Planned Omission Is a Design, Not a Mechanism
A study may deliberately leave some questionnaire items unasked or assign different participants to different measurement forms. This can reduce participant burden, cost, or fatigue. Because the omissions are intentional, it may be tempting to label them “MCAR by design.” That conclusion does not follow from the word planned. The mechanism depends on the rule that assigns the omissions: which information determines who is asked each item, and whether that information is observed.
The key question is therefore not simply whether the omission was planned. It is whether the probability of an item being missing depends on the item’s unobserved value, on observed information, or on neither. The assignment protocol can answer that question when it is recorded and followed as specified.
Let \(Y\) denote the variables that may be observed or missing, and let \(Z\) denote fully observed design variables, such as randomized form assignment or a baseline characteristic measured for every participant. For a particular response pattern \(r\), write \(Y_{\mathrm{obs}(r)}\) for the components observed under that pattern. In this setting, the relevant distinctions are:
These definitions concern the assignment rule, not the investigator’s intention. If an item is omitted by a lottery independent of all study variables, the omission is MCAR. If assignment probabilities vary by a fully observed group, the omission can be MAR but not MCAR. If assignment depends on the value that would have been recorded for the omitted item, the mechanism is not MAR merely because the protocol was written in advance.
How the Assignment Rule Determines the Classification
Start by listing the variables used to assign forms or omit items. Separate variables known for everyone from values that are absent precisely because an item was omitted. Then express the probability of a response pattern conditional on those variables. For a single item, the question is whether \(P(R_j=0\mid Y,Z)\) changes with \(Y_j\), with observed variables, or with neither.
State which indicator corresponds to each item, and what 1 and 0 mean. Distinguish a value not collected by design from a value that was intended to be collected but was not.
Name every variable used to allocate forms or decide which items are asked. Use the protocol and implementation records, not just the study’s stated intention.
If assignment depends only on fully observed variables, its dependence is on observed data. If it depends on a value that is missing, the rule may depend on unobserved data.
Probabilities constant across all study values satisfy MCAR. Probabilities that vary with observed values but not unobserved values satisfy MAR.
Worked Example: Random Assignment to Two Forms
Suppose a hypothetical survey measures a common variable \(A\) for everyone. A fair random assignment then gives each participant one of two forms: Form 1 asks \(B\), and Form 2 asks \(C\). Thus \(B\) is missing for everyone on Form 2, while \(C\) is missing for everyone on Form 1. Assume the form lottery is implemented independently of the participants’ values of \(A\), \(B\), and \(C\).
Let \(R_B\) and \(R_C\) indicate whether \(B\) and \(C\) are observed. The two possible patterns have probabilities
The probabilities are the same for every possible value of \(A\), \(B\), and \(C\). For example, if a participant has \(A=12\), \(B=7\), and \(C=4\), the probability of assignment to Form 1 is still \(0.5\); the same is true for \(B=2\) or \(C=10\). The response pattern is therefore independent of the study values, so this planned omission is MCAR. It is also MAR, since MCAR is a special case of MAR.
This conclusion relies on the stated independence of the lottery from participant data. If staff instead assign forms using a participant characteristic, the probability calculation must include that characteristic.
Worked Example: Omission Rates Differ by an Observed Group
In a hypothetical study, every participant reports age group, classified as younger or older. A planned form omits an income item for 20% of younger participants and 60% of older participants. Within each age group, form assignment is random and does not use participants’ income.
Let \(Z\) denote age group and let \(R_{\mathrm{inc}}=1\) mean income is observed. The omission probabilities are
The probabilities do not vary with income within either age group. Because age group is observed for everyone, the rule depends on observed information only, so the mechanism is MAR. But the probabilities differ by age group: \(0.20\ne0.60\). Therefore the mechanism is not MCAR.
This example shows why an overall omission rate is not enough to classify the mechanism. If the two groups were equally large, the overall omission rate would be \((0.20+0.60)/2=0.40\), but that average hides the different assignment probabilities by observed age group.
Worked Example: A Planned Rule That Depends on the Target Value
Imagine a hypothetical protocol that assigns a detailed symptom questionnaire only to participants whose true symptom severity exceeds a specified threshold. Suppose the severity value is not otherwise measured for participants who do not receive the questionnaire. The protocol is planned, but its assignment rule uses the value being omitted.
Let \(Y\) be symptom severity and \(R=1\) indicate that the detailed questionnaire is administered. For illustration, suppose the rule is \(R=1\) when \(Y>5\), and \(R=0\) otherwise. Then
The probability of observation changes with the unobserved target value. It cannot be written as a function only of values observed when \(R=0\), so this rule is not MAR. It is also not MCAR. The fact that the decision rule was specified in advance does not change that classification.
In an actual study, the rule might instead use a screening measure observed for everyone. If so, that screening measure—not the unobserved severity value—would be the relevant assignment variable, and the classification could differ. The precise recorded rule matters.
Why the MAR Conclusion Follows
For a protocol that assigns omissions using only fully observed design variables \(Z\), the justification is direct. If assignment is randomized according to probabilities that depend on \(Z\), but not on any component of \(Y\), then for every response pattern \(r\),
Since \(Z\) is observed for everyone, it is part of the observed information under every response pattern. Thus \(P(R=r\mid Z)\) is a function only of observed data, which gives
That is the MAR condition. If the probabilities also do not vary with \(Z\), then \(P(R=r\mid Z)=P(R=r)\), giving MCAR. Conversely, if probabilities vary with \(Z\), the mechanism is not MCAR, even though it remains MAR under the stated conditions.
The argument depends on the design being followed: the variables used to allocate forms must be known, the relevant design covariates must actually be observed, and the assignment probabilities must not depend on omitted values after conditioning on those covariates. If the protocol is only partly implemented, or staff alter assignments using additional information, classify the mechanism from the rule that generated the recorded data rather than the rule that was intended.
Diagnostics, Comparisons, and Limitations
For planned omissions, useful checks begin with the design records. Verify which form each participant was assigned, compare the recorded assignment with the protocol, and tabulate omission rates across the variables that were supposed to govern assignment. Unexpected differences can indicate implementation problems or an unrecorded assignment factor. These checks describe observed assignments; they do not establish that the omission probability is unrelated to an unobserved item value.
A planned design can make an MAR assumption more credible when the assignment algorithm is documented and uses fully observed variables. But MAR is a statement about the mechanism, not a guarantee that every later analysis is valid. The analysis must still use suitable models, account for important variables associated with assignment and the incomplete measurements, and have enough observed information to estimate the quantities of interest. If a form is never assigned within a relevant group, the missing values for that group cannot be recovered from that group’s observed item responses without additional assumptions.
| Assignment rule | Mechanism classification | Reason |
|---|---|---|
| Random form lottery independent of study values | MCAR (and MAR) | Pattern probabilities are constant across values. |
| Random assignment with probabilities varying by a fully observed design variable | MAR, not MCAR | Assignment depends on observed information, not the omitted value. |
| Assignment depends on the value of the item that is omitted | Not MAR; not MCAR | The probability of observation depends on an unobserved value. |
A common mistake is to reason, “The omission was planned, so it is MCAR.” The correction is to write the assignment probabilities. A planned random lottery independent of participant data can be MCAR; a planned lottery stratified by a fully observed characteristic is generally MAR but not MCAR; and a planned rule using an unobserved target value is not MAR. The label “planned” describes how the omission arose, not which mechanism applies.
In an imputation analysis, recorded form assignment and the observed variables used to determine it are important information about why item values are missing. Including them can help represent the assignment process. However, the MAR classification alone does not ensure that an imputation model is correctly specified or that information is sufficient for every analysis.
Check Your Understanding
For each question, classify the mechanism under the stated assumptions and compare your reasoning with the solutions.
- A fair lottery assigns one of two forms, independently of all participant characteristics. One form omits a particular item. What is the mechanism for that item?
- A fully observed region indicator is used to assign an item with different omission probabilities in each region. Assignment within region is random and does not use the item value. Is the mechanism MCAR, MAR, or neither?
- A protocol omits a measurement whenever its unobserved value would be below a threshold. Does calling this rule planned make it MAR?
- What should be checked if the study protocol says forms are assigned randomly, but the observed omission rates differ unexpectedly across baseline groups?
Solutions. (1) MCAR, because the lottery is independent of the study values; it is also MAR. (2) MAR but not MCAR: assignment depends on the fully observed region, and its probabilities differ across regions. (3) No. The assignment probability depends on the value that is missing, so the rule is not MAR; being planned does not alter that dependence. (4) Check assignment and implementation records, compare observed allocations with the protocol, and investigate whether another variable influenced assignment. The observed difference alone does not establish dependence on unobserved item values, but it is a reason to verify the actual rule.