Why an Observed Outcome Can Matter
Suppose a study records an outcome for every participant but fails to record a predictor for some participants. The chance that the predictor is missing may differ with the outcome that was recorded. That dependence does not, by itself, make the predictor’s missingness nonignorable: if response depends on the observed outcome but not on the missing predictor value once the outcome is held fixed, the mechanism is MAR.
This distinction is conditional. Comparing participants with different outcomes can reveal different predictor response rates. The key question is whether, within a given outcome value, predictor response still depends on the predictor value that is unavailable for nonresponders.
This definition assumes that \(Y\) really is observed for every participant in the analysis and that the relevant response probabilities are defined for the outcome values being considered. If \(Y\) is also missing, it cannot simply be treated as an observed conditioning variable for every record. The definition also concerns the response process, not whether \(X\) and \(Y\) are associated in the population.
Conditional Response Rates: A Worked Check
For a binary predictor and binary outcome, the condition can be checked conceptually by comparing response rates across predictor values separately within each outcome category. Equal rates across \(X\) within each \(Y\) category are consistent with MAR in that table; rates that differ by \(X\) within a category indicate dependence on the missing predictor value.
Worked Example: Response Depends on the Observed Outcome
Consider a hypothetical study of a binary outcome \(Y\) and a binary predictor \(X\). The outcome is recorded for everyone. The table gives the number with \(X\) observed out of the number in each \(X,Y\) group.
| Outcome \(Y\) | Predictor \(X\) | \(X\) observed | Total | Response rate |
|---|---|---|---|---|
| 0 | Low | 8 | 10 | 0.80 |
| 0 | High | 16 | 20 | 0.80 |
| 1 | Low | 6 | 10 | 0.60 |
| 1 | High | 12 | 20 | 0.60 |
For \(Y=0\), the response rates are \(8/10=0.80\) for low \(X\) and \(16/20=0.80\) for high \(X\). For \(Y=1\), they are \(6/10=0.60\) and \(12/20=0.60\), respectively. Thus, in each outcome category, the rates do not vary by \(X\). But they do vary by \(Y\): response is lower when \(Y=1\).
The pattern is consistent with a MAR mechanism in which response depends on the observed outcome. For example, the response probabilities are \(P(R_X=1\mid Y=0)=0.80\) and \(P(R_X=1\mid Y=1)=0.60\), with no further dependence on \(X\) in either outcome category. This does not show that MAR is true in a real study; it demonstrates the conditional pattern that the MAR definition describes.
Why This Condition Supports Imputing the Predictor
Under MAR, the distribution of \(X\) among participants with missing \(X\) matches the distribution of \(X\) among all participants with the same observed \(Y\). This follows directly from the definition, provided there is a positive chance of missingness at that value of \(Y\).
The second line uses the MAR condition. The result says that, within an outcome value, missing cases have the same predictor distribution as the cases represented by the conditional distribution \(X\mid Y=y\). This is why an imputation model for \(X\) can use the observed outcome \(Y\). Omitting a relevant outcome from that model can fail to reflect the very variation in response that makes the predictor MAR.
The same point can be seen from the observed-data contribution when \(X\) is missing. Write \(f(x,y)\) for the joint data distribution and \(g(r\mid x,y)\) for the response mechanism. When \(R_X=0\), \(X\) is unobserved and must be summed or integrated out. Under MAR, \(g(0\mid x,y)=g(0\mid y)\), so
The observed outcome remains available, while the unobserved predictor is integrated out. This factorization is a justification for treating response as depending on observed information in this setup; it is not evidence that the MAR assumption holds.
Worked Example: Dependence on the Missing Predictor Is Different
Now consider another hypothetical study with the same binary outcome and predictor. The outcome is observed for everyone, but the response rates are as follows.
| Outcome \(Y\) | Predictor \(X\) | \(X\) observed | Total | Response rate |
|---|---|---|---|---|
| 0 | Low | 9 | 10 | 0.90 |
| 0 | High | 10 | 20 | 0.50 |
| 1 | Low | 8 | 10 | 0.80 |
| 1 | High | 8 | 20 | 0.40 |
Within \(Y=0\), the response rate is \(9/10=0.90\) for low \(X\) and \(10/20=0.50\) for high \(X\). Within \(Y=1\), it is \(8/10=0.80\) and \(8/20=0.40\). In both outcome categories, response depends on \(X\). For instance, at \(Y=0\), the two conditional response probabilities are not equal:
Because \(X\) is missing for nonresponders, this residual dependence is on information that is not observed for those records. Under the stated setup, the mechanism is MNAR, not MAR. Knowing that response also varies with the observed outcome does not remove this dependence on \(X\).
MAR Does Not Mean No Marginal Association
A subtle consequence is that \(R_X\) and \(X\) can be associated before conditioning on \(Y\), even when the mechanism is MAR. This can happen if the distribution of \(Y\) differs across values of \(X\), and response rates differ across values of \(Y\). Conditional independence is not the same as marginal independence.
Worked Example: A Marginal Difference Under MAR
Suppose, hypothetically, that response probability is \(0.80\) for \(Y=0\) and \(0.50\) for \(Y=1\), regardless of \(X\). Suppose also that \(P(Y=1\mid X=\text{Low})=0.20\), while \(P(Y=1\mid X=\text{High})=0.80\). The response probabilities are constant across \(X\) within each outcome category, so this mechanism satisfies MAR.
For low \(X\), the marginal response probability is
For high \(X\), it is
Thus, the marginal response rates are \(0.74\) and \(0.56\), despite MAR. The difference arises because low- and high-\(X\) participants have different outcome distributions, and the outcome predicts response. It would be incorrect to infer MNAR from this marginal difference alone; the relevant comparison is conditional on the observed \(Y\).
Assumptions, Diagnostics, and Limits
The conditional interpretation depends on the stated setup: the outcome is measured for all records being considered, the predictor’s response indicator is correctly defined, and the variables used for conditioning are genuinely observed. In an imputation analysis, a suitable model must also represent the relationship between \(X\) and the observed information used to predict it. MAR is a statement about the missingness mechanism, not a guarantee that any particular imputation model is well specified.
Observed response rates by outcome category can be useful diagnostics. If rates vary across observed outcome values, the data are inconsistent with MCAR for this predictor, which requires response not to depend on observed data. Such a pattern is compatible with MAR, but does not establish it. The missing \(X\) values cannot be directly compared across responders and nonresponders, so observed data alone generally cannot rule out dependence on the unobserved \(X\).
A useful comparison is:
| Mechanism in this setup | Response condition | Interpretation |
|---|---|---|
| MCAR | \(P(R_X\mid X,Y)=P(R_X)\) | Response does not depend on either \(X\) or \(Y\). |
| MAR | \(P(R_X\mid X,Y)=P(R_X\mid Y)\) | Response may depend on observed \(Y\), but not on \(X\) after conditioning on \(Y\). |
| MNAR | Response still depends on \(X\) given \(Y\). | Response depends on the missing predictor value even among records with the same observed outcome. |
These labels are conditional on which variables are observed and included. If the outcome is not observed for some records, the simple MAR condition above no longer applies unchanged to those records. And even when MAR is plausible, sparse outcome categories can make conditional response rates unstable, while an omitted observed variable associated with response can make the assumed conditioning set inadequate.
A Common Mistake
The reverse mistake is also possible: finding different response rates by outcome and calling the mechanism MNAR. Outcome-dependent response is compatible with MAR when there is no remaining dependence on the missing predictor within outcome values. The conditional comparison, rather than the presence of any association, determines the classification.
Check Your Understanding
Answer each question using the conditional definition. Full solutions follow.
- If \(P(R_X=1\mid X=x,Y=0)=0.7\) for every \(x\), and \(P(R_X=1\mid X=x,Y=1)=0.4\) for every \(x\), what mechanism does this describe?
- At \(Y=0\), response probabilities are \(0.8\) for low \(X\) and \(0.5\) for high \(X\). Is the mechanism MAR under the stated setup?
- Can response be marginally associated with \(X\) under MAR? Give the reason.
- Why can the observed outcome be included when imputing a missing predictor in this setup?
- Does observing different response rates across outcome categories prove MAR? Explain.
Solutions. (1) The pattern is MAR: response varies with observed \(Y\), but is constant across \(X\) within each value of \(Y\). (2) No. At the same observed outcome \(Y=0\), response depends on the predictor value, so the MAR equality fails. (3) Yes. If \(X\) and \(Y\) have different distributions and response depends on \(Y\), averaging over \(Y\) can produce different response rates by \(X\), even though response is conditionally independent of \(X\) given \(Y\). (4) Under MAR, \(P(X\mid Y,R_X=0)=P(X\mid Y)\), so the observed outcome carries relevant information for the predictor distribution among missing cases. (5) No. It shows that response may depend on observed information and is inconsistent with MCAR if the difference is genuine, but it cannot rule out additional dependence on the unobserved predictor value within outcome categories.