Observed-Data Distributions and Identification Targets

Why Separate the Observed Law From the Complete-Data Law?

A study with missing outcomes produces a distribution of what was actually recorded, not a direct sample from the distribution that would have been available if every outcome had been measured. Keeping those distributions distinct is essential. A response rate and an outcome distribution among respondents are features of the observed data. The overall outcome distribution, which includes outcomes that were not observed, is a feature of the complete data and may require assumptions beyond the recorded information.

This distinction is not a technicality about notation. It determines what the data alone can establish. The observed-data distribution describes the probabilities of observed records. A complete-data target, such as the population mean outcome, is identified only if its value is determined by that observed distribution under the model assumptions being used.

Defining the Observed-Data Distribution

Let \(X\) denote fully observed covariates, \(Y\) an outcome that may be missing, and \(R\) its response indicator, with \(R=1\) if \(Y\) is observed and \(R=0\) otherwise. A complete-data record is \((X,Y,R)\). The recorded outcome can be written \(Y_{\mathrm{obs}}\): it equals \(Y\) when \(R=1\) and is marked missing when \(R=0\). Thus an observed-data record is \(O=(X,R,Y_{\mathrm{obs}})\). This definition does not require that missingness be random or that any particular model for \(R\) hold.

For a discrete outcome, the observed-data probabilities follow directly from marginalization. For each covariate value \(x\), the probability of observing \(Y=y\) with \(R=1\) is the joint probability \(P(X=x,Y=y,R=1)\). The probability of \(X=x\) and \(R=0\) combines all possible values of the missing outcome:

For a continuous outcome, replace the sum over missing \(y\) by an integral. More generally, if \(R\) indexes a pattern of observed and missing outcomes and \(Y_R\) denotes the components observed in pattern \(R\), then the observed-data probability or density is obtained by integrating the complete-data joint law over the missing components \(Y_{\bar R}\):

For discrete missing components, the integral is replaced by a sum. The resulting observed-data law includes the probabilities of response patterns as well as the distributions of outcomes observed within those patterns. It does not include the unrecorded values themselves.

What Does “Identified” Mean Here?

Identification is a statement about what follows from the observed-data distribution under a specified set of assumptions. It is not a statement about whether a particular sample is large or whether an estimator is precise. To make the statement precise, let \(\mathcal{M}\) be the set of complete-data distributions allowed by the model assumptions. Each law in \(\mathcal{M}\) induces an observed-data distribution by marginalization.

The observed-data distribution itself is, by definition, the distribution of the recorded record. Its features are therefore the natural starting point for identification. Examples include the probability of each response pattern and the distribution of \(Y\) among records with \(R=1\). A complete-data target such as \(P(Y=1)\) or \(E(Y)\) is different: it includes contributions from outcomes when \(R=0\). Whether that target is identified depends on whether the model assumptions determine those contributions from the observed law.

This definition is relative to \(\mathcal{M}\). A target can be unidentified when only the observed-data law is specified, yet identified under a more restrictive model. The identifying restriction is then an assumption added to the observed information; it is not another observed-data feature. The distinction between what the records determine and what a model stipulates should remain explicit.

Deriving the Marginalization and Reading Its Consequences

The observed-data formula is a direct consequence of adding the probabilities of complete records that produce the same observed record. For example, when \(X=x\) and \(R=0\), the recorded information does not distinguish among the possible values of \(Y\). The disjoint events \(\{X=x,Y=y,R=0\}\), over all possible \(y\), make up the event \(\{X=x,R=0\}\). Additivity of probability gives:

The same argument applies to multiple missing outcomes: the observed record corresponds to all complete records that agree on the observed components, and their probabilities are combined. This derivation explains both what marginalization retains and what it loses. The observed law retains the total probability assigned to each observed record, but it does not reveal how that probability is divided among complete records that differ only in missing outcomes.

In an independent sample, the likelihood for the observed records is built from their probabilities under \(P_{\mathrm{obs}}\). A record with a missing outcome contributes the probability of its observed pattern and covariates, with the missing outcome marginalized out. It does not contribute a probability evaluated at an unobserved value. This is why an observed-data likelihood is not simply the complete-data likelihood with blank outcomes treated as known values.

Covariates and Pattern-Specific Features

When covariates are observed for everyone, the observed-data law can describe response and outcome distributions conditionally on \(X\). These conditional features can be important: response rates may differ across covariate groups, and the respondent outcome distribution may also differ across groups. Conditioning does not, by itself, turn a respondent distribution into the outcome distribution for everyone in that group.

Assumptions, Diagnostics, and Common Mistakes

The mapping from the complete-data law to the observed-data law is definitional. It requires no missing-at-random assumption. Assumptions become necessary when the goal is to relate the observed law to features of the complete-data law that include unobserved outcomes. Before interpreting such a target, state the target and the restrictions being used. The earlier material on factorizing joint distributions helps express a joint law in a convenient order; changing the factorization does not change which records were observed.

Useful checks begin with the observed record itself. Verify that the response indicator and missingness codes are defined consistently, tabulate response patterns, and check that each reported outcome summary uses the correct observed subgroup as its denominator. With covariates, inspect pattern frequencies within covariate groups; an empty observed pattern in a group can prevent estimation of features that require that pattern. These checks can reveal coding problems or sparse data. They cannot verify the values of outcomes that were never recorded.

A particularly common mistake is to reason, “The mean among respondents is 5, so the population mean is 5.” The correction is to name the conditional target: the calculation estimates \(E(Y\mid R=1)\), not necessarily \(E(Y)\). Equating these quantities needs a justified assumption that connects response and outcome, or other identifying information. A second mistake is to treat an assumption as if it were a feature visible in the observed-data distribution. Assumptions can restrict the model and thereby identify a target, but their validity is not established merely because the observed records fit the model.

The main alternatives differ in what they target and how they bridge missing information. A complete-case summary uses only records with observed outcomes and directly describes that selected subset. An observed-data likelihood uses the full distribution of recorded patterns and outcomes, with missing components marginalized. A model-based complete-data analysis may additionally impose assumptions about the relation between response and the unobserved outcome. These approaches should not be confused: a more elaborate model does not change the observed-data law, though it may make additional targets identifiable conditional on its assumptions.