Interactions as Part of the Analysis Model

Why an Interaction Belongs to the Analysis Model

A regression model with an interaction allows the association between one predictor and the outcome to vary with another predictor. This matters for imputation because an analysis model asks for a particular relationship, not merely a collection of separate associations. If the analysis estimand includes a joint effect, imputations that fail to preserve the information needed for that effect can change the analysis being carried out.

The term “interaction” is specific to a model and its scale. In a linear regression, a product term allows a predictor’s slope to vary across values of another predictor. In a model with a different link function, an interaction in the linear predictor describes variation on that link scale. It does not automatically imply the same form of interaction on every other scale.

Here \(Y\) is the outcome, \(X\) and \(Z\) are predictors, and the \(\beta\) values are regression coefficients. The terms \(\beta_1X\) and \(\beta_2Z\) are the constituent main effects, while \(\beta_3XZ\) is the product term. When a product term is included, the main-effect coefficients remain part of the model: \(\beta_1\) describes the association of \(X\) with \(Y\) when \(Z=0\), and \(\beta_2\) describes the association of \(Z\) with \(Y\) when \(X=0\). Those reference values may or may not be substantively meaningful. Centering a predictor changes the reference value represented by zero, but does not remove the underlying interaction.

Deriving the Conditional Effect

The meaning of the interaction follows directly from the model. For a fixed value of \(Z=z\), collect the terms involving \(X\). The resulting expression shows the conditional slope of \(Y\) with respect to \(X\).

Therefore, the conditional slope for \(X\) is \(\beta_1+\beta_3z\). Comparing this slope at two values \(z_1\) and \(z_2\) gives:

The difference in slopes is exactly the interaction coefficient multiplied by the difference between the two moderator values. Thus, \(\beta_3\) is the change in the \(X\) slope for a one-unit increase in \(Z\), in the units of the outcome per unit of \(X\). If \(\beta_3=0\), the slope for \(X\) is constant across \(Z\) in this linear model.

For conventional regression inference, a common working model assumes that observations are independent, the conditional mean has the stated form, and errors have mean zero given the predictors. Exact small-sample \(t\)- and \(F\)-based inference additionally relies on normally distributed errors; constant conditional error variance is a standard assumption for the usual standard errors. These assumptions concern the analysis model and its inference. They do not, by themselves, establish that an imputation model is compatible with it.

What Compatibility Requires

In MICE, variables with missing values are imputed through a sequence of conditional models. The analysis model is the regression model ultimately fitted to each completed data set. Compatibility concerns whether the imputation model or models can represent relationships consistent with that analysis model; congeniality is a related but distinct issue discussed in the earlier material on Congeniality Between Imputation and Analysis and Compatibility Versus Congeniality.

An interaction makes the joint role of its constituent predictors part of the target analysis. In the model above, \(XZ\) is not a third unrelated predictor: its value depends on both \(X\) and \(Z\), and its coefficient determines how their joint configuration relates to the outcome. Preserving only separate summaries of \(X\) and \(Z\) does not, in general, guarantee preservation of the relationship involving \(XZ\). The association of the outcome with \(X\) may differ across \(Z\), and the analysis model explicitly represents that variation.

This point matters especially when either constituent predictor is incomplete. For records with missing \(X\) or \(Z\), the product \(XZ\) cannot be evaluated from observed values alone. The completed data used for analysis must have values for both constituents, and the resulting joint patterns affect the product term and its estimated coefficient. Compatibility is therefore assessed against the whole analysis formula, including its interaction, rather than by checking only whether each variable appears somewhere in an imputation model.

This is a statement about the target relationship, not a recipe for specifying particular conditional imputation models. The exact construction depends on the variables, their measurement scales, missingness patterns, and analysis model. The essential requirement here is to recognize that an interaction adds a joint feature to the analysis target, so imputation must not silently reduce that target to an additive one.

Model Scale, Interpretation, and Diagnostics

An interaction is interpreted on the scale of the model term. For the continuous-outcome linear model, the product coefficient describes a change in slope measured in outcome units per unit of \(X\) per unit of \(Z\). In a logistic regression, a product term is an interaction on the log-odds scale; it should not be described as a constant interaction in probabilities without additional analysis. An apparent interaction can vary with scale, so the intended estimand and interpretation should be stated before fitting or imputing.

Useful checks include examining whether the interaction is scientifically motivated, whether the data contain adequate observations across the relevant joint ranges of the predictors, and whether the model’s predicted patterns are plausible. Sparse combinations, extreme predictor values, influential observations, and strong collinearity between product and constituent terms can make an interaction estimate unstable. With missing data, also examine whether missingness is concentrated in particular predictor ranges or combinations, because the information supporting the joint effect may then be limited.

A poor fit or weak data support is not solved merely by adding a product term. Conversely, a non-significant estimated interaction does not prove the conditional effects are identical; uncertainty and precision matter. In completed-data analyses, review whether estimates of the interaction and its uncertainty behave plausibly across imputations. Large differences can indicate sensitivity to missing-data assumptions, sparse support, or model instability; they are reasons to investigate, not automatic proof of one specific failure.

Common Pitfall and Practical Limits

Wrong reasoning: “Both \(X\) and \(Z\) appear in the imputation process, so their interaction in the analysis is necessarily preserved.” This does not follow. Representing each constituent variable is not the same as representing the joint outcome relationship encoded by \(XZ\). The correction is to treat the interaction as part of the analysis target when considering compatibility, then assess whether the imputation specification can support that target.

A second pitfall is interpreting \(\beta_1\) as the effect of \(X\) for everyone. With an interaction, it is the conditional slope at \(Z=0\). If zero is outside the meaningful range, centering \(Z\) at a substantively useful reference can make that coefficient easier to interpret, though it does not change the fitted conditional relationship when the model is reparameterized correctly.

Interactions also cost information: they require estimating an additional coefficient and can be imprecise, particularly when joint predictor patterns are rare. Compatibility does not guarantee that the model is correctly specified, that data support is sufficient, or that the missing-data assumptions are credible. It is a necessary modeling consideration, not a substitute for subject-matter judgment, diagnostics, or sensitivity analysis.