Why Multiple Partially Observed Variables Need Pattern-by-Pattern Reasoning
When several variables can be missing, there is no single “observed set” that applies to every person. One person might have \(Y_1\) and \(Y_3\) recorded but not \(Y_2\); another might have \(Y_2\) alone recorded. A missingness mechanism is MAR only if its probability behaves appropriately for each missingness pattern. A predictor that is observed in one pattern may be unobserved in another, so the same dependence can be acceptable in one pattern and disqualifying in another.
The practical task is to take a proposed mechanism, list the values available under each pattern, and check whether the probability of that pattern changes when only its missing values are changed. This is a classification of a mechanism, not a test that can prove MAR from the observed data.
This condition concerns the joint event \(R=r\), not merely one indicator considered in isolation. An overall MAR classification requires the condition to hold for every pattern that can occur (on the support of the data). If even one pattern’s probability depends on a value missing in that pattern, the mechanism is not MAR.
A Direct Check for One Pattern
For a particular pattern, hold its observed values fixed and vary its missing values. If the probability of that pattern stays the same, the pattern passes the MAR check. If the probability changes, it fails. This gives a simple way to see the definition’s equivalence: MAR says that two complete-data vectors \(y\) and \(y'\) with the same observed values under \(r\) must satisfy \(P(R=r\mid Y=y)=P(R=r\mid Y=y')\).
Worked Example: Checking a Pattern With Two Observed Variables
Suppose \(Y_1,Y_2,Y_3\) are partly observed, and consider the pattern \(r=(1,0,1)\): \(Y_1\) and \(Y_3\) are observed, while \(Y_2\) is missing. A hypothetical mechanism assigns this pattern probability \(0.12+0.03Y_1+0.02Y_3\), where the variables in this example are coded 0 or 1.
Take \(Y_1=1\) and \(Y_3=0\), and compare complete-data records that differ only in \(Y_2\). When \(Y_2=0\), the probability is \(0.12+0.03(1)+0.02(0)=0.15\). When \(Y_2=1\), it is again \(0.12+0.03(1)+0.02(0)=0.15\). Thus this pattern’s probability does not depend on the value missing under this pattern; it passes the MAR check.
Now suppose the proposed probability instead includes a term \(0.04Y_2\). With the same observed values, it is \(0.15\) at \(Y_2=0\) and \(0.19\) at \(Y_2=1\), since \(0.04(1)-0.04(0)=0.04\). That pattern fails the MAR check: its probability depends on an unobserved value. Passing this check for one pattern does not, by itself, establish that the whole mechanism is MAR.
Work Through All Patterns, Not Just One Indicator
For \(p\) binary response indicators there can be up to \(2^p\) distinct patterns. In principle, each joint pattern probability must be checked. The relevant observed set changes from row to row in this check. In particular, a variable’s value can be a permissible predictor for one pattern, because it is observed there, and an impermissible predictor for another, because it is missing there.
Worked Example: A Two-Variable Mechanism That Is Not MAR
Let \(Y_1\) and \(Y_2\) each be coded 0 or 1. The following hypothetical table gives the probability of each response pattern conditional on the complete values. Here, 1 in the pattern means observed.
| Pattern \(r\) | Variables observed | \(P(R=r\mid Y_1,Y_2)\) |
|---|---|---|
| 11 | \(Y_1,Y_2\) | 0.40 |
| 10 | \(Y_1\) | \(0.20+0.04Y_1\) |
| 01 | \(Y_2\) | \(0.20+0.04Y_2\) |
| 00 | None | \(0.20-0.04Y_1-0.04Y_2\) |
The probabilities are valid: for any \(Y_1,Y_2\), they sum to 1. The smallest probability in the table is 0.12, which occurs for pattern 00 when \(Y_1=Y_2=1\); none is negative. Pattern 10 depends only on \(Y_1\), which is observed in that pattern, and pattern 01 depends only on observed \(Y_2\). Pattern 11 has no missing values, so its probability meets the MAR condition.
Pattern 00 is different: both variables are missing, so its probability must not depend on either value for MAR to hold. At \((Y_1,Y_2)=(0,0)\), its probability is \(0.20-0.04(0)-0.04(0)=0.20\). At \((1,1)\), it is \(0.20-0.04(1)-0.04(1)=0.12\). The observed set is empty in both cases, but the probabilities differ. Therefore pattern 00 fails the condition, and the overall mechanism is not MAR.
This example also illustrates why a mechanism cannot be called MAR just because some pattern probabilities depend only on observed variables. MAR is an all-pattern condition. A failure in even one possible pattern is sufficient to rule out an overall MAR classification.
Worked Example: The Same Predictor Can Pass in One Pattern and Fail in Another
Consider two pattern probabilities in a hypothetical three-variable mechanism. For \(r=(1,0,1)\), suppose the probability is \(0.10+0.02Y_1+0.05Y_3\). Since \(Y_1\) and \(Y_3\) are observed under this pattern, this expression satisfies the pattern-specific MAR condition. For example, at \(Y_1=1,Y_3=0\), the probability is \(0.10+0.02(1)+0.05(0)=0.12\), whether missing \(Y_2\) is 0 or 1.
For \(r=(1,1,0)\), suppose the probability is \(0.22+0.03Y_1+0.04Y_3\). Here \(Y_3\) is missing. Holding observed \(Y_1=1\) fixed, the probability is \(0.25\) at \(Y_3=0\) and \(0.29\) at \(Y_3=1\). The change is \(0.04(1)-0.04(0)=0.04\). Thus this second pattern fails the MAR condition. The fact that \(Y_3\) was an allowed predictor for the first pattern does not make it an allowed predictor for the second.
These calculations classify the two specified pattern probabilities; a claim about the full mechanism would still require checking every other possible pattern.
How MAR Differs From MCAR and MNAR
The same pattern-by-pattern comparison clarifies the neighboring mechanism definitions. MCAR requires the probability of each response pattern to be independent of all complete-data values. MAR permits dependence on the values observed under that particular pattern, but not on its missing values. MNAR describes dependence on at least one value that is missing under the relevant pattern. A mechanism may contain some patterns that meet MAR’s condition and another that does not.
| Mechanism | Pattern-specific probability may depend on | Pattern-specific probability may not depend on |
|---|---|---|
| MCAR | No values of \(Y\) | Any values of \(Y\) |
| MAR | Values observed under that pattern | Values missing under that pattern |
| MNAR | At least one value missing under that pattern | Not applicable as a restriction for that pattern |
These labels describe the mechanism’s dependence, not whether the observed missingness rates look large or small. They also do not identify the mechanism from the observed records alone. For any particular incomplete record, values missing in that record are unavailable for a direct comparison. In practice, MAR plausibility must be argued using subject-matter knowledge, study design, and measured auxiliary variables; it cannot be confirmed just by finding no visible association in the observed data.
Common Misreadings and Practical Checks
- Checking only one variable’s response indicator. A condition on \(P(R_j=1\mid Y)\) is not, by itself, the definition of MAR for the joint response pattern. Check the probability of each joint pattern against that pattern’s observed values.
- Using one fixed observed set for everyone. The set \(Y_{\mathrm{obs}}(r)\) depends on \(r\). Re-list it for every pattern being assessed.
- Calling any observed predictor acceptable everywhere. A variable is available as a predictor only in patterns where it is observed. Dependence on it can be MAR in one pattern and MNAR in another.
- Stopping after a pattern passes. One passing pattern does not establish overall MAR. Check all patterns that can occur.
- Treating a data check as proof of MAR. The condition involves complete-data values, including values that are missing. Observed data alone cannot verify that dependence on those values is absent.
When there are many variables, an explicit pattern table helps prevent these errors. For each row, write the pattern, its observed set, its missing set, and the complete-data variables appearing in the pattern-probability expression. Any dependence on a member of that row’s missing set is a warning that the row fails MAR. If some patterns have zero probability under the mechanism, assess the condition on the mechanism’s support rather than treating impossible patterns as observed cases.
Check Your Understanding
For each question, use the observed set specified by the pattern. Solutions follow the questions.
- With \(Y=(Y_1,Y_2,Y_3)\), pattern \(r=(0,1,1)\) leaves \(Y_1\) missing. Which values may a MAR pattern probability depend on?
- For pattern \(r=(1,0)\), suppose \(P(R=r\mid Y_1,Y_2)=0.25+0.06Y_2\), with \(Y_2\) coded 0 or 1. Does this pattern pass the MAR check?
- A two-variable mechanism has three patterns that meet the MAR condition and one possible pattern whose probability depends on a value missing under that pattern. Can the overall mechanism be classified as MAR?
- Why does observing that missingness is associated with an observed variable not, by itself, show that a mechanism is MNAR?
Solutions. (1) It may depend on \(Y_2\) and \(Y_3\), which are observed; it may not depend on missing \(Y_1\). (2) No. Pattern 10 observes \(Y_1\), not \(Y_2\); the probability changes from \(0.25\) at \(Y_2=0\) to \(0.31\) at \(Y_2=1\), a difference of \(0.06\). (3) No. Overall MAR requires every possible pattern to pass. (4) Dependence on a value observed under the relevant pattern is permitted under MAR. The issue is whether the pattern probability depends on values missing under that pattern.