Observed-Data Equivalence Of MAR And MNAR

Why Different Missingness Models Can Look the Same

When an outcome is missing, the record tells us that it was not observed, but not what its value would have been. That distinction can make two different explanations of missingness indistinguishable from the data we actually see. In particular, a model in which missingness does not depend on the unobserved outcome and a model in which it does can assign the same probabilities to every possible observed record.

The example here uses one binary outcome and no auxiliary variables. This deliberately simple setting makes the source of the ambiguity visible: for a person whose outcome is missing, the observed record does not reveal whether the outcome was 0 or 1. The equality of observed-data probabilities is a mathematical feature of the example, not evidence that either missingness assumption is true.

Definition: Let \(Y\in\{0,1\}\) be a binary outcome and let \(R=1\) indicate that \(Y\) is observed, with \(R=0\) indicating that it is missing. A missingness model is MAR if the probability distribution of \(R\), conditional on the observed information, does not depend on the missing values. It is MNAR if that distribution still depends on a missing value after conditioning on the observed information. With no other variables in this example, MAR requires the response probability not to depend on \(Y\); a response probability that differs by \(Y\) is MNAR.

For each unit, the observable record has three possible categories: \(R=1,Y=0\), \(R=1,Y=1\), or \(R=0\). In the third category, \(Y\) is unknown. Thus, the observed record does not distinguish between a missing outcome of 0 and a missing outcome of 1. The full-data distribution has four categories, since it specifies both \(Y\) and \(R\); the observed-data distribution combines two of those categories into the single record \(R=0\).

Assume units are sampled from the same population, each has one outcome and one response indicator, and there are no other measured variables. For the numerical examples, probabilities describe this population; the sample-likelihood example additionally assumes independent sampling across units. These assumptions simplify the illustration. They do not make the missing outcomes observable.

Two Models With Identical Observable Probabilities

Suppose the observable probabilities are \(0.32\) for a recorded 0, \(0.18\) for a recorded 1, and \(0.50\) for a missing outcome. They sum to 1. The following two full-data models both produce exactly those probabilities.

Worked Example: A MAR Model and an MNAR Model

MAR model. Suppose \(P(Y=1)=0.36\) and the response probability is \(P(R=1\mid Y=0)=P(R=1\mid Y=1)=0.50\). Because response does not depend on \(Y\), this model is MAR (and, in this no-covariate setting, also MCAR). The observable probabilities are

P(R=1,Y=0)=P(Y=0)P(R=1\mid Y=0) =(1-0.36)(0.50)=0.64(0.50)=0.32,

P(R=1,Y=1)=P(Y=1)P(R=1\mid Y=1) =(0.36)(0.50)=0.18, \qquad P(R=0)=1-0.32-0.18=0.50.

MNAR model. Instead, suppose \(P(Y=1)=0.60\), \(P(R=1\mid Y=1)=0.30\), and \(P(R=1\mid Y=0)=0.80\). Both response probabilities are between 0 and 1, so they define a valid model. Since response depends on the value of \(Y\), which is unobserved when \(R=0\), this model is MNAR. Its observable probabilities are

P(R=1,Y=1)=(0.60)(0.30)=0.18, \qquad P(R=1,Y=0)=(0.40)(0.80)=0.32,

$$ P(R=0)=P(Y=1,R=0)+P(Y=0,R=0) =(0.60)(0.70)+(0.40)(0.20)=0.42+0.08=0.50. $$

Both models therefore assign probability \(0.32\) to a recorded 0, \(0.18\) to a recorded 1, and \(0.50\) to a missing outcome. Yet they imply different outcome prevalences: \(0.36\) under the MAR model and \(0.60\) under the MNAR model. They also imply different distributions among missing outcomes. The observed records alone do not show which full-data distribution generated them.

Key result: Two full-data models are observed-data equivalent when they assign the same probability to every possible observable record. Observed-data equivalence does not require the models to agree about missing outcome values, the prevalence of \(Y\), or the dependence of response on \(Y\).

Deriving the Range of Compatible Full-Data Models

The example can be generalized without choosing a particular missingness model. Write \(a=P(R=1,Y=1)\), \(b=P(R=1,Y=0)\), and \(c=P(R=0)\), where \(a+b+c=1\). The observed data reveal \(a\), \(b\), and \(c\). They do not reveal how much of the missing group had \(Y=1\).

Let \(u=P(Y=1,R=0)\). The remaining missing probability must then be \(P(Y=0,R=0)=c-u\). To form valid probabilities, \(u\) must satisfy \(0\leq u\leq c\). The full-data probabilities and outcome prevalence are consequently

\begin{aligned} P(Y=1,R=1)&=a, & P(Y=0,R=1)&=b,\\ P(Y=1,R=0)&=u, & P(Y=0,R=0)&=c-u,\\ P(Y=1)&=a+u. \end{aligned}

Every allowed value of \(u\) gives the same observed probabilities \(a\), \(b\), and \(c\), because the two missing-outcome probabilities combine to \(u+(c-u)=c\). But changing \(u\) changes the full-data prevalence \(P(Y=1)=a+u\). This is the basic algebra behind observed-data equivalence.

Worked Example: A Continuum of Compatible Prevalences

Suppose the observed probabilities are \(a=0.18\), \(b=0.32\), and \(c=0.50\). Any \(u\) between 0 and 0.50 gives a valid full-data distribution. If \(u=0.18\), then the missing group has \(0.18\) probability of \(Y=1\) and \(0.32\) probability of \(Y=0\), so \(P(Y=1)=0.18+0.18=0.36\). This is the MAR model from the first example: within the missing group, the proportion with outcome 1 is \(0.18/0.50=0.36\), the same as among respondents.

If \(u=0.42\), then \(P(Y=1)=0.18+0.42=0.60\), and \(P(Y=0,R=0)=0.50-0.42=0.08\). This is the MNAR model already constructed. Both choices give the same observable probabilities, but they produce different values of the prevalence. Across all allowed allocations, the prevalence ranges from \(a=0.18\) to \(a+c=0.68\). The observed records alone do not select a particular value in that range.

Why the Observed-Data Likelihood Cannot Choose Between Them

The same equivalence appears in a sample. A likelihood based only on observed records uses the probability of each observable category, not the unobserved split of \(c\) into \(u\) and \(c-u\). If \(n_{10}\) outcomes are recorded as 0, \(n_{11}\) are recorded as 1, and \(n_0\) are missing, the likelihood, apart from a factor common to the models, is

L \propto b^{n_{10}}a^{n_{11}}c^{n_0}.

There is no separate factor for the number of missing outcomes that were actually 1, because that number is not in the observed data. Therefore, if two models have the same \(a\), \(b\), and \(c\), the observed-data likelihood is identical under them.

Worked Example: Identical Likelihood for a Hypothetical Sample

Imagine a hypothetical independent sample of 200 units with 64 recorded outcomes of 0, 36 recorded outcomes of 1, and 100 missing outcomes. The observed proportions are \(64/200=0.32\), \(36/200=0.18\), and \(100/200=0.50\), matching both models above. Under either model, the observed-data likelihood, omitting the same multinomial coefficient, is

L\propto (0.32)^{64}(0.18)^{36}(0.50)^{100}.

Its log is \(64\ln(0.32)+36\ln(0.18)+100\ln(0.50)\), which is approximately \(-203.97\). This value is the same for the MAR and MNAR models because the observable cell probabilities are the same. The sample provides no likelihood-based preference between these two particular models when only these observed categories are used. It does not follow that the missing outcomes are equally likely to be 0 or 1, or that the two models imply the same prevalence.

Interpretation, Diagnostics, and Limits

The important conclusion is about what the observed data can identify. In this example, observed records identify the three observable category probabilities, but do not identify the distribution of \(Y\) among those with \(R=0\). A MAR assumption supplies one way to fill in that missing distribution; an MNAR model can supply another. Choosing between them requires assumptions or information beyond the observed categories used here.

Diagnostics based only on the observed records can check features of those records, such as the recorded response proportion or the distribution of \(Y\) among respondents. They cannot reveal the actual \(Y\) values for nonrespondents. Thus, even perfect agreement between an observed-data model and the records does not verify MAR or rule out MNAR. This is consistent with the earlier material on why observed data cannot verify MAR and on nonidentifiability of MNAR mechanisms.

The example is intentionally limited. It has one binary outcome and no observed predictors; real analyses may have several variables, covariates, or missingness patterns. Those features can change the model structure, but they do not make unobserved values directly observable. In practice, conclusions that depend on the missing outcomes should be examined under defensible alternative assumptions, rather than treating a fit to observed records as proof of a particular mechanism.

Common pitfall: “The response rate and the observed outcome proportions fit a MAR model, so the data show that missingness is MAR.” This is incorrect. The MNAR model above fits those same observable probabilities exactly. The correction is that observed-data fit can be compatible with MAR without distinguishing it from MNAR.

The central comparison is not between a model that fits and one that fails to fit. It is between different full-data explanations that are observationally indistinguishable in the available records. This is why assumptions about missing values matter: they can determine conclusions that the observed data alone do not settle.

Check Your Understanding

Answer each question using the observable-category probabilities and the allocation \(u=P(Y=1,R=0)\). Solutions follow the questions.

If \(P(R=1,Y=1)=0.25\), \(P(R=1,Y=0)=0.35\), and \(P(R=0)=0.40\), what is the range of compatible values of \(P(Y=1)\)?
For these same observed probabilities, set \(u=0.20\). Find \(P(Y=1)\), \(P(Y=0,R=0)\), and the full-data probability of each of the four \((Y,R)\) combinations.
Why can no diagnostic based only on the three observed categories determine whether the missingness mechanism is MAR or MNAR in this example?
Two models assign probabilities \(0.25\), \(0.35\), and \(0.40\) to the observable categories. For a sample with 50 recorded 1s, 70 recorded 0s, and 80 missing outcomes, write their observed-data likelihoods, omitting the common multinomial coefficient. Are they different?

Solutions. (1) Here \(a=0.25\) and \(c=0.40\), so \(0\leq u\leq0.40\) and \(P(Y=1)=a+u\). The compatible prevalence ranges from \(0.25\) to \(0.65\). (2) With \(u=0.20\), \(P(Y=1)=0.25+0.20=0.45\), and \(P(Y=0,R=0)=0.40-0.20=0.20\). The four probabilities are \(P(Y=1,R=1)=0.25\), \(P(Y=0,R=1)=0.35\), \(P(Y=1,R=0)=0.20\), and \(P(Y=0,R=0)=0.20\); they sum to 1. (3) The diagnostic sees only the combined missing probability \(u+(c-u)=c\), not its division between missing 1s and missing 0s. Different divisions, including ones compatible with MAR and MNAR models, produce the same observable records. (4) Under either model the likelihood is proportional to \((0.25)^{50}(0.35)^{70}(0.40)^{80}\). The likelihoods are identical because each model gives the same probability to every observed category.