A Pitfall: Treating MAR Or MNAR As A Dataset Property

Why the Label Needs a Reference

A follow-up dataset is often described simply as “MAR” or “MNAR.” That shorthand hides an essential question: MAR or MNAR for which outcome, relative to which observed information? A response process can depend on a prior outcome that is available for everyone. It can then be MAR when that history is conditioned on, yet look nonignorable if the analysis leaves the history out. Likewise, changing the outcome from a detailed score to a coarser category can change which dependencies matter.

The classification is a statement about a specified probability model, not a permanent feature stamped onto a file of records. To make the statement precise, name the outcome being treated as potentially missing, the response indicator or pattern, and the variables included in the conditioning set. The earlier material on missingness mechanisms establishes the relevant MAR and MNAR ideas; the emphasis here is on keeping their reference specification explicit.

Definition: Let \(Y\) denote the outcome under study, \(R\) its response indicator or response pattern, and \(X\) a specified set of conditioning variables. Write \(Y_{\mathrm{obs}}(R)\) and \(Y_{\mathrm{mis}}(R)\) for the components of \(Y\) observed and missing under pattern \(R\). The mechanism is MAR relative to \(Y\) and \(X\) if, for each response pattern with relevant support, the conditional distribution of \(R\) given \(Y\) and \(X\) depends on \(Y\) only through \(Y_{\mathrm{obs}}(R)\). It is MNAR relative to that specification if this condition fails because the mechanism depends on a component missing under the pattern, even after conditioning on the specified observed information.

This is a conditional definition: the variables in \(X\) matter. In a longitudinal study, \(X\) might include baseline covariates, treatment group, and observed earlier outcomes. A variable can be used as conditioning information only when it is observed for the relevant records and is part of the stated analysis specification. A response process depending on \(X\) can still be MAR conditional on \(X\); MAR does not mean that response is unrelated to every measured variable.

The labels are also relative to the chosen outcome. If the target is a single final measurement, an earlier measurement may be treated as conditioning history. If the target is an entire trajectory, that same earlier measurement is a component of the outcome vector—but it can still be observed when the later component is missing. What matters is whether the response probability depends on the missing component after the observed components and \(X\) have been taken into account.

One Follow-Up Process, Two Classifications

Consider one hypothetical two-visit study. Let \(H\) indicate whether a participant had a high symptom score at the first visit, and let \(Y\) indicate whether the score is high at the second visit. Everyone has \(H\) recorded; \(Y\) is recorded only when \(R=1\). Suppose the response process and outcome association follow these probabilities:

First-visit history \(H\)	\(P(Y=1\mid H)\)	\(P(R=1\mid H,Y)\)
0 (not high)	0.20	0.80 for either value of \(Y\)
1 (high)	0.80	0.40 for either value of \(Y\)

Assume \(P(H=0)=P(H=1)=0.50\). These are hypothetical probabilities defining a mechanism, not estimates from a real study. Within each value of \(H\), the chance of response is the same for both values of the current outcome. Thus, conditional on \(H\), response does not depend on the potentially missing \(Y\).

Worked Example: Omitting Observed History Changes the Label

Classify the response process first conditional on the observed history \(H\), then for an analysis of \(Y\) that does not condition on \(H\).

Conditional analysis. For \(H=0\), \(P(R=1\mid H=0,Y=0)=P(R=1\mid H=0,Y=1)=0.80\). For \(H=1\), both corresponding probabilities are 0.40. Within either history stratum, response is unchanged across values of \(Y\). The mechanism is therefore MAR for the potentially missing second-visit outcome conditional on \(H\).

Outcome-only analysis. Marginally, \(P(Y=1)=0.50(0.20)+0.50(0.80)=0.50\). By averaging over the history distribution conditional on each outcome,

P(R=1\mid Y=1) =\frac{0.50(0.20)(0.80)+0.50(0.80)(0.40)}{0.50} =\frac{0.24}{0.50} =0.48.

Similarly,

P(R=1\mid Y=0) =\frac{0.50(0.80)(0.80)+0.50(0.20)(0.40)}{0.50} =\frac{0.36}{0.50} =0.72.

The response probabilities differ by \(Y\) when \(H\) is omitted: 0.48 versus 0.72. Relative to an outcome-only specification that excludes \(H\), the response process is not MAR. This calculation does not show that the underlying process changed; it shows that removing observed history from the conditioning set changed the classification.

The Marginalization Behind the Contrast

The contrast follows from averaging over a variable that predicts both response and outcome. If response depends on \(H\), then the outcome-only response probability is

P(R=1\mid Y=y) =\sum_h P(R=1\mid Y=y,H=h)\,P(H=h\mid Y=y).

Even when \(P(R=1\mid Y=y,H=h)\) is constant in \(y\) within each \(h\), the marginal probability can vary with \(y\), because \(P(H=h\mid Y=y)\) varies with \(y\). In the example, high current symptoms are more common when the prior history is high, and response is lower in that history group. Consequently, high current symptoms have a lower marginal response rate. This is a standard consequence of marginalization, not evidence that response directly depends on the unobserved current symptom after history is held fixed.

For a longitudinal outcome vector \((H,Y)\), the same process can also be described as MAR: \(H\) is observed for everyone, and the missingness of \(Y\) depends only on that observed component. The classification is not contradicted by calling the process non-MAR for the narrower outcome-only specification that excludes \(H\). Those statements refer to different conditioning sets.

Worked Example: Re-describing the Same Follow-Up Data

Use the hypothetical two-visit process above. An investigator defines the target as the second-visit endpoint \(Y\), but includes the first-visit indicator \(H\) as observed history. A second investigator uses the same endpoint but does not include \(H\).

For the first specification, the relevant comparison is within history strata. The response probability is 0.80 for \(H=0\), whether \(Y=0\) or \(Y=1\), and 0.40 for \(H=1\), whether \(Y=0\) or \(Y=1\). Hence the process is MAR relative to \(Y\) conditional on \(H\).

For the second specification, the relevant probabilities are the marginal values already calculated: \(P(R=1\mid Y=1)=0.48\) and \(P(R=1\mid Y=0)=0.72\). They differ, so the process is non-MAR for \(Y\) under that outcome-only specification. The two investigators have not found different properties in the records. They have made different conditional statements.

The distinction matters for an analysis because leaving out a useful observed predictor may change the assumptions needed to describe the missingness process and may also discard information relevant to the outcome. Calling one analysis “MAR” does not, by itself, establish that its outcome model or its conditioning choices are adequate.

Changing the Outcome Can Change the Question

The outcome definition determines which distinctions the analysis retains. Suppose a hypothetical clinical score \(Y\) takes values 0, 1, 2, or 3, each with probability 0.25. A score is observed with probabilities 0.20, 0.80, 0.80, and 0.20, respectively. Let \(R=1\) mean the score is observed. The response probability plainly varies with the exact score, so response depends on the potentially missing \(Y\): it is MNAR relative to this four-level score with no additional observed conditioning variables.

Worked Example: Coarsening a Score Changes Its Classification

Now change the outcome to the binary endpoint \(B=I(Y\geq 2)\), and suppose \(B\) is missing whenever the score is missing. Calculate the response probability in each binary category.

For \(B=0\), the possible scores are 0 and 1. They have equal probability, so

P(R=1\mid B=0)=\frac{0.20+0.80}{2}=0.50.

For \(B=1\), the possible scores are 2 and 3, also equally probable, giving

P(R=1\mid B=1)=\frac{0.80+0.20}{2}=0.50.

Thus, for the binary outcome \(B\), response is independent of the missing binary value in this specified distribution; the mechanism is MAR (indeed, independent of \(B\)). For the detailed score \(Y\), it is MNAR. The binary analysis has not recovered the missing score information: it has asked a coarser question that does not distinguish scores within each category. This example also relies on the stated equal probabilities within categories; different score distributions could yield different binary-category response rates.

What the Data Can and Cannot Diagnose

Observed data can reveal useful patterns in response rates across recorded history, but they generally cannot determine whether response depends on the values that are missing. For participants with \(R=0\), the missing \(Y\) is not available to compare directly with their response status. The probability calculations above describe hypothetical full-data mechanisms; in a real study, their dependence on missing outcomes cannot usually be checked from the observed records alone.

Diagnostics should therefore be used to assess the plausibility and adequacy of a specified model, not to certify MAR. Useful checks include:

Tabulate response rates across observed prior outcomes, baseline covariates, and other scientifically relevant history. Sparse or empty groups make conditional comparisons unstable or impossible.
Check whether the chosen conditioning variables are recorded for both respondents and nonrespondents, and whether their timing makes them available for the intended analysis.
Look for omitted predictors that could be associated with both response and the outcome. A strong association is a reason to examine the conditioning set, not proof of MNAR.
Assess whether the response mechanism has adequate overlap. If a history group has essentially no respondents or no nonrespondents, comparisons and model-based adjustments may rely on unsupported extrapolation.

These checks have limits. Similar observed response rates across recorded groups do not rule out dependence on missing outcomes. Conversely, response rates that differ across observed history do not establish MNAR: such differences are compatible with MAR when the history is included. A plausible MAR specification is an assumption supported by subject-matter knowledge and diagnostics, not a fact demonstrated by a nonsignificant test.

Comparison, Limitations, and a Common Pitfall

The main alternatives are not interchangeable labels for the same specification. MCAR is the stronger claim that response is independent of the outcome (and any variables covered by the stated MCAR definition); MAR permits dependence on observed outcomes and conditioning information; MNAR allows dependence on missing outcomes after the specified observed information is accounted for. In practice, analysts may choose a MAR-based likelihood analysis or use an explicitly MNAR model as a sensitivity analysis. The classification alone does not select a model, guarantee correct estimation, or settle which assumptions are scientifically credible.

A common mistake is: “Dropout is associated with the final outcome in the data, so this dataset is MNAR.” The correction is that an observed association, or an association under a marginal analysis, does not identify dependence on the missing outcome conditional on relevant observed history. In the worked follow-up process, the marginal response rates were 0.48 and 0.72 by current outcome, yet response was independent of that outcome within prior-history strata. The mechanism was MAR conditional on that observed history.

A second mistake is to say “the dataset is MAR” without naming the target outcome or conditioning set. The correction is to write a complete claim, such as “the response process for the second-visit symptom indicator is MAR conditional on the first-visit symptom indicator and baseline covariates.” If the target, available history, or conditioning variables change, revisit the classification. A different classification does not necessarily mean the records or the response process changed; it may mean the question changed.

Finally, adding variables to a conditioning set is not automatically beneficial. Variables must be observed where needed, appropriately timed, and supported by the data; careless conditioning can complicate interpretation or require extrapolation. The point is not to condition on everything, but to state and justify the information relative to which the mechanism is being described.

Key takeaway: MAR and MNAR are relative classifications. Always specify the outcome, response pattern, and conditioning information before assigning a label. Observed-data diagnostics can challenge a specification, but cannot generally prove that dependence on missing outcomes is absent.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Response depends on a recorded prior score \(H\), but not on the missing current outcome \(Y\) within values of \(H\). How should the mechanism be classified for \(Y\) conditional on \(H\)?
In the two-visit example, why did the outcome-only response rates differ by current outcome even though response did not depend on \(Y\) within history strata?
A mechanism is MAR for a continuous score but independent of a binary category formed from that score. Does this necessarily contradict itself?
Can a table of response rates by observed history establish that a mechanism is MAR? Give one reason for your answer.

Solutions. (1) MAR relative to \(Y\) conditional on \(H\), because response depends on observed history and not on \(Y\) after conditioning on that history. (2) The distribution of \(H\) differs across values of \(Y\), and response rates differ across \(H\); averaging over \(H\) therefore gives different marginal response rates. (3) No. The classifications can differ because the binary category retains less outcome information; the classification is relative to the specified outcome. (4) No. Such a table can show response patterns across observed variables, but missing outcome values are unavailable for nonrespondents, so it cannot generally test whether response depends on those missing values.