Imputation With an Outcome by Predictor Interaction

Why an Interaction Changes Predictor Imputation

When an analysis model includes an interaction, the association between its outcome and a predictor depends on another predictor. If that predictor is incomplete, imputing it requires attention to how the observed outcome informs its missing values. A model that uses the outcome only as a main effect may fail to preserve the varying association the analysis is designed to estimate.

Let \(Y\) be an outcome, \(X\) a predictor that may be missing, and \(Z\) another predictor. Suppose the substantive analysis uses a linear conditional mean with an \(X\)-by-\(Z\) interaction. The interaction means that the expected change in \(Y\) for a one-unit increase in \(X\) is allowed to vary with \(Z\). When imputing \(X\), the corresponding outcome association may therefore need to vary with \(Z\) as well.

The two product terms serve different roles. The analysis includes \(X Z\), because it models the outcome relationship. The imputation model for \(X\) may include \(Y Z\), because it models how outcome information relates to the missing predictor across levels of \(Z\). The latter is not obtained by mechanically copying the analysis formula; it reflects which conditional relationships must be represented when \(X\) is imputed.

From the Analysis Relationship to the Imputation Relationship

Write the analysis conditional mean as

At a fixed value \(Z=z\), the coefficient of \(X\) in this mean is \(\beta_X+\beta_{XZ}z\). Thus the outcome relationship with \(X\) differs across \(z\) whenever \(\beta_{XZ}\ne 0\). If \(X\) is missing, a model for its conditional distribution given \(Y\) and \(Z\) needs to be able to represent a corresponding variation in how \(Y\) informs \(X\).

A commonly useful term structure for a continuous \(X\) is

This is a conditional-mean specification, not a universal formula for the exact conditional distribution. Its key feature is that the association with \(Y\) is \(\gamma_Y+\gamma_{YZ}z\) at \(Z=z\). If \(\gamma_{YZ}=0\), the model restricts that association to be the same for every \(z\). Whether additional terms or a different distribution are needed depends on the substantive model, the distributions of the predictors, and the variable types. Including \(Y Z\) is an important structural consideration, but by itself it does not prove that the imputation model is fully compatible with the analysis model.

Why the Product Term Matters

For binary \(Z\), the role of \(Y Z\) can be seen directly by writing a single regression that permits different outcome slopes in the two groups. Suppose the imputation conditional mean is linear in \(Y\) within each group, with intercept \(a_0\) and slope \(a_1\) when \(Z=0\), and intercept \(b_0\) and slope \(b_1\) when \(Z=1\). The expression

equals \(a_0+a_1Y\) for \(Z=0\). For \(Z=1\), it equals

Thus, with binary \(Z\), the main effect of \(Z\) permits the group intercepts to differ, while \(Y Z\) permits the slopes of \(X\) on \(Y\) to differ. Without the product term, the model forces \(a_1=b_1\). This is precisely the restriction that may conflict with an analysis whose \(X\) association varies by group.

Specification, Assumptions, and Alternatives

The term structure must match the variable types and the intended conditional model. The examples above use a continuous \(X\) and a conditional mean linear in \(Y\) and \(Z\); they do not establish that a normal linear imputation model is appropriate for every \(X\). A binary \(X\), for example, needs a suitable categorical imputation model, with its linear predictor able to represent the outcome-by-\(Z\) relationship. For a continuous \(Z\), the interaction is \(Y Z\), but a straight-line change across \(Z\) may be inadequate if the substantive relationship is more complex.

As with other MICE conditional models, the imputation specification also needs to be supported by the data: relevant predictors should be available, the chosen functional form should be plausible, and the model must be estimable in the observed cases. Standard missing-data assumptions and appropriate handling of the missingness process remain important. A correctly placed interaction term cannot compensate for an unsupported missingness assumption, severe model misspecification, or too little information in a subgroup.

There are three practical approaches to representing the varying association:

• Use an explicit product term. Include \(Y Z\) in the conditional model for \(X\), alongside suitable main effects. This is compact and can be used for continuous or categorical modifiers with appropriate coding.
• Fit group-specific relationships. For a binary \(Z\), separate conditional models by group can allow different \(Y\) slopes. This uses more parameters and may be unstable when few observed \(X\) values occur in one group.
• Use a model tailored to the substantive model. This can address compatibility more directly when the simple conditional-mean form is inadequate, but it requires a carefully specified model and is not guaranteed merely by adding one product term.

A main-effects-only conditional model is the simpler alternative, but it imposes a shared outcome slope. It is appropriate only when that restriction is defensible for the imputation target; it should not be chosen just because it is easier to fit. Conversely, adding every possible product term can create sparse cells, unstable estimates, or an overfit model. Preserve the interaction needed by the analysis while checking whether the available data can support it.

Diagnostics and Common Mistakes

Diagnostics should ask whether the outcome relationship for the incomplete predictor varies across the modifier as the model permits, and whether the data support that variation. Useful checks include comparing observed \(X\)-versus-\(Y\) patterns across levels or ranges of \(Z\), checking conditional predictions for implausible values, and inspecting convergence and stability of the chained-equation procedure. For a binary modifier, group-specific plots or summaries can reveal a slope difference that a pooled main-effects model hides. These checks do not prove compatibility, but they can expose obvious restrictions or failures.

• Mistake: “The outcome is included, so the imputation model is adequate.” Including \(Y\) only as a main effect forces the same \(Y\)-to-\(X\) association at all \(Z\). If that association needs to vary, include the corresponding interaction or use another defensible specification.
• Mistake: “Copy \(X Z\) directly into the model for imputing \(X\).” Since \(X\) is the target being imputed, \(X Z\) is not the outcome-dependent term that expresses how \(Y\) informs \(X\). The relevant term in the illustrated conditional model is \(Y Z\).
• Mistake: “Adding \(Y Z\) guarantees compatibility.” The term allows one important varying association, but the conditional distribution, other required terms, and assumptions must still be appropriate.
• Mistake: “A non-significant interaction can be omitted without consequence.” A test in the observed cases may have low power, particularly with sparse groups. Whether to represent a substantive interaction is a modeling decision, not a conclusion that should rest only on a significance test.

The central limitation is that the needed conditional distribution may be more complicated than a linear mean with one product term. The exact requirements depend on the full joint structure and on the outcome and predictor types. Here, the goal is to identify the outcome-dependent relationship to preserve; deriving a particular full conditional distribution is a separate step.