What Goes Wrong When an Interaction Is Omitted

Why an Omitted Interaction Matters

A regression interaction says that the association between a predictor and an outcome changes with another predictor. If that interaction is part of the analysis model but omitted when imputing a predictor, the imputed values may fail to preserve the subgroup-specific relationships the analysis is meant to estimate. The consequence can be more than a small loss of precision: the fitted interaction itself can be distorted.

The direction and size of the distortion depend on the data distribution, missingness pattern, imputation model, and analysis model. Omitting an interaction does not guarantee attenuation, inflation, or even bias in a particular direction. The central issue is that the imputation model may impose conditional relationships that are incompatible with the analysis model.

Definition: An imputation model omits an analysis interaction when it does not represent the interaction-dependent conditional relationship needed to preserve the analysis estimand. For example, if the analysis includes \(X Z\), but the model imputing \(X\) from \(Y\) and \(Z\) uses only additive terms, it omits a possible \(Y Z\) dependence in the conditional distribution of \(X\).

Consider the continuous-outcome regression

E(Y\mid X,Z)=\beta_0+\beta_X X+\beta_Z Z+\beta_{XZ}XZ.

For binary \(Z\), the slope of \(Y\) on \(X\) is \(\beta_X\) when \(Z=0\), and \(\beta_X+\beta_{XZ}\) when \(Z=1\). Thus, \(\beta_{XZ}\) is the difference between those two slopes. The earlier material on interactions as part of the analysis model and imputing predictors under an interaction model establishes why analysis terms must be considered when specifying predictor imputations. Here we examine what can happen when that structure is absent.

How the Imputation Misspecification Enters

Suppose \(X\) is incomplete and \(Y\) is observed for everyone. An additive regression imputation might model the conditional mean of \(X\) as

E(X\mid Y,Z)=\gamma_0+\gamma_Y Y+\gamma_Z Z,

whereas the needed conditional mean may vary with \(Y Z\) as well:

E(X\mid Y,Z)=\gamma_0+\gamma_Y Y+\gamma_Z Z+\gamma_{YZ}YZ.

If the latter term matters, the additive model cannot reproduce the way the \(X\)-versus-\(Y\) relationship changes across values of \(Z\). Its predictions for missing \(X\) can therefore place observations in the wrong locations relative to the subgroup-specific regression lines. Subsequent analysis of the completed data may then estimate the \(X Z\) interaction poorly.

This argument does not say that the omitted term must create a particular numerical bias in every sample. For instance, if there are no missing \(X\) values, the imputation model has no effect on the analysis. Nor does an omitted term necessarily matter when its true coefficient is zero or when the missing cases do not involve the part of the covariate distribution where the misspecification matters. Those are substantive and design-dependent conditions, not automatic protections.

Key distinction: Multiple imputation can represent uncertainty about missing values under the model that is specified. It does not, by pooling multiple draws, correct a systematically misspecified imputation model. More imputations reduce Monte Carlo error in the pooled calculation; they do not remove bias caused by omitting an important relationship.

Worked Example: How Completed Values Change the Fitted Interaction

Suppose a hypothetical analysis uses \(Y\) as the outcome, continuous \(X\) as a predictor, and binary \(Z\). The data-generating regression has intercept 1 and slope 2 for \(Z=0\), and intercept 2 and slope 5 for \(Z=1\). The true interaction, the difference in slopes, is therefore \(5-2=3\) outcome units per unit of \(X\).

For a small illustrative set of six rows, four observed \(X\) values are \((0,2)\) in each group. The other two \(X\) values are missing, while \(Y\) and \(Z\) are observed. Suppose an additive imputation model fitted using a larger observed portion of the hypothetical data gives the same conditional-mean prediction, \(1.4828\), for each of those two missing values. We use these fixed predictions only to isolate the geometry of one completed data set; this is not a claim that deterministic mean substitution is a suitable multiple-imputation procedure.

The resulting completed pairs are \((X,Y)=(0,1),(2,5),(1.4828,5)\) for \(Z=0\), and \((0,2),(2,12),(1.4828,9)\) for \(Z=1\). Fit a separate intercept and slope in each group. The slope is \(S_{XY}/S_{XX}\), where \(S_{XY}=\sum XY-(\sum X)(\sum Y)/n\) and \(S_{XX}=\sum X^2-(\sum X)^2/n\).

In both groups, \(\sum X=3.4828\), \(\sum X^2=6.19869584\), and \(n=3\). Therefore,

S_{XX}=6.19869584-\frac{(3.4828)^2}{3} =6.19869584-\frac{12.12989584}{3} =2.15539723.

For \(Z=0\), \(\sum Y=11\) and \(\sum XY=17.414\). Hence,

S_{XY}=17.414-\frac{(3.4828)(11)}{3} =17.414-12.77026667 =4.64373333, \qquad \hat\beta_{X\mid Z=0}=\frac{4.64373333}{2.15539723}\approx2.1545.

For \(Z=1\), \(\sum Y=23\) and \(\sum XY=37.3452\). Thus,

S_{XY}=37.3452-\frac{(3.4828)(23)}{3} =37.3452-26.70146667 =10.64373333, \qquad \hat\beta_{X\mid Z=1}=\frac{10.64373333}{2.15539723}\approx4.9382.

The fitted interaction is the difference in fitted slopes:

\hat\beta_{XZ}=4.9382-2.1545=2.7837.

In this completed data set the fitted interaction is about 2.784, rather than the data-generating value 3. The example shows how a particular set of imputed values changes the fitted interaction; by itself, it does not establish the average bias over repeated samples or imputations. The numerical calculation also shows why the centered sums must be computed from the displayed pairs, rather than inferred from rounded intermediate values.

What Pooling Can and Cannot Do

With \(m\) completed data sets, let \(Q_j\) be the analysis estimate of the interaction in imputation \(j\), and let \(U_j\) be its estimated variance. Rubin’s rules define the pooled estimate \(\bar Q\), the average within-imputation variance \(W\), the between-imputation variance \(B\), and the total variance \(T\) as follows:

\bar Q=\frac{1}{m}\sum_{j=1}^{m}Q_j,\qquad W=\frac{1}{m}\sum_{j=1}^{m}U_j,\qquad B=\frac{1}{m-1}\sum_{j=1}^{m}(Q_j-\bar Q)^2,\qquad T=W+\left(1+\frac{1}{m}\right)B.

The within-imputation component represents estimated sampling variance within completed data sets. The between-imputation component captures variation in estimates across imputations. The factor \(1+1/m\) accounts for the finite number of imputations. These rules combine uncertainty estimates under the imputation and analysis procedures used; they do not test whether the imputation model correctly represents the needed relationships.

Worked Example: Pooling Estimates From an Omitted-Interaction Imputer

Suppose five hypothetical completed data sets use an imputation model that omits a needed interaction. Their fitted interaction estimates are \(2.4, 2.6, 2.5, 2.7,\) and \(2.3\), and each analysis reports a standard error of \(0.70\). The pooled estimate is

\bar Q=\frac{2.4+2.6+2.5+2.7+2.3}{5}=2.5.

Each within-imputation variance is \(0.70^2=0.49\), so \(W=0.49\). The squared deviations from 2.5 are \(0.01,0.01,0,0.04,\) and \(0.04\), giving

B=\frac{0.01+0.01+0+0.04+0.04}{5-1}=0.025, \qquad T=0.49+\left(1+\frac{1}{5}\right)(0.025)=0.52.

The pooled standard error is \(\sqrt{0.52}\approx0.7211\). Using a large-sample normal critical value of 1.96 for illustration, the approximate 95% interval is

2.5\pm1.96(0.7211)=[1.087,\ 3.913].

The interval is on the scale of the interaction: outcome units per unit of \(X\), comparing the \(Z=1\) slope with the \(Z=0\) slope. If the data-generating interaction were known to be 3, as in the preceding hypothetical example, this pooled interval includes 3. That does not establish that the procedure is unbiased or well calibrated. The estimates may be systematically displaced under repeated sampling even though this particular interval includes the target.

Diagnosing and Responding to the Problem

No single diagnostic proves compatibility. Instead, examine whether the imputation model can reproduce the relationships required by the analysis, and assess the impact of plausible alternatives. Useful checks include:

Compare model terms. List the terms in the analysis model and check whether the imputation model for an incomplete predictor can represent the dependence associated with them.
Inspect conditional patterns. Among observed cases, examine plots or summaries of \(X\) against \(Y\) separately by \(Z\). A change in the slope across groups is evidence against an additive conditional mean for \(X\) given \(Y\) and \(Z\).
Compare completed-data relationships. Check whether imputed and observed \(X\) values yield plausible subgroup-specific patterns and whether the fitted interaction changes sharply across imputations.
Run sensitivity analyses. Compare pooled results from a model that includes the needed interaction structure with results from a simpler model. Large changes in the estimate or its uncertainty flag sensitivity to the imputation specification.

These checks have limits. The observed cases may not represent the covariate relationships among cases with missing \(X\), and a plot cannot verify an assumption about unobserved values. Missingness mechanisms and model compatibility are distinct considerations: diagnostics can reveal conflicts or implausibility, but cannot directly show the unobserved values.

The main alternative is to use an imputation model that preserves the relevant interaction, or a joint model compatible with the analysis. A simpler additive imputation model may be easier to fit, but can impose the wrong conditional structure. A richer model can better preserve the target relationship, but may require more data and careful specification. The appropriate choice depends on the estimand, available information, and the stability of the fitted models.

Worked Example: Distinguishing Between-Imputation Variation From Model Bias

Consider a second hypothetical analysis in which an imputation model designed to preserve the interaction produces five estimates \(2.8,3.1,3.0,3.2,\) and \(2.9\), each with standard error \(0.70\). Their mean is

\bar Q=\frac{2.8+3.1+3.0+3.2+2.9}{5}=3.0.

The squared deviations from 3.0 are \(0.04,0.01,0,0.04,\) and \(0.01\), so \(B=0.10/4=0.025\). As before, \(W=0.49\) and \(T=0.49+1.2(0.025)=0.52\), giving \(SE(\bar Q)=\sqrt{0.52}\approx0.7211\). The approximate large-sample 95% interval is \(3.0\pm1.96(0.7211)=[1.587,4.413]\).

This example has the same between-imputation variance as the previous one but a different pooled center. The common \(B\) does not mean that the two imputation strategies are equally appropriate: between-imputation variance measures variation across the particular imputations, not displacement of the estimator from its target. Under repeated sampling, the two strategies could have different biases even when their \(B\) values happen to match.

Common Pitfalls

“Many imputations will fix the omitted interaction.” More imputations make the Monte Carlo approximation more stable; they do not make a misspecified conditional model correct.
“A small \(B\) shows the imputation model is adequate.” The imputations can agree closely and still produce estimates centered away from the target.
“Omitting an interaction always attenuates it.” The direction of distortion is not universal. It depends on the joint distribution, missingness pattern, and model details.
“A pooled interval containing the true value proves compatibility.” One interval cannot establish repeated-sampling calibration or absence of bias.

A practical rule is to treat a mismatch between analysis and imputation terms as a modeling concern to investigate, not as a problem automatically solved by pooling. The target interaction should guide imputation-model construction, and sensitivity to plausible specifications should be reported when it matters to the scientific conclusion.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

In \(E(Y\mid X,Z)=\beta_0+\beta_X X+\beta_Z Z+\beta_{XZ}XZ\), what does \(\beta_{XZ}\) represent when \(Z\) is binary?
Why can increasing the number of imputations fail to correct bias from an omitted interaction?
For the first worked example, what is the fitted interaction if the \(Z=0\) slope is 2.1545 and the \(Z=1\) slope is 4.9382?
Can a small between-imputation variance establish that an imputation model is compatible with the analysis? Why or why not?

Solutions: (1) It is the difference between the \(X\) slopes for \(Z=1\) and \(Z=0\). (2) More imputations reduce Monte Carlo error in pooling but retain the model’s misspecified conditional structure. (3) \(4.9382-2.1545=2.7837\), approximately 2.784 outcome units per unit of \(X\). (4) No. \(B\) measures variation among imputation-specific estimates, not bias or whether their common center is correct.