Missingness Mechanisms and Causal Identification

Why Missingness Mechanisms Matter for Causal Effects

A missing outcome is not merely an absent entry in a data table. The reason it is absent can affect what the observed data reveal about the outcomes that were not recorded. If outcome measurement is unrelated to the outcome and treatment, observed outcomes may represent the full target population. If measurement depends on observed characteristics, identification may still be possible by accounting for those characteristics. If it depends on the missing outcome itself, the unobserved outcomes can remain systematically different in ways the observed data cannot determine.

The key distinction is between the causal assumptions that connect treatment assignment to potential outcomes and the missingness assumptions that connect observed records to unobserved values. Neither set of assumptions substitutes for the other. For example, random treatment assignment does not by itself guarantee that outcomes missing after assignment can be recovered from the outcomes that were recorded.

Definition: For a variable \(Z\), let \(R_Z=1\) if \(Z\) is observed and \(R_Z=0\) if it is missing. For an outcome \(Y\), \(R_Y\) is its observation indicator. Let \(A\) denote treatment and \(X\) denote fully observed pretreatment covariates. The complete data include values such as \(A,X,Y\), whether or not all of them are observed for every unit.

The definitions below describe relationships between the observation indicators and data values. They are not labels for how large a fraction is missing: a variable can be mostly observed under any of the three mechanisms.

Three Missingness Mechanisms

Definition: Missingness is MCAR (missing completely at random) when the observation pattern is independent of the complete data. For one outcome, this means \(R_Y\) is independent of \(A,X,Y\).

Missingness is MAR (missing at random) when, conditional on the observed data, the probability of the observation pattern does not depend on the values that are missing. With \(A\) and \(X\) observed and only \(Y\) potentially missing, this is \(R_Y \perp Y \mid A,X\).

Missingness is MNAR (missing not at random) when the missingness mechanism still depends on missing values after conditioning on observed data. In the same setting, this means \(R_Y\) is not independent of \(Y\) given \(A,X\).

For multiple variables that may be missing, MAR is defined pattern by pattern: given the values observed for a unit under a particular pattern, the probability of that pattern does not depend on that unit’s unobserved values. Thus, the classification depends on what is observed and included in the conditioning set. A mechanism can be MAR relative to a rich, observed set of covariates but appear MNAR if some of those covariates are not recorded.

A causal diagram can make the assumed relationships visible. For an outcome that may be missing, a diagram with arrows \(A \to R_Y\) and \(X \to R_Y\), alongside \(A,X \to Y\), can represent MAR when \(A\) and \(X\) are observed and account for the dependence between \(R_Y\) and \(Y\). An arrow \(Y \to R_Y\), not blocked by observed variables, represents dependence of outcome observation on the missing outcome and is characteristic of MNAR. MCAR corresponds to no dependence of \(R_Y\) on the complete-data variables. These diagrams describe assumptions; data with missing values alone generally cannot establish which mechanism is true.

Worked Example: Classifying an Outcome-Observation Rule

Suppose a hypothetical trial records treatment \(A\), baseline severity \(X\), and a 12-month outcome \(Y\). Investigators believe follow-up is more likely for participants with severe baseline symptoms and for participants assigned to the new treatment. Conditional on \(A\) and \(X\), they assume follow-up does not depend on the unrecorded 12-month outcome. Classify the mechanism under this assumption.

The proposed observation rule depends on observed variables: \(P(R_Y=1\mid A,X,Y)=P(R_Y=1\mid A,X)\). Therefore \(R_Y \perp Y\mid A,X\), which is MAR, not MCAR. The mechanism is not MCAR because observation probabilities vary with \(A\) and \(X\).

A corresponding diagram has \(A\to R_Y\), \(X\to R_Y\), and arrows from \(A\) and \(X\) to \(Y\), but no remaining dependence of \(R_Y\) on \(Y\) after conditioning on \(A,X\). The classification relies on the stated conditional-independence assumption; observing that follow-up rates differ across severity groups would not, by itself, prove that assumption.

What Identification Requires

To make the identification question precise, consider a binary treatment \(A\), a fully observed pretreatment covariate \(X\), and potential outcomes \(Y^1\) and \(Y^0\). The average treatment effect is \(E[Y^1-Y^0]\). The observed outcome under treatment \(a\) is \(Y=Y^a\) for units with \(A=a\), by consistency. A standard causal identification argument also assumes conditional exchangeability, \(Y^a\perp A\mid X\), and treatment positivity, \(P(A=a\mid X=x)>0\) for covariate values in the target population.

With a missing outcome, additional conditions are needed. In this setup, MAR requires \(R_Y\perp Y\mid A,X\), and response positivity requires \(P(R_Y=1\mid A=a,X=x)>0\) in the relevant treatment and covariate groups. These assumptions imply that the observed outcomes within an \(A,X\) group have the same conditional mean as all outcomes in that group.

E[Y\mid A=a,X=x,R_Y=1] = E[Y\mid A=a,X=x].

This equality is the missing-data part of the argument. The causal assumptions then connect the conditional observed-outcome mean to the potential-outcome mean. Under consistency and exchangeability, \(E[Y\mid A=a,X=x]=E[Y^a\mid X=x]\). Averaging over the target population’s distribution of \(X\) gives the identification formula:

E[Y^a] = \sum_x E[Y\mid A=a,X=x,R_Y=1]P(X=x).

For continuous \(X\), the sum is replaced by an integral. Subtracting the formula for \(a=0\) from that for \(a=1\) identifies the average treatment effect. This derivation shows why MAR alone is not enough: treatment exchangeability, treatment positivity, consistency, and response positivity are also required. Nor does MAR guarantee useful precision; if response is rare in a covariate-treatment group, the observed data may provide little information about its mean.

Worked Example: Standardizing Under MAR

Consider a hypothetical randomized trial with a binary baseline severity variable \(X\), where 60% of participants have \(X=0\) and 40% have \(X=1\). The table gives the number with observed outcomes and the mean outcome among those observed in each group. The outcome is measured in points.

Treatment \(A\)	Baseline severity \(X\)	Participants	Outcomes observed	Observed mean outcome
1	0	60	54	8
1	1	40	24	12
0	0	60	48	6
0	1	40	32	9

Assume the outcome is MAR given \(A,X\), response positivity holds in all four groups, and treatment was randomized. Randomization supplies exchangeability, and both treatments occur in each severity group. Under the MAR assumption, each observed group mean estimates the corresponding full group’s mean. Standardizing to the trial’s 60%/40% severity distribution gives:

E[Y^1]=0.60(8)+0.40(12)=4.8+4.8=9.6\text{ points},

E[Y^0]=0.60(6)+0.40(9)=3.6+3.6=7.2\text{ points}.

The estimated average treatment effect is \(9.6-7.2=2.4\) points. This is an identification calculation under the stated assumptions, not proof that the assumptions hold. In particular, if participants with unusually poor outcomes are less likely to return even after conditioning on \(A,X\), the MAR condition fails and these observed means need not represent the group means.

Why MNAR Usually Leaves the Effect Unidentified

When observation depends on the missing outcome even after conditioning on observed variables, the observed records do not reveal how the unobserved outcomes are distributed. Without additional information or assumptions, different distributions among the missing cases can produce the same observed data but different treatment effects. This is a failure of identification: the observed-data distribution does not determine a unique value of the target effect.

In a diagram, an arrow \(Y\to R_Y\) indicates that the outcome itself influences whether it is recorded. Among the observed cases, \(R_Y=1\), outcome values can then be overrepresented or underrepresented. Conditioning on observed outcomes cannot tell us how many unobserved participants had each outcome value. A causal diagram helps state this concern, but an arrow is not a numerical correction and does not by itself supply the missing distribution.

Worked Example: Two MNAR Models, One Observed Distribution

Suppose a hypothetical randomized trial has a binary outcome. In each treatment arm, half of outcomes are observed. Among observed participants, the outcome-positive fraction is 0.60 for \(A=1\) and 0.40 for \(A=0\). The observed data therefore have \(P(R_Y=1\mid A=a)=0.50\) in both arms and the stated positive fractions among respondents.

Let \(q_a=P(Y=1\mid A=a,R_Y=0)\), the positive fraction among outcomes not observed. The full outcome mean in arm \(a\) must satisfy:

E[Y\mid A=a] = P(R_Y=1\mid A=a)E[Y\mid A=a,R_Y=1] + P(R_Y=0\mid A=a)q_a.

One completion of the missing outcomes sets \(q_1=0.20\) and \(q_0=0.80\). The full means are \(0.50(0.60)+0.50(0.20)=0.40\) for treatment and \(0.50(0.40)+0.50(0.80)=0.60\) for control, giving a risk difference of \(-0.20\).

A different completion sets \(q_1=0.80\) and \(q_0=0.20\). The full means are \(0.50(0.60)+0.50(0.80)=0.70\) and \(0.50(0.40)+0.50(0.20)=0.30\), respectively, giving a risk difference of \(0.40\). Both completions reproduce the same observed response rates and observed outcome fractions. Each also permits outcome-dependent observation: the missing and observed outcome fractions differ within treatment arms. Because treatment was randomized, these full-mean differences are the corresponding average causal effects under consistency, yet they disagree. The observed data alone therefore do not identify the effect.

Reading Mechanisms Without Overclaiming

A useful way to interpret a missingness diagram is to ask what variables are observed, what predicts the observation indicator, and whether any path links the indicator to a missing value after conditioning on observed variables. The absence of a direct arrow from \(Y\) to \(R_Y\) is not, on its own, enough to establish MAR: an open path through an unobserved common cause can still make observation informative about \(Y\). Conversely, dependence on an observed treatment or covariate does not make the mechanism MNAR if conditioning on those observed variables removes dependence on the missing outcome.

Mechanism assumptions are substantive, not merely technical. MCAR is a strong condition because it rules out dependence on observed as well as unobserved data. MAR allows dependence on observed information but still makes a claim about values that were not observed. MNAR explicitly permits residual dependence on those values. The observed-data distribution can describe response patterns, but it generally cannot distinguish MAR from MNAR on its own.

Common mistake: “The outcome is missing more often in one treatment arm, so the data are MNAR.”
Correction: Different observation rates by treatment can be MAR if treatment is observed and accounts for the dependence. MNAR concerns dependence on missing values after conditioning on observed information.
Common mistake: “The trial was randomized, so missing outcomes cannot bias the treatment effect.”
Correction: Randomization addresses treatment assignment, not outcome observation. If observation depends on the outcome, the observed treatment groups can still fail to represent their full assigned groups.

Key takeaway: MCAR, MAR, and MNAR classify how observation depends on complete and observed data. A causal effect is identified only when the missingness assumptions and the causal assumptions jointly connect the observed records to the target potential-outcome means. Under MNAR, that connection is generally not determined by observed data alone.

Check Your Understanding

For each question, state the assumptions that matter rather than inferring a mechanism from response rates alone.

Suppose an outcome’s observation indicator is independent of treatment, baseline covariates, and the outcome. Which missingness mechanism is described?
Outcome observation depends on recorded baseline severity, but is independent of the outcome conditional on treatment and severity. Is this MCAR, MAR, or MNAR? Explain.
Why does randomized treatment assignment not, by itself, identify an average treatment effect when some outcomes are MNAR?
State the additional conditions, beyond MAR, used in the standardization formula for \(E[Y^a]\).
In the binary-outcome example, why do the two possible completions show failure of identification rather than merely uncertainty about a single identified effect?