Why the Missingness Mechanism Matters
A regression data set may be incomplete because some outcomes, predictors, or both were not recorded for some individuals. To describe the consequences, it is useful to separate two questions: what values are missing, and how did the chance of their being missing relate to the data? The second question concerns the missingness mechanism.
Three standard labels are missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). They differ according to whether the probability of a particular missingness pattern can depend on the complete data, and, if so, whether it can depend on values that are themselves missing in that pattern. These labels describe assumptions about a data-generating process; they are not descriptions of how many entries are absent.
Representing Missingness With Response Indicators
Suppose the complete data for individual \(i\) would contain an outcome \(Y_i\) and \(p\) predictors \(X_{i1},\ldots,X_{ip}\). Write the complete data vector as \(Z_i=(Y_i,X_{i1},\ldots,X_{ip})\). “Complete” here means the values that would be available if nothing had gone unrecorded; it does not mean every value is observed in the actual data set.
For each component, define a response indicator. For example, \(R_{iY}=1\) if \(Y_i\) is observed and \(R_{iY}=0\) if it is missing. Define \(R_{ij}\) in the same way for predictor \(X_{ij}\). Taken together, these indicators form the missingness pattern \(R_i\). A pattern \(r\) specifies which components are observed and which are missing.
These definitions are pattern-specific. For example, if a pattern has an observed outcome but a missing predictor, the predictor is a missing component for that pattern even though the outcome is available. The relevant question is always whether the probability of that pattern depends on values that would be unobserved under it.
Missing Completely at Random
MCAR says that the probability of each missingness pattern is the same for all complete data values. Formally, for any pattern \(r\) and any complete values \(z\) and \(z'\),
A simple MCAR mechanism might be an equipment failure occurring independently of the person’s outcome and predictor values. MCAR is a strong condition: it rules out a relationship between missingness and any part of the complete data, whether that part is observed or missing. Thus, MCAR does not merely mean that missingness is unrelated to the outcome.
Worked Example: A Random Recording Failure
Imagine a hypothetical data set with outcome \(Y\) and predictor \(X\). A recording device loses the outcome with probability \(0.10\), independently of both values; otherwise it records the outcome. The predictor is always recorded. For the pattern where only \(Y\) is missing, suppose
for every \(y\) and \(x\). For instance, at \(Y=12\) and \(X=4\), the probability is \(0.10\); at \(Y=28\) and \(X=9\), it is still \(0.10\). The calculation makes the lack of dependence explicit:
The missingness probability is unchanged across the complete values, so this example satisfies MCAR. The example describes a hypothetical mechanism; it does not claim that real recording failures usually behave this way.
Missing at Random
MAR allows missingness to be related to the data, provided that for each pattern the dependence can be accounted for by the values observed under that pattern. In symbols, for any \(z\) and \(z'\) that agree on all components in \(O(r)\),
The two complete data vectors may differ on missing components in \(M(r)\); MAR requires that those differences do not change the probability of pattern \(r\). Equivalently, for that pattern the probability can be written as a function of the observed values, \(P(R_i=r\mid Z_i=z)=P(R_i=r\mid Z_{i,O(r)})\). The observed values may still predict missingness, so MAR is not the same as MCAR.
Worked Example: Missing Outcomes Related to Observed Age
Consider a hypothetical regression data set where age is recorded for everyone, but an outcome score \(Y\) is sometimes missing. Suppose the probability of a missing score depends on age: it is \(0.05\) for people under 50 and \(0.20\) for people aged 50 or older. Within each age group, it does not depend on the score that would have been recorded. For the pattern with \(Y\) missing and age observed, the rule is
At age 62, the probability is \(0.20\) whether the unobserved score would have been 10 or 30:
The chance of missingness differs across observed ages, so the mechanism is not MCAR. But for the pattern in which age is observed and \(Y\) is missing, it does not depend on the unobserved \(Y\) once age is fixed. Under the stated rule, this is MAR.
Missing Not at Random
MNAR means that MAR does not hold: for at least one missingness pattern, the probability of that pattern depends on a value that is missing under the pattern, even after the observed values are held fixed. For instance, people with larger values of an outcome might be less likely to report that outcome. The missing values themselves then help determine which values are absent.
Worked Example: Higher Outcomes Less Likely to Be Reported
Suppose a hypothetical survey asks for a numeric outcome \(Y\), and every individual’s predictor \(X\) is recorded. Imagine that the probability of \(Y\) being missing is \(0.05\) when \(Y<20\) and \(0.35\) when \(Y\geq20\), regardless of \(X\). For the pattern with \(Y\) missing, this rule gives
For the same observed predictor value \(X=7\), the missingness probabilities differ depending on the unobserved outcome:
Because \(Y\) is missing in this pattern, the difference between \(Y=12\) and \(Y=25\) cannot be accounted for by the observed data in the record. The mechanism is MNAR. In practice, the observed records do not reveal the missing \(Y\) values directly; the example specifies a hypothetical mechanism to show what the assumption means.
A Short Logical Relationship
MCAR is a special case of MAR. To see this, suppose the missingness probability is constant across complete data values. Then fixing the observed values cannot make it vary with the missing values: for any fixed observed values and any two choices of missing values, the probability remains the same constant.
This implication does not run in reverse. In the age example, missingness varies with an observed value, so it can be MAR without being MCAR. MNAR describes a departure from MAR, not simply a mechanism with a large fraction of missing values. A small amount of missing data can be MNAR, and a large amount can be MCAR; the labels concern dependence, not quantity.
How to Read the Assumptions Carefully
For a particular pattern, classify the mechanism by holding the observed values fixed and asking what changes the probability of that pattern:
- If no part of the complete data changes the probability, the mechanism is MCAR.
- If observed values may change it, but changing missing values while holding observed values fixed cannot, the mechanism is MAR.
- If changing a missing value can change it even with observed values fixed, the mechanism is MNAR.
The same data collection process can have different mechanisms for different patterns. For example, missing outcomes and missing predictors may have different causes. A statement such as “the data are MAR” summarizes the assumption being made across relevant patterns; it should not be inferred from a single explanation of one missing variable.
A common mistake is to say, “Missingness depends on age, so it must be MNAR.” That conclusion does not follow. If age is observed for the records whose outcomes are missing, and missingness depends on age but not on the missing outcomes after age is fixed, the mechanism can be MAR. The correction is to identify the missingness pattern first, then distinguish dependence on observed values from dependence on values missing in that pattern.
Check Your Understanding
For each question, identify which values are observed and which are missing in the stated pattern before deciding what the mechanism implies.
- A predictor is missing with the same probability for every combination of outcome and predictor values. Which missingness assumption does this describe?
- An outcome is missing more often for older individuals. Age is observed whenever the outcome is missing, and the probability does not depend on the unobserved outcome after age is fixed. Is this MCAR, MAR, or MNAR?
- In a pattern where an outcome is missing, its probability of being missing rises with the outcome’s value, even after observed predictors are fixed. Which assumption applies?
- Explain why MAR does not mean that missingness is unrelated to every variable in the data set.
- For a pattern with an observed outcome and a missing predictor, which values count as observed and missing when applying the definition of MAR?