Why an Outcome-Free Imputation Can Seem Plausible
When a predictor \(X\) is missing, it may seem reasonable to impute it from other predictors \(Z\) alone. Perhaps the outcome \(Y\) is not the scientific focus, its association with \(X\) looks weak, or the imputation model is intended only to reproduce the distribution of the predictors. These arguments can be relevant, but none by itself establishes that the resulting imputations are appropriate for the analysis.
The key question is not simply whether \(Y\) is a predictor of \(X\) in the full population. It is whether the proposed model reproduces the distribution of \(X\) among the cases whose \(X\) values are missing, conditional on the information available for those cases. That distribution can differ both from the population distribution and from the distribution among observed cases.
In particular, fitting a model to cases with observed \(X\) estimates \(P(X\mid Z,R_X=1)\), not automatically \(P(X\mid Z)\) and not automatically \(P(X\mid Z,R_X=0)\). This distinction matters when inclusion in the observed-\(X\) subset is related to \(Y\), and \(X\) is associated with \(Y\).
What MAR Does and Does Not Justify
For a clear comparison, suppose \(Y\) and \(Z\) are observed for everyone, and assume missingness in \(X\) is MAR conditional on \(Y\) and \(Z\). In this setting, MAR means \(R_X\) is conditionally independent of \(X\) given \(Y,Z\). Also assume positivity: within the relevant \(Y,Z\) patterns, there is a nonzero probability of observing \(X\). Then an imputation model that includes \(Y,Z\) can use observed cases to estimate the conditional distribution needed for missing cases.
This equality does not imply that \(Y\) can be omitted. To see the distinction, average over \(Y\) among cases with missing \(X\):
By contrast, the population distribution \(P(X\mid Z)\) averages over \(P(Y\mid Z)\), and the observed-case distribution averages over \(P(Y\mid Z,R_X=1)\). If the distribution of \(Y\) differs across response groups and \(X\) depends on \(Y\) after conditioning on \(Z\), those averages need not agree.
This is a sufficient condition, not a requirement that every defensible outcome-omitting model must satisfy in precisely that form. A model could be justified by a different known structure or a direct argument about \(P(X\mid Z,R_X=0)\). The burden is to establish the target distribution, rather than to assume that an outcome-free model estimates it.
Worked Example: Missingness Selected Through the Outcome
Suppose \(Z\) is absent, \(Y\) is binary with \(P(Y=1)=0.5\), and \(X=Y\). Let the probability that \(X\) is observed be 0.2 when \(Y=1\), and 0.8 when \(Y=0\). Missingness is MAR given \(Y\), because response depends on \(Y\) and not additionally on \(X\) once \(Y\) is known.
For cases with missing \(X\), the joint probabilities of outcome and missingness are \(P(Y=1,R_X=0)=0.5(0.8)=0.4\) and \(P(Y=0,R_X=0)=0.5(0.2)=0.1\). Thus \(P(R_X=0)=0.4+0.1=0.5\). Since \(X=Y\),
Among observed cases, \(P(Y=1,R_X=1)=0.5(0.2)=0.1\) and \(P(Y=0,R_X=1)=0.5(0.8)=0.4\), so \(P(R_X=1)=0.5\) and
The population value is \(P(X=1)=0.5\). Therefore, neither the observed-case proportion, 0.2, nor the population proportion, 0.5, equals the missing-case proportion, 0.8. A model fitted to observed \(X\) values without \(Y\) would not reproduce the distribution needed for these missing cases.
When Omission Can Be Defensible
The strongest simple justification is conditional independence: after conditioning on \(Z\), \(X\) does not depend on \(Y\). If missingness is MAR given \(Y,Z\), this makes the outcome irrelevant to the conditional distribution of \(X\). A second possible justification is that the specific distribution required for the missing cases is otherwise known or correctly represented by the proposed outcome-free model. These are substantive or model-based claims that require support; they do not follow merely because \(Y\) is not the primary scientific endpoint.
An outcome association that is small after adjusting for \(Z\) may make the consequences of omission small for some purposes, but “small” is an approximation, not a general guarantee. It should be assessed in relation to the analysis target and the amount and pattern of missingness. Likewise, a low overall percentage of missing \(X\) may limit the practical effect on some estimates, but does not turn an incorrect imputation model into a correct one.
The relevant estimand also matters. A model may be adequate for a limited target, such as a marginal summary of \(X\), yet fail to preserve the joint distribution of \(X,Y,Z\) needed for a regression coefficient or other association measure. Conversely, if the analysis does not use a relationship involving \(X\), a particular failure to preserve that relationship may not affect that analysis target. The question is always which features of the data distribution the chosen estimand depends on.
Worked Example: A Conditional Independence Case
Suppose \(Z\) is fixed, \(X\) and \(Y\) are independent given \(Z\), and \(P(X=1\mid Z)=0.3\). Let \(P(Y=1\mid Z)=0.5\), with \(P(R_X=1\mid Y=1,Z)=0.2\) and \(P(R_X=1\mid Y=0,Z)=0.8\). Assume the response probabilities do not depend on \(X\) once \(Y,Z\) are fixed.
The probability of missing \(X\), conditional on \(Z\), is \(0.5(1-0.2)+0.5(1-0.8)=0.5\). The joint probability that \(X=1\) and \(X\) is missing is, using \(X\perp Y\mid Z\), \(0.3[0.5(0.8)+0.5(0.2)]=0.3(0.5)=0.15\). Hence
For observed cases, \(P(R_X=1\mid Z)=0.5(0.2)+0.5(0.8)=0.5\), and \(P(X=1,R_X=1\mid Z)=0.3[0.5(0.2)+0.5(0.8)]=0.15\). Therefore \(P(X=1\mid Z,R_X=1)=0.15/0.5=0.3\) as well. Here an outcome-free model for \(X\) given \(Z\) reproduces both response-group distributions. The justification comes from the conditional independence assumption, not from the fact that missingness depends on \(Y\) alone.
Which Analysis Targets Can Be Changed?
A basic regression estimand illustrates why preserving the predictor distribution alone is not enough. Consider the population linear regression of \(Y\) on a binary \(X\) with an intercept. Its slope is \(\operatorname{Cov}(X,Y)/\operatorname{Var}(X)\). If imputation weakens the association between \(X\) and \(Y\), it can change this slope even when all observed outcomes remain unchanged.
Worked Example: Omission Alters a Regression Slope
Use the hypothetical setup from the first example: \(P(Y=1)=0.5\), \(X=Y\), and response probabilities 0.2 for \(Y=1\) and 0.8 for \(Y=0\). The true population has \(E(X)=E(Y)=0.5\), \(E(XY)=0.5\), and \(\operatorname{Var}(X)=0.5(0.5)=0.25\). Its regression slope is
Now suppose missing \(X\) values are imputed independently of \(Y\) using the observed-case proportion \(P(X=1\mid R_X=1)=0.2\), as calculated above. In the full population, observed cases contribute mass 0.1 to \(X=1,Y=1\) and 0.4 to \(X=0,Y=0\). Among missing cases, the \(Y=1\) mass is 0.4 and the \(Y=0\) mass is 0.1. Imputation assigns \(X=1\) to mass \(0.4(0.2)=0.08\) in the \(Y=1\) group and \(0.1(0.2)=0.02\) in the \(Y=0\) group.
The completed distribution therefore has \(E(X)=0.1+0.08+0.02=0.20\), \(E(Y)=0.50\), and \(E(XY)=0.1+0.08=0.18\). Its variance for \(X\) is \(0.20(0.80)=0.16\), and its covariance is \(0.18-(0.20)(0.50)=0.08\). The resulting slope is
In this limiting distribution, the slope is 0.5 rather than the true value 1. If missing \(X\) were instead imputed from the correct \(P(X\mid Y,R_X=0)\), then \(X=Y\) would be restored for the missing cases and the slope would be 1. The calculation demonstrates that preserving the observed outcomes while imputing \(X\) without \(Y\) need not preserve the regression estimand.
This example uses an extreme deterministic relationship to make the discrepancy transparent. In applied analyses, the extent and direction of any change depend on the response mechanism, the conditional relationship, the model fitted, and the estimand. The example is not a universal formula for bias.
Assessing the Argument and Choosing an Approach
Before excluding \(Y\), state the intended analysis estimand and the distribution the imputation model must reproduce. Then identify what assumptions would make the proposed outcome-free model valid for that distribution. In particular, examine whether \(X\) and \(Y\) remain associated conditional on \(Z\), and whether response in \(X\) varies with \(Y\). These questions concern the modeled relationships, not just the overall missingness percentage.
| Approach | What it models | Key consideration |
|---|---|---|
| Include observed \(Y\) in predictor imputation | \(P(X\mid Y,Z)\) | Under MAR given \(Y,Z\), positivity, and a correctly specified model, this targets the conditional distribution needed for missing \(X\). |
| Omit \(Y\), fitting on observed \(X\) cases | Typically \(P(X\mid Z,R_X=1)\) | It is suitable only if this model reproduces \(P(X\mid Z,R_X=0)\) for the target analysis. |
| Use a justified outcome-free model | A specified model for \(P(X\mid Z,R_X=0)\) | Requires a defensible condition or external structure; it is not justified by omission alone. |
Including a fully observed outcome is therefore the usual protection for relationships between an incomplete predictor and the outcome. It is not a universal remedy: the imputation model still needs suitable predictors, adequate support in relevant patterns, and a form capable of representing the needed conditional distribution. If \(Y\) is also missing, its missingness and imputation require additional modeling rather than a simple rule to exclude it.
Diagnostics can flag trouble but cannot prove that omission is harmless. Compare observed-case distributions across outcome levels, inspect whether the predictor–outcome relationship remains after conditioning on \(Z\), and assess how the analysis estimate changes under plausible outcome-including and outcome-omitting specifications. Such comparisons are sensitivity checks, not tests that establish a missingness assumption from observed data alone. If the observed data provide little information for a particular \(Y,Z\) pattern, model-based extrapolation is especially important and should be reported as a limitation.
Common Mistake
The introductory material on preserving the analysis estimand emphasizes that imputation should retain the relationships needed by the planned analysis. Here the practical consequence is specific: a decision to omit \(Y\) is a claim about the missing-case distribution and the estimands it preserves, not merely a simplification of the predictor list.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- With \(Z\) observed, what is the target distribution for imputing \(X\) when \(Y\) is omitted?
- Under MAR given \(Y,Z\), does that assumption alone imply that an observed-case model for \(X\mid Z\) is valid for missing cases?
- State a sufficient condition under which \(Y\) can be omitted from the conditional model for \(X\) in the setting considered here.
- In a linear regression of \(Y\) on \(X\), which joint-distribution feature can be altered if imputation weakens the \(X,Y\) association?
Solutions: (1) The target is \(P(X\mid Z,R_X=0)\), unless a different model is shown to reproduce it. (2) No. MAR given \(Y,Z\) gives \(P(X\mid Y,Z,R_X=0)=P(X\mid Y,Z,R_X=1)\), but averaging over \(Y\) can differ across the missing and observed groups. (3) \(X\perp Y\mid Z\) is sufficient under MAR given \(Y,Z\); then the conditional distribution of \(X\) given \(Z\) is the same in both response groups. (4) The covariance \(\operatorname{Cov}(X,Y)\), and potentially \(\operatorname{Var}(X)\) as well, can change; because the slope is their ratio, the regression estimand can change.