Classifying the Response Process
In a longitudinal trial, a participant may provide some scheduled outcomes but not others. Classifying the response process means asking a precise question: after accounting for the information actually observed, does the chance that an outcome is recorded still depend on an unobserved value? The answer distinguishes MCAR, MAR, and MNAR. It must be based on the process that generated the missing values, not merely on the appearance of the completed dataset.
Let \(Y_t\) denote the outcome scheduled at visit \(t\), and let \(R_t=1\) if that outcome is observed and \(R_t=0\) otherwise. Let \(W\) denote all trial information that could be relevant to response, including outcomes and covariates, and let \(W_{\mathrm{obs}}\) denote the parts of that information observed for a participant. When classifying a particular visit, include the response history and all information available by the time the response decision is made. A covariate recorded only at the missed visit is not an observed predictor for that decision unless it was obtained by another means.
These are claims about a data-generating process, not labels that can usually be read directly from a dataset. In the examples below, “depends on” means that the response probability changes with the stated information when other listed predictors are held fixed. For visit-level classification, ask what information was available when the decision to attend, collect, or record the outcome was made.
Assumptions Behind the Labels
For clarity, use the strict convention that MCAR means independence from all trial data, including treatment assignment. If response probabilities differ by randomized treatment arm, response depends on an observed trial variable: this is not strict MCAR, although it can be MAR conditional on arm. Some discussions use “MCAR within arm” as shorthand; that is a conditional statement, not the strict definition used here.
MAR is likewise relative to the information observed for the participant. A prior outcome can explain a later missed visit and still be compatible with MAR if that prior outcome was recorded. A current outcome that is absent cannot serve as an observed explanation merely because it would predict attendance if it were known. This distinction is particularly important for time-varying covariates: they support a MAR classification only when their values are available at the relevant response decision.
For a visit \(t\), a useful way to express the classification is to compare the response probability given observed history with the probability given both observed history and the full data. Write \(H_t\) for the observed information available at the decision, including relevant baseline variables, previously observed outcomes, available covariates, treatment, and response history. Then the MAR condition is that adding the unobserved values does not change the response distribution:
MCAR is more restrictive: response is independent of the full data, rather than merely independent of unobserved values after conditioning on what is observed. MNAR describes a failure of the displayed MAR condition. These definitions concern response probabilities; they do not assert that a particular observed dataset must show a visible association.
Worked Classifications
Worked Example: Randomly Selected Visit Records
Hypothetical trial. At each follow-up visit, a computer independently selects each participant’s record for successful outcome capture with probability \(0.85\). The selection is generated independently of treatment, prior outcomes, current outcomes, covariates, and earlier response. Classify the mechanism.
Let \(R_t=1\) indicate successful capture. By the stated design, for every possible full-data history, \(P(R_t=1)=0.85\). For example, participants with a low prior score and those with a high prior score have the same capture probability, and the same holds for every possible current outcome.
Thus, the response probability does not change when any observed or unobserved trial value is supplied. The mechanism is MCAR. The value \(0.85\) is the probability of capture under this hypothetical process; it is not an estimate of an outcome effect.
The classification depends on the generating rule, not on whether the realized records happen to show equal capture rates in every group. A small sample could show more missing records among participants with high scores by chance without contradicting this MCAR design.
Worked Example: Prior Outcome and an Observed Time-Varying Covariate
Hypothetical trial. Before visit \(t\), a symptom score \(Y_{t-1}\) from the previous visit has been recorded. A daily activity measure \(L_t\) is obtained from a monitoring device even when the participant does not attend the clinic. Suppose the visit-response probability follows
Assume both predictors are standardized and consider participants with \(L_t=1\). For a participant with \(Y_{t-1}=0\), the linear predictor is \(-0.4+0.7(0)-0.5(1)=-0.9\), so
For a participant with \(Y_{t-1}=2\), the linear predictor is \(-0.4+0.7(2)-0.5(1)=0.5\), giving
The probabilities differ by approximately \(0.6225-0.2891=0.3334\). This dependence does not make the process MNAR: both predictors are observed even for participants who miss the clinic visit. Under the stated model and availability assumptions, the mechanism is MAR conditional on the observed history. If \(L_t\) were available only by measuring it at the missed visit, this reasoning would not establish MAR.
Worked Example: Dependence on an Unobserved Current Outcome
Hypothetical trial. Let \(Y_t\) be a standardized benefit score at the current visit, with higher values representing greater benefit. Suppose the chance of attending the visit follows
where all components of the observed history \(H_t\) are held fixed. If \(Y_t=0\), then the linear predictor is \(-1\), and
If \(Y_t=2\), the predictor is \(-1+1.2(2)=1.4\), so
The response probabilities differ by \(0.8021839-0.2689414=0.5332425\), or approximately 0.5332. Because \(Y_t\) is unobserved when \(R_t=0\), and response still depends on it after the observed history is held fixed, the mechanism is MNAR. Here, better current benefit is associated with a greater probability of attending under the hypothetical model. This is a statement about the specified response process, not evidence that the model is true in an actual trial.
Missed Visits, Covariates, and Response History
A missed visit is an event, not a mechanism classification. Suppose a participant misses visit \(t\), returns at \(t+1\), and provides an outcome then. The outcome at \(t\) remains missing, and its response mechanism still depends on what drove the absence at \(t\). If an observed prior score or a recorded transportation disruption explains the absence, that is compatible with MAR, provided no unobserved outcome also affects the response probability after conditioning on the observed information. If an unrecorded symptom at \(t\) drives the absence, the process is MNAR with respect to that outcome.
The same care applies to time-varying covariates. A value collected before the response decision can be part of the observed history even if it changes over time. A value that exists only for participants who attend the visit is not generally available to explain nonattendance at that visit. Also, if response depends on prior response history, that history must be included in the conditioning information when assessing a sequential MAR claim.
Treatment arm may also matter. If participants in one arm are less likely to attend, then response depends on an observed variable, treatment assignment. Conditional on arm, the mechanism may be MAR if no unobserved outcome affects response beyond the observed information. It would be incorrect to call the mechanism MCAR under the strict definition above.
A Short Justification and Practical Checks
The distinction in the logistic examples follows directly from the MAR condition. If the response model can be written using observed information alone, then adding the unobserved outcomes leaves the response probability unchanged. Conversely, if the response probability changes with an unobserved outcome while observed history is fixed, the MAR equality fails.
This check classifies a stated model; it does not identify an unknown mechanism from observed records alone. Associations between response and recorded outcomes or covariates can show that strict MCAR is implausible, but they cannot prove MAR. In particular, the unobserved outcomes needed to test whether response still depends on them are missing by definition.
For a practical review, write down the response decision, the information available at that time, and any proposed driver of nonresponse. Then check whether each proposed driver is observed for participants who miss the outcome. This simple audit often exposes mistaken classifications before an analysis is chosen.
Comparing the Classifications and Their Limits
| Mechanism | What may drive response? | Key classification check |
|---|---|---|
| MCAR | Neither observed nor unobserved trial values | Response is independent of the full data |
| MAR | Observed information, including available history and covariates | After conditioning on observed information, unobserved values do not further affect response |
| MNAR | At least one unobserved value still affects response after conditioning | Response probability changes with an unobserved value at fixed observed history |
The categories are useful because they clarify what assumptions a missing-data analysis is making. They are not diagnostics that can ordinarily be proven from the observed data. A response model based on observed history can be a defensible MAR working assumption, but the data alone generally cannot rule out an alternative process in which missing outcomes also affect response. Conversely, dependence on observed history does not by itself establish MNAR.
The main alternative is not a different label for the same known process, but a different assumption about the unobserved values. An analysis under MAR uses observed information to account for response; an MNAR analysis must additionally specify how response relates to values that were not recorded. MCAR is a stronger special case that rules out dependence on observed predictors as well. No classification, by itself, guarantees a correct outcome model or resolves other study-design problems.
Common Mistakes and Diagnostics
- “Response is predictable, so it is MNAR.” That is wrong because recorded prior outcomes, treatment, and observed covariates can predict response under MAR. The correction is to ask whether an unobserved value still matters after conditioning on the observed history.
- “The participant returned, so the missed outcome is MAR.” Later return does not reveal what drove the earlier absence. Classify the response decision for the missed visit using information available at that time.
- “No observed association proves MCAR.” A small or noisy dataset may fail to show a real association. MCAR is a claim of independence from the full data, not a conclusion established by a nonsignificant comparison.
- “The covariate is measured in the trial, so it is observed for everyone.” It may be unavailable precisely when a visit is missed. Check its timing and availability for nonresponders before using it to support MAR.
- “MAR or MNAR is a permanent property of the dataset.” The classification depends on the specified response process and observed information set. Different response decisions or conditioning sets may require different assessments.
A useful diagnostic record lists, for each type of nonresponse, the visit, whether absence is temporary or permanent, what was known before the missed measurement, which outcomes were later collected, and the proposed reason for nonresponse. Such records help assess the plausibility of assumptions; they do not reveal missing outcomes or prove the mechanism.
Check Your Understanding
For each situation, classify the stated response process and identify the key condition. Solutions follow the questions.
- At each visit, a random number generator selects participants for outcome capture independently of treatment and every trial measurement. What is the mechanism?
- Attendance depends on the previous recorded outcome and on a device-based activity measure available even when the clinic visit is missed. No current unobserved outcome affects attendance after conditioning on these variables. What is the mechanism?
- A participant misses visit \(t\), returns at \(t+1\), and the probability of missing \(t\) depended on an unrecorded current symptom score. Does the later return make the missing outcome MAR?
- Response differs between treatment arms, but within each arm it depends only on observed history. Is this strict MCAR, MAR, or MNAR under the convention in this tutorial?
Solutions. (1) MCAR: the capture process is independent of the full trial data. (2) MAR: both predictors are observed when response is determined, and the stated condition rules out additional dependence on unobserved values. (3) No. The later return does not change the earlier response process; dependence on the unobserved current symptom makes that visit’s mechanism MNAR. (4) It is not strict MCAR because response depends on treatment arm. It is MAR conditional on treatment and the observed history, given the stated absence of further dependence on unobserved outcomes.