictions. If the target is invariant across all compatible laws in the specified model, it is identified in that model.

A useful comparison is between direct observed-data quantities and quantities that require assumptions. Response rates and observed-case means are directly determined when the relevant response groups are present. A full-population mean may be identified under a specified missingness model, but that model-based conclusion must be reported with its assumptions. If the observed data and assumptions still permit several target values, point identification has not been achieved; partial-identification methods address a different question by describing a range of compatible values.

Diagnostics, Failure Modes, and Limitations

There is no diagnostic based only on the observed records that can reveal the actual outcomes of nonrespondents. A useful check is instead logical: write down what is observed, identify which part of the target depends on missing values, and determine whether the proposed assumptions constrain that part. Sensitivity to plausible changes in missing-outcome assumptions signals that conclusions rely on those assumptions, but it does not identify which assumption is true.

• Confusing a computable statistic with an identified target. A respondent average is computable, but it is not automatically the population average.
• Treating a response rate as information about missing outcomes. Knowing how many outcomes are missing does not reveal their values or distribution.
• Calling an assumption-identified quantity identified without qualification. State the restriction that supports the conclusion and distinguish it from identification using the observed-data distribution alone.
• Ignoring unavailable groups. A formula using respondent means within covariate groups requires observed outcomes in the groups being averaged over; otherwise those terms cannot be obtained from respondent data.

Pattern-mixture and selection models can express restrictions on how outcomes and response are related in different ways. Choosing a parameterization does not, by itself, make a target identified: identification depends on whether the restrictions distinguish among complete-data laws that otherwise fit the same observed records. Likewise, a close fit to observed data cannot settle how the unobserved outcomes are distributed.