Respecting Model Hierarchy During Imputation

Why hierarchy matters for imputation

An interaction says that the relationship between one variable and the outcome depends on another variable. A model containing an interaction therefore needs enough terms to represent both that changing relationship and the variables’ lower-order contributions. This principle is called model hierarchy. It is not merely a convention about how to write a regression formula: omitting lower-order terms can change the model’s predictions, and an imputation model with those omissions may fail to preserve relationships required by the analysis.

For MICE, the key question is how the analysis model’s interaction structure is reflected in the conditional model used to impute a variable. If \(X\) is incomplete and the analysis includes \(X Z\), the conditional distribution of \(X\) given the outcome \(Y\) and \(Z\) will generally involve terms in \(Y\), \(Z\), and \(Y Z\). The exact form depends on the assumed distributions, but an imputation regression that drops relevant lower-order terms may not represent that conditional distribution.

Hierarchy in an analysis model and its conditional

Consider the linear analysis model

With strong hierarchy, the interaction \(XZ\) appears alongside both \(X\) and \(Z\). If \(X\) is missing, however, the imputation model is not simply a copy of this equation with \(X\) as its response: it models \(X\) conditionally on observed information, including \(Y\) and \(Z\). In general, conditioning on \(Y\) changes the functional form. The interaction can induce an interaction between \(Y\) and \(Z\) in the conditional mean of \(X\).

The following derivation makes this point under explicit assumptions. Suppose \(Z\) is binary, \(X\mid Z=z\sim N(0,1)\), and

Assume these distributions are correctly specified, the residual variation in \(Y\) is independent of \(X\) given \(Z\), and the stated variances are known. The conditional distribution of \(X\) given \(Y=y,Z=z\) follows by multiplying the normal prior density for \(X\) by the normal likelihood for \(Y\).

To see the mean, expand the exponent’s quadratic terms in \(x\): the coefficient of \(x^2\) is \(-(1+a_z^2)/2\), and the linear term is \(a_z(y-\alpha_z)x\). Completing the square gives a normal distribution with the mean and variance shown. Variation in \(\alpha_z\) can induce a \(Z\) term, while variation in \(a_z\) can induce a \(YZ\) term; both may vary.

What lower-order omissions do

If the imputation conditional is restricted to an interaction term alone, such as \(X\) regressed on \(YZ\) without an intercept or the main effects, it cannot reproduce the example’s conditional means. At \(Z=0\), \(YZ=0\) for every value of \(Y\), so this restricted model predicts the same conditional mean for every \(Y\), while the derived mean is \(0.5Y\). At \(Y=4,Z=0\), the derived mean is \(2\), but an interaction-only model predicts zero.

Even omitting just \(Z\) can matter. A model with an intercept and predictors \(Y\) and \(YZ\), but not \(Z\), predicts the same mean at \(Y=0\) for both groups: both product terms vanish, leaving only the intercept. The derived means at that value are \(0\) for \(Z=0\) and \(0.6\) for \(Z=1\). The omitted \(Z\) term prevents the conditional model from representing this difference.

Hierarchy choices and practical specification

Strong hierarchy is a clear default for interactions: retain each component main effect in the model where it is a predictor. Weak hierarchy is less restrictive and may be used in some modeling contexts, but retaining only one main effect can still exclude meaningful patterns. For imputation, the more important issue is whether the conditional model can represent the relationships implied by the analysis. A hierarchical formula is often a useful safeguard, but it is not a proof of compatibility; the conditional distributions and their functional forms still matter.

For an incomplete \(X\) and a binary \(Z\), a practical specification informed by an analysis interaction might include \(Y\), \(Z\), and \(Y Z\) as predictors when imputing \(X\), subject to appropriate model types for the variables. If imputing \(Z\), the relevant conditional needs to be considered afresh; simply reusing the \(X\)-conditional formula is not justified. MICE fits a sequence of conditional models, and the choices for each incomplete variable must be checked against the intended analysis relationships.

The main alternative to a deliberately hierarchical conditional model is a reduced additive specification that drops one or more terms, or an interaction-only specification. Such models are simpler, but can fail to represent the conditional means needed by the analysis. A joint model or a substantive-model-compatible imputation approach offers a different way to address model compatibility; neither makes the need to understand the target interaction disappear.

Diagnostics, limitations, and common pitfalls

Diagnostics should ask whether the imputed data preserve the interaction structure, not only whether the univariate distributions of imputed variables look plausible. Useful checks include comparing conditional summaries across levels of the interacting variable, inspecting the fitted imputation terms, and comparing observed and imputed values across relevant outcome and predictor strata. If an imputation model omits a lower-order term, look for systematic differences in those conditional summaries that the model cannot express.

• Wrong reasoning: “The product term is in the imputation formula, so the interaction is preserved.” Correction: An interaction-only model may exclude group-specific offsets or trends. Check whether its lower-order terms allow it to reproduce the conditional relationship.
• Wrong reasoning: “Strong hierarchy guarantees valid MICE imputations.” Correction: Hierarchy is a useful specification principle, not a guarantee of correct conditional distributions, compatibility among chained models, or adequate fit.
• Wrong reasoning: “Copy the analysis terms unchanged into every imputation regression.” Correction: Derive or reason about each conditional model. When \(X\) is the response, the analysis term \(XZ\) may imply \(YZ\) in the conditional for \(X\), as the derivation showed.
• Wrong reasoning: “If a lower-order coefficient is not statistically significant, its term can always be removed.” Correction: Removing a term can change the model’s structure and conditional predictions. A significance test alone does not establish that the reduced imputation model preserves the analysis relationship.

These considerations do not solve every challenge in interaction imputation. Sparse combinations of predictor values can make estimates unstable; poor functional-form choices can persist even when hierarchy is respected; and chained conditional models need not correspond to a single coherent joint distribution. Hierarchy helps avoid a particular class of omissions, but diagnostics and substantive judgment remain necessary.