Why the Conditioning Set Matters
A response model may include several variables, but the variable names alone do not determine whether the mechanism is MAR or MNAR. The key question is whether the model depends on a value that is missing for the response pattern being considered. A value can help explain nonresponse when it is observed, while dependence on that same value can make the mechanism MNAR when it is missing.
The distinction is pattern-specific. In a data set with several partially observed variables, one response pattern may have a particular value available and another pattern may not. Classification therefore requires identifying the pattern first, then checking which values are observed within it.
This definition asks about conditional dependence, not merely which variables appear in a written model. A variable might appear in a formula but have no effect on the response probability, or its effects might cancel. Conversely, a mechanism may depend on a missing value through an interaction or nonlinear term even when a simple main-effect coefficient is absent.
MCAR is the stronger special case in which the response-pattern probability does not depend on any data values. Thus, a mechanism depending only on fully observed values can be MAR without being MCAR. For MAR, “observed” is determined by the pattern under examination; it does not mean that a variable is observed for everyone in the data set.
A Direct Test for a Proposed Mechanism
For a particular pattern, the practical test is to hold the observed values fixed and compare the response-pattern probability across possible values of the missing variables. If that probability is constant across those missing values, the pattern satisfies MAR. If it changes, the pattern is MNAR.
The equality can be justified directly. Fix a pattern \(r\) and a set of observed values \(y_{O(r)}\). If the probability is the same for every possible missing value \(y_{M(r)}\), it can be written using the observed values alone, as in the MAR definition. If two possible missing values produce different probabilities while the observed values remain fixed, no function of the observed values alone can represent both probabilities. The condition fails for that pattern.
In a specified logistic model, this comparison can sometimes be read from its terms. For example, if \(\operatorname{logit}P(R=1\mid A,Y)=\beta_0+\beta_A A+\beta_Y Y\), and \(A\) is observed while \(Y\) is missing when \(R=0\), a nonzero \(\beta_Y\) makes response depend on the missing value \(Y\), given \(A\). But this shortcut is model-specific: the actual criterion is whether the probability changes with missing values after observed values are held fixed.
Worked Example: Response Depends on Missing Income
Suppose a hypothetical survey records age \(A\) for everyone but income \(Y\) only when the respondent answers the income question. For respondents aged 40, the proposed probabilities of answering are 0.80 at an income of $30,000, 0.60 at $60,000, and 0.30 at $90,000. Classify the mechanism for cases with missing income.
For the pattern \(R=0\), income is missing and age is observed. The relevant comparison fixes age at 40 and varies income. The response probabilities shown are for answering, so the probabilities of missing income are their complements: $$ P(R=0\mid A=40,Y=30{,}000)=1-0.80=0.20, $$ $$ P(R=0\mid A=40,Y=60{,}000)=1-0.60=0.40, \qquad P(R=0\mid A=40,Y=90{,}000)=1-0.30=0.70. $$
The probability of the missing-income pattern changes from 0.20 to 0.70 while the observed age is fixed. The change is \(0.70-0.20=0.50\), so the response mechanism depends on a value missing in this pattern. It is MNAR for nonresponse on income. These hypothetical probabilities illustrate a mechanism; they are not estimates from observed data.
Worked Example: Response Depends Only on Observed Age
In a separate hypothetical survey, suppose income response rates are 0.90 for respondents under 50 and 0.65 for respondents aged 50 or older. Within each age group, assume the rate is the same at every income value. Is the mechanism MAR for missing income?
For a respondent under 50, the probability of missing income is \(1-0.90=0.10\) at every income value. For a respondent aged 50 or older, it is \(1-0.65=0.35\) at every income value. Therefore, fixing observed age and comparing possible missing incomes gives $$ P(R=0\mid A<50,Y=y)=0.10 \quad\text{for every }y, $$ and $$ P(R=0\mid A\geq 50,Y=y)=0.35 \quad\text{for every }y. $$
The missing-income probability varies by age, but age is observed and the probability does not vary with income once age is fixed. This mechanism is MAR, not MCAR: it depends on an observed value, but not on the missing income value. The classification relies on the stated assumption that response does not vary by income within age groups.
Different Patterns Can Have Different Classifications
When more than one variable can be missing, the same variable may be observed in one pattern and missing in another. It is then essential to classify the probability of each response pattern using that pattern’s own observed and missing values. The mechanism as a whole is MAR only if the MAR condition holds for every relevant pattern.
Worked Example: The Same Variable Is Observed in One Pattern and Missing in Another
Consider hypothetical income \(I\) and health status \(H\), each of which may be observed or missing. The four patterns are written \((R_I,R_H)\). Suppose the following probabilities apply for either health value, so they do not vary with \(H\):
| Income value | \(P(R=(1,1)\mid I,H)\) | \(P(R=(1,0)\mid I,H)\) | \(P(R=(0,1)\mid I,H)\) | \(P(R=(0,0)\mid I,H)\) |
|---|---|---|---|---|
| Low | 0.30 | 0.30 | 0.10 | 0.30 |
| High | 0.10 | 0.50 | 0.30 | 0.10 |
Each row sums to 1.00, as required for probabilities of the four mutually exclusive patterns. For pattern \((1,0)\), income is observed and health is missing. The pattern probability is 0.30 at low income and 0.50 at high income, but income is part of the observed values for this pattern. The probability does not vary with missing health. Thus, this pattern satisfies MAR.
For pattern \((0,1)\), income is missing and health is observed. Its probability is 0.10 at low income and 0.30 at high income, even with the observed health value held fixed. It therefore depends on missing income, so this pattern is MNAR. Because at least one pattern violates MAR, this overall mechanism is not MAR. The income dependence is harmless for the first pattern’s MAR check but violates MAR for the second pattern’s check.
Assumptions, Diagnostics, and Limits
A classification from a proposed model is conditional on the model being an adequate description of the response probabilities and on correctly identifying which variables are observed for each pattern. The comparison must cover the relevant possible values in the data’s support; checking only one pair of values may miss a dependence elsewhere. If a proposed probability is zero for some values, comparisons must be restricted to values for which the conditional probabilities are defined and relevant.
There is no general diagnostic using observed data alone that can prove MAR or rule out MNAR. For cases with a missing value, its value is unavailable, so the response probabilities cannot be directly compared across those unobserved values. Observed-data summaries can reveal patterns worth investigating—for example, different response rates across observed age groups—but such differences do not establish that missing values have no additional effect.
Useful checks when reviewing a proposed mechanism include:
- Write down the response pattern being classified and mark each variable as observed or missing in that pattern.
- Translate the proposed model into a response-pattern probability, rather than relying only on a description such as “response depends on income.”
- Hold the observed values fixed and compare that probability across plausible values of the missing variables.
- Check interactions and nonlinear terms as well as main effects; any term that changes the probability with a missing value can violate MAR.
- State what assumptions or substantive knowledge support the proposed mechanism, since observed data cannot settle the MAR-versus-MNAR question by itself.
A frequent error is to classify a mechanism as MNAR because its formula contains a variable that is sometimes missing. That rule ignores the response pattern: the variable may be observed for the pattern at issue. The opposite error is to call a mechanism MAR because it includes observed predictors, while overlooking a missing variable that also affects response. The correction in both cases is the same: condition on the values observed for the pattern, then check whether changing any missing value changes the pattern probability.
MCAR, MAR, and MNAR are assumptions about how response relates to complete data, not labels that can always be read directly from a table of observed cases. MCAR requires no dependence on data values; MAR permits dependence on observed values; MNAR includes dependence on values missing in the relevant pattern. A proposed mechanism is useful only when its pattern-specific conditioning sets and dependencies are made explicit.
Check Your Understanding
For each item, identify the relevant pattern and compare probabilities while holding its observed values fixed. Brief solutions follow.
- For a missing test score \(Y\), a response model gives \(P(R=0\mid A,Y)=0.25\) for every \(Y\) when observed age \(A=20\). Is this pattern MAR or MNAR?
- For missing blood pressure \(B\), nonresponse probability is 0.10 when observed sex \(S\) is one category and 0.30 when it is another. Within each category, it does not vary with \(B\). Is the mechanism MCAR, MAR, or MNAR?
- In pattern \((R_X,R_Z)=(1,0)\), the probability of the pattern changes with \(X\) but is constant across \(Z\) when \(X\) is fixed. Classify this pattern.
- A model includes a missing variable as a predictor, but its response probability is constant across all values of that variable when observed values are fixed. Does the variable’s appearance alone establish MNAR?
Solutions. (1) MAR: at fixed observed age, the probability is constant across missing scores. (2) MAR but not MCAR: response depends on observed sex, not missing blood pressure. (3) MAR: \(X\) is observed in this pattern, while the probability does not depend on missing \(Z\). (4) No. The relevant question is whether the probability actually changes with the missing value; a variable’s name in a formula is not sufficient.