Deriving Bias From Omitting the Outcome

Why Omitting the Outcome Can Change the Estimated Association

When a predictor has missing values, an imputation model that uses only the predictor’s marginal distribution can reproduce its overall mean and variance while failing to reproduce its association with the outcome. If the analysis then regresses the outcome on this completed predictor, the resulting association may be too weak. This tutorial derives that effect in a simple continuous-outcome setting. It makes precise the issue raised in Why the Outcome Belongs in Predictor Imputation.

The derivation is deliberately specific: one predictor is partly missing, the outcome is fully observed, missingness is completely at random, and missing predictor values are drawn independently from the predictor’s marginal distribution. Under these assumptions, the completed-data slope is the true slope multiplied by the fraction of predictor values that were observed. The formula is not a universal correction for every imputation procedure or missingness mechanism.

Definition: Let \(X\) be a predictor and \(Y\) a continuous outcome. Let \(R=1\) when \(X\) is observed and \(R=0\) when \(X\) is missing. Write \(r=P(R=1)\) for the proportion observed. For a missing value, let \(\tilde X\) denote a random draw from the marginal distribution of \(X\), made independently of \(Y\). The completed predictor is \(X^*=RX+(1-R)\tilde X\).

Model and Assumptions

Assume the outcome follows the linear model

Y=\alpha+\beta X+\varepsilon, \qquad E(\varepsilon\mid X)=0, \qquad \operatorname{Var}(X)=\sigma_X^2>0.

Here \(\beta\) is the population regression slope: the expected change in \(Y\) per one-unit increase in \(X\), under this model. The condition \(E(\varepsilon\mid X)=0\) ensures that the population linear regression slope of \(Y\) on \(X\) is \(\beta\). For the attenuation derivation, also assume:

Outcome observed: \(Y\) is available for every individual in the analysis.
Missing completely at random: \(R\) is independent of \((X,Y)\), so the observed fraction does not depend on the predictor or outcome.
Marginal imputation: each \(\tilde X\) has the same mean and variance as \(X\), and is independent of \(Y\) and of the missingness indicator. In particular, the imputation does not use the outcome.
Population calculation: the result describes population moments, or their large-sample counterparts. Finite-sample estimates fluctuate around these moments.

A normal distribution for \(X\) is not required for this covariance result. It would be needed for some particular normal-theory imputation models, but not for the calculation below. If missingness depends on observed covariates, or if the imputation model uses predictors besides \(X\), the simple factor \(r\) need not apply without a separate derivation.

Deriving the Attenuation

The key point is that the observed predictor values retain their association with \(Y\), while the independently drawn missing values do not. Because the marginal draw has the same mean as \(X\), replacing missing values this way preserves the predictor’s population variance. The numerator of the regression slope shrinks, but its denominator does not.

\begin{aligned} \operatorname{Cov}(X^*,Y) &=\operatorname{Cov}\{RX+(1-R)\tilde X,Y\}\\ &=r\operatorname{Cov}(X,Y)+(1-r)\operatorname{Cov}(\tilde X,Y)\\ &=r\operatorname{Cov}(X,Y), \end{aligned}

The second equality uses independence of \(R\) from \((X,Y)\); the last uses independence of \(\tilde X\) and \(Y\). For the variance, both the observed values and the marginal draws have mean \(\mu_X\) and variance \(\sigma_X^2\). Since the choice between them is independent of their values, the completed predictor has that same marginal distribution of moments:

\operatorname{Var}(X^*)=r\sigma_X^2+(1-r)\sigma_X^2=\sigma_X^2.

The population regression slope from regressing \(Y\) on \(X^*\) is covariance divided by predictor variance. Also, the original model gives \(\operatorname{Cov}(X,Y)=\beta\operatorname{Var}(X)=\beta\sigma_X^2\). Therefore:

\beta^* =\frac{\operatorname{Cov}(X^*,Y)}{\operatorname{Var}(X^*)} =\frac{r\beta\sigma_X^2}{\sigma_X^2} =r\beta.

Result: Under the stated assumptions, marginally imputing missing predictor values with draws independent of the outcome yields a population regression slope \(\beta^*=r\beta\). The bias is \(\beta^*-\beta=-(1-r)\beta\), so the slope is attenuated toward zero when \(0<r<1\). This is a result about this imputation scheme and these assumptions, not a general law for all missing-data analyses.

Worked Examples

Worked Example: Calculating the Attenuated Slope

Consider a hypothetical population in which \(X\) is measured in years and has mean 10 years and variance \(4\) years squared. Suppose \(Y\), measured in outcome units, follows \(Y=30+3(X-10)+\varepsilon\), where \(E(\varepsilon\mid X)=0\) and \(\operatorname{Var}(\varepsilon)=5\). Assume 60% of predictor values are observed and the missing values are drawn from the marginal distribution of \(X\), independently of \(Y\).

The true slope is \(\beta=3\) outcome units per year. The true covariance is \( \operatorname{Cov}(X,Y)=\beta\operatorname{Var}(X)=3(4)=12 \) outcome-unit-years. After marginal imputation, the completed-data covariance is \( r\operatorname{Cov}(X,Y)=0.60(12)=7.2 \) outcome-unit-years, while the completed predictor variance remains \(4\) years squared. Thus the completed-data slope is

\hat\beta_{\text{population}}^* =\frac{7.2}{4} =1.8\ \text{outcome units per year}.

The slope is 1.8 rather than 3, a reduction of \(3-1.8=1.2\) outcome units per year. This calculation describes the population target under the assumptions, not a guarantee that a finite sample estimate will equal 1.8 exactly.

Worked Example: Interpreting Bias for a Negative Slope

In a second hypothetical setting, suppose a fully specified linear model has a true slope of \(-2.4\) mmHg per year of age. Assume 75% of age values are observed, missingness is completely at random, and missing ages are drawn independently of blood pressure from the marginal age distribution.

Here \(r=0.75\), so the expected completed-data slope under this imputation scheme is

\beta^*=r\beta=0.75(-2.4)=-1.8\ \text{mmHg per year}.

The bias is \(\beta^*-\beta=-1.8-(-2.4)=0.6\) mmHg per year. Because the true slope is negative, a positive bias moves the estimate toward zero: the imputed analysis indicates a weaker negative association. “Toward zero” describes the direction of this particular bias; it does not mean every imputation model that omits the outcome must produce this exact factor.

Worked Example: Using the Outcome in the Predictor Imputation

Return to the first hypothetical model, with \(E(X)=10\), \(E(Y)=30\), \(\operatorname{Var}(X)=4\), \(\operatorname{Var}(Y)=41\), and \(\operatorname{Cov}(X,Y)=12\). These values follow from the model: \(\operatorname{Var}(Y)=3^2(4)+5=41\). For a bivariate normal model, the conditional distribution of \(X\) given \(Y\) has mean

E(X\mid Y) =10+\frac{12}{41}(Y-30).

For a hypothetical person with \(Y=38\), the conditional mean is \(10+(12/41)(38-30)=10+96/41\approx12.34\) years. The conditional variance is \(4-12^2/41=4-144/41\approx0.4878\) years squared. A draw from this conditional distribution uses the observed outcome to retain the predictor–outcome relationship; it also includes conditional residual variation rather than assigning every missing age its conditional mean.

This calculation illustrates the relevant contrast, not a complete prescription for every MICE analysis. The conditional model must be suitable for the analysis and data structure. Simply substituting the conditional mean would omit residual variation and would not be equivalent to drawing from the conditional distribution.

Diagnostics, Alternatives, and Limits

A useful diagnostic is to inspect the predictor–outcome relationship in the completed data, separating observed predictor values from imputed values. Scatterplots, group summaries, or regressions with the two groups distinguished can reveal whether imputed values carry little association with the outcome. Also check whether the imputed predictor distribution has plausible spread and whether it differs sharply from the observed values. These checks can expose an omitted outcome or other model mismatch, but they cannot verify the unobserved true predictor values. Agreement in the overall mean and variance alone is not evidence that the relationship needed by the analysis has been preserved.

In multiple imputation, repeating marginal draws does not by itself fix this structural problem. Each draw still lacks the outcome-dependent component of the missing predictor distribution. Averaging estimates across completed data sets cannot restore that omitted conditional association merely by increasing the number of imputations. Parameter uncertainty in an imputation model and the choice of variables in that model address different issues.

Approach	What it does to the predictor–outcome link	Important qualification
Marginal random draws that omit the outcome	Under the assumptions derived here, the completed-data slope is \(r\beta\).	The exact factor depends on the specified assumptions and imputation distribution.
Draws from a suitable outcome-conditional model	Can preserve the relationship by making missing predictor draws depend on the observed outcome.	The conditional model and its assumptions must be appropriate for the analysis.
Deterministic mean imputation	Does not retain individual-level variability among imputed values.	In a simple unadjusted linear regression under MCAR, its population slope need not equal the \(r\beta\) result above; variance and uncertainty problems remain.
Complete-case analysis	Uses only records with observed predictor values.	It discards incomplete records; its validity depends on the missingness mechanism and analysis target.

The deterministic mean-imputation distinction matters. In a centered simple regression, replacing each missing \(X\) by its mean makes the completed predictor equal to \(RX\). Under MCAR, both its covariance with \(Y\) and its variance are multiplied by \(r\), so their ratio can remain \(\beta\). That does not make mean imputation generally sound: it creates a point mass at the mean and misrepresents uncertainty and the predictor distribution. It also shows why the attenuation formula above must be tied to the stated random-draw procedure rather than applied indiscriminately.

The derivation has further limits. If missingness depends on \(X\) or \(Y\), then \(R\) is not independent of \((X,Y)\), and the covariance calculation changes. If the analysis adjusts for other variables, the relevant quantity is a conditional or residual association; the unadjusted factor \(r\) cannot simply be assumed. Interactions, nonlinear terms, and complex outcome models also require imputation models that preserve the relationships represented in the analysis. Those cases need their own compatibility assessment.

Common Mistake

Common mistake: “The imputed \(X\) values have the right mean and variance, so the estimated association with \(Y\) must be preserved.” Matching the marginal distribution does not preserve the joint distribution. Under the assumptions here, independent marginal draws leave \(\operatorname{Var}(X^*)\) unchanged but multiply \(\operatorname{Cov}(X^*,Y)\) by \(r\), which gives the slope \(r\beta\). Check the predictor’s conditional relationship with the outcome, not only its marginal summaries.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Under the assumptions of the derivation, \(r=0.80\) and the true slope is \(5\) outcome units per predictor unit. What is the completed-data population slope?
The true covariance is \(6\) outcome-unit-predictor-unit, the predictor variance is \(3\) predictor-units squared, and \(r=0.50\). Calculate the true and imputed slopes.
Why does deterministic mean imputation not follow the same \(r\beta\) derivation in a simple centered regression?
Name one diagnostic that may reveal the omitted-outcome problem, and explain why it cannot prove that the imputations are correct.

Solutions. (1) \(\beta^*=r\beta=0.80(5)=4\) outcome units per predictor unit. (2) The true slope is \(6/3=2\) outcome units per predictor unit. The completed covariance is \(0.50(6)=3\), and the marginal-draw imputation preserves predictor variance at 3, so the imputed slope is \(3/3=1\) outcome unit per predictor unit. (3) With centered \(X\), mean-imputed values give \(X^*=RX\). Under MCAR, both \(\operatorname{Cov}(X^*,Y)\) and \(\operatorname{Var}(X^*)\) are multiplied by \(r\); their ratio need not be attenuated by \(r\). The imputed distribution and uncertainty are nevertheless distorted. (4) For example, compare predictor–outcome scatterplots or associations for observed and imputed predictor values. A weak association among imputed values can flag the problem, but the missing true predictor values are unavailable for direct comparison, so the diagnostic cannot establish that the imputations reproduce their true conditional distribution.