Why Mechanism Labels Depend On The Conditioning Set

Why the Conditioning Set Matters

A mechanism label is a statement about what information dropout depends on. The phrase “dropout is MAR” is therefore incomplete unless we specify what was observed and available when dropout occurred. A baseline covariate, for example, might explain why participants with different later outcomes have different dropout rates. If that covariate was recorded, dependence on it can be consistent with MAR; if it is omitted from the conditioning set, the same pattern can look like dependence on an unobserved outcome.

The key distinction is between the full observed information and a narrower set of variables used in a particular description or model. A narrower set can be useful, but a label relative to it should not be confused with the mechanism’s classification relative to everything observed. This tutorial uses a visit-specific dropout decision to make that distinction explicit.

Definition: Consider a participant who is still under follow-up immediately before visit $t$. Let $D_t=1$ indicate dropout at that decision and $D_t=0$ indicate continued follow-up. Let $C_t$ be a specified set of information observed and available at that decision, and let $U_t$ denote the relevant full-data information not observed then. The mechanism is MAR relative to $C_t$ if $$ P(D_t=1\mid C_t,U_t)=P(D_t=1\mid C_t) $$ for the possible values of $C_t$ and $U_t$. It is MNAR relative to $C_t$ if the dropout probability still depends on $U_t$ after conditioning on $C_t$. For the usual MAR classification, $C_t$ must include the full information observed under the realized dropout pattern, not merely a selected subset used in one model.

For example, let $Y_{t-1}$ be the last observed outcome, $X$ a recorded baseline covariate, and $Y_t$ an outcome not observed for a participant who drops out before its measurement. If the dropout probability depends on $X$ but not on $Y_t$ once $X$ and the available history are fixed, the mechanism can be MAR given that full observed information. A model that conditions on $Y_{t-1}$ but leaves out $X$ is asking a narrower question. Its conditional independence statement may fail even though the full-information MAR statement holds.

Omitting an Observed Covariate

The arithmetic behind this distinction is marginalization. Suppose a recorded covariate $X$ is left out of the conditioning set. To calculate the dropout probability given an outcome $Y$, we average over the possible values of $X$ among participants with that outcome:

P(D=1\mid Y=y) = \sum_x P(D=1\mid X=x,Y=y)P(X=x\mid Y=y).

Even if $D$ and $Y$ are independent conditional on $X$, the weights $P(X=x\mid Y=y)$ can differ across values of $Y$. The resulting averages can therefore differ. This does not establish that dropout depends on an unobserved outcome after conditioning on the full observed information; it can reflect the omitted, observed covariate.

Worked Example: An Omitted Baseline Covariate Creates Apparent Dependence

Suppose a hypothetical study has a binary baseline covariate $X$, with $P(X=1)=P(X=0)=0.5$. A later outcome $Y$, which may be unobserved after dropout, satisfies $P(Y=1\mid X=1)=0.8$ and $P(Y=1\mid X=0)=0.2$. Among participants at risk of dropout, suppose $P(D=1\mid X=1,Y)=0.6$ and $P(D=1\mid X=0,Y)=0.2$, with the dropout probability not otherwise depending on $Y$ once $X$ is fixed. Find the dropout probabilities conditional on $Y$ alone.

First, $P(Y=1)=0.5(0.8)+0.5(0.2)=0.5$. Thus $P(X=1\mid Y=1)=0.8$, while $P(X=1\mid Y=0)=0.2$. Averaging over $X$, $$ P(D=1\mid Y=1)=0.8(0.6)+0.2(0.2)=0.52, $$ and $$ P(D=1\mid Y=0)=0.2(0.6)+0.8(0.2)=0.28. $$

The dropout probabilities differ by $0.52-0.28=0.24$, or 24 percentage points, when compared by $Y$ alone. Yet for either value of $X$, the specified dropout probability is the same at $Y=1$ and $Y=0$. If $X$ is observed and included in the full conditioning set, this mechanism is MAR relative to that set. The marginal association with $Y$ arises because $X$ predicts both $Y$ and dropout. A $Y$-only description misses that explanation.

The example does not show that every omitted-variable association is harmless. It shows why the classification must name its conditioning set. If an observed variable belongs to the full observed history, dependence on that variable is allowed under MAR. A reduced model may still be inadequate for describing or adjusting for the observed dropout pattern.

Adding Information Can Change the Label

The same logic applies when an observed covariate is added to a conditioning set. Suppose a recorded covariate $Z$ predicts both dropout and an outcome that may be missing. Comparing dropout across outcome values without $Z$ can show an association. Comparing participants at the same value of $Z$ may remove that association. In this case, adding $Z$ changes the conditional statement being assessed; it does not change what happened to the participants.

Worked Example: Conditioning on a Recorded Covariate

In a hypothetical follow-up study, let $Z$ be a binary symptom measure recorded before the dropout decision, and let $Y$ be an outcome not observed after dropout. Assume $P(Z=1)=0.5$, $P(Y=1\mid Z=1)=0.75$, and $P(Y=1\mid Z=0)=0.25$. Suppose dropout probabilities are 0.70 for $Z=1$ and 0.10 for $Z=0$, with no additional dependence on $Y$ once $Z$ is fixed. Calculate dropout probabilities by $Y$, then describe the mechanism.

Here $P(Y=1)=0.5(0.75)+0.5(0.25)=0.5$, so $P(Z=1\mid Y=1)=0.75$ and $P(Z=1\mid Y=0)=0.25$. Therefore, $$ P(D=1\mid Y=1)=0.75(0.70)+0.25(0.10)=0.55, $$ whereas $$ P(D=1\mid Y=0)=0.25(0.70)+0.75(0.10)=0.25. $$ The difference is $0.55-0.25=0.30$, or 30 percentage points. But within either value of $Z$, dropout has the same probability for $Y=1$ and $Y=0$. Because $Z$ is observed before the decision, the mechanism is MAR relative to a conditioning set that includes $Z$. A set omitting $Z$ does not capture the conditional explanation.

An expanded conditioning set is not automatically better in every practical sense: variables must be measured at the relevant time, their values must genuinely be available, and the conditional model must be estimable. Nor does adding observed variables convert every mechanism to MAR. If dropout still depends on relevant unobserved information after conditioning on the full observed history, the MAR condition fails.

Worked Example: An Observed Covariate Does Not Remove All Dependence

Consider a hypothetical setting with a recorded binary covariate $X$, an unobserved outcome $Y$ for those who drop out, and the following dropout probabilities at either fixed value of $X$: $P(D=1\mid X,Y=1)=0.70$ and $P(D=1\mid X,Y=0)=0.20$. What does conditioning on $X$ establish?

At each value of $X$, the conditional dropout probabilities differ by $0.70-0.20=0.50$, or 50 percentage points. Thus, $$ P(D=1\mid X,Y=1)\ne P(D=1\mid X,Y=0). $$ Including $X$ in the conditioning set does not make dropout independent of $Y$. If $Y$ is unobserved for those who drop out and is relevant to the mechanism, the mechanism is not MAR relative to the full observed information in this example. The observed covariate alone does not account for the remaining dependence.

How to State and Assess a Mechanism Label

A clear mechanism statement identifies the decision, the information set, and the variables still unobserved. One useful sequence is:

Specify the dropout decision.
State the visit and who is at risk. Clarify which measurements were available before the decision.

Name the conditioning set.
List the observed outcomes and covariates included. Do not treat a convenient subset as if it were automatically the full observed history.

Identify what remains unobserved.
For the relevant dropout pattern, distinguish missing outcomes or other unavailable information from recorded variables.

State the conditional claim.
Ask whether dropout still depends on the unobserved information after conditioning on the stated observed set. Qualify the label by that set.

The familiar categories can be compared as follows. MCAR is the strongest restriction in this table: it rules out dependence on both observed and unobserved data. MAR allows dependence on observed data but not on unobserved data after conditioning on the full observed information. MNAR allows residual dependence on unobserved information.

Label	Conditional description	Implication for the conditioning set
MCAR	Dropout is independent of the relevant data.	No dependence on observed or unobserved values.
MAR	Given the full observed information, dropout does not depend on unobserved values.	Dependence on recorded outcomes or covariates is permitted.
MNAR	Dropout retains dependence on unobserved values after conditioning on the full observed information.	Adding observed variables may not remove the residual dependence.

Observed-data diagnostics can help reveal whether dropout rates vary across recorded histories or covariate values. Such differences can indicate that a description omitting those variables is too crude. However, a pattern in observed data cannot by itself establish that dropout is independent of outcomes that are missing precisely because of dropout. The distinction between MAR and MNAR involves unobserved values, so it cannot generally be settled by checking observed dropout rates alone.

Common Pitfalls and Limitations

Calling a mechanism MNAR because a reduced model shows association. If dropout and an unobserved outcome are associated after conditioning on only one observed variable, another recorded covariate may explain that association. Check the full observed history before making the MAR-versus-MNAR claim.
Calling dropout MAR because it is predictable. Predictability from observed variables is compatible with MAR, but it is not enough to establish the required conditional independence from unobserved values.
Adding a variable that was not actually observed at the decision. A variable known only later, or unavailable for participants who dropped out, cannot simply be treated as observed history for those decisions.
Treating a label as a permanent property of a dataset. The conditional statement changes when the information set changes. The data collection process is the same; the description and its conditioning set differ.

The main alternative to a full-history description is a deliberately reduced conditional description. That can be useful for a particular analysis, but it should be named as such and should not be mistaken for the standard full-observed-data MAR condition. When scientific knowledge suggests dependence on missing values, the limitation is more fundamental: observed data alone may not distinguish the proposed mechanism from alternatives. The label therefore records assumptions about how dropout relates to unobserved information, not a fact that can always be verified from the observed sample.

Key takeaway: State the conditioning set whenever assigning a mechanism label. Dependence on a recorded covariate can produce an apparent outcome association when that covariate is omitted; dependence that remains after conditioning on the full observed history is not explained away by relabelling a reduced model.

Check Your Understanding

For each item, identify the conditioning-set issue and compare your reasoning with the solutions below.

A recorded baseline variable $X$ predicts dropout and an outcome that is sometimes missing. A dropout comparison by outcome alone shows different rates. Does that alone prove MNAR?
At a fixed value of every recorded variable available before dropout, the dropout probability still differs across values of a missing outcome. Which mechanism description is appropriate?
A symptom measure was collected before a visit, but an analyst leaves it out of the conditioning set. What should be specified before interpreting an outcome-related dropout pattern?
Why can observed dropout rates across recorded histories not generally prove MAR?

Solutions. (1) No. The association may arise because $X$ predicts both dropout and the outcome; assess the relationship after conditioning on the full observed information. (2) The mechanism is MNAR relative to that full observed set, because dependence on unobserved information remains. (3) State that the description is conditional on a reduced set, identify the omitted recorded symptom measure, and distinguish this from the full-history MAR claim. (4) MAR concerns independence from values that are unobserved for some participants; observed rates alone do not reveal those missing values or verify that independence.