Preserving Cross-Level Effects Over Time

Why the Interaction Matters

In longitudinal data, the association between time and an outcome may differ across clusters, and a cluster-level characteristic may explain some of that difference. For example, the rate of change in a clinical outcome might depend on a treatment assigned to a clinic. If an imputation model represents only a common time slope, it can smooth over that pattern: values imputed for clusters with different characteristics are pulled toward a shared trajectory.

The key modeling term is the interaction between time and the cluster-level predictor. In chained equations, including that term in the conditional model for a repeatedly measured variable allows its imputed values to reflect different expected trajectories. It does not, by itself, ensure that every aspect of the multilevel data structure or missingness process is modeled correctly.

Modeling Cluster-Specific Trajectories

For a continuous outcome, one useful linear specification is

Here \(\beta_1\) is the fixed time slope when \(W_j=0\), and \(\beta_3\) is the change in that slope for a one-unit increase in \(W_j\). The cluster-specific slope is \(\beta_1+\beta_3W_j+b_{1j}\), where \(b_{1j}\) represents cluster-specific deviation from the fixed slope. The random intercept \(b_{0j}\) allows clusters to differ in their starting levels. A model may also include person-level random effects or additional predictors where the design and analysis require them.

A conventional model-based interpretation assumes the conditional mean is adequately represented by the specified terms, the random effects and residual structure are appropriately modeled, and the observed data provide information across the relevant combinations of time and \(W_j\). For example, a random-effects model often assumes normally distributed random effects with a specified covariance matrix. Repeated outcomes may also require a residual structure that accounts for serial dependence. The appropriate assumptions depend on the imputation model and data; adding \(tW_j\) does not replace them.

For chained equations, the conditional imputation model for a missing \(Y_{ijt}\) should contain time, \(W_j\), and their product when the analysis model or substantive knowledge indicates that the time trend differs by \(W_j\). Other relevant observed outcomes, covariates, and cluster structure should also be represented as appropriate. To reflect imputation uncertainty, the procedure should draw model parameters and residual or random-effect contributions rather than imputing only fitted conditional means.

When the outcome is binary or otherwise noncontinuous, the same design principle applies, but the conditional model uses a suitable link and outcome distribution. The interaction then modifies the time association on the model’s link scale; it is not automatically an interaction of the same size on the outcome-probability scale.

Why the Pooled Slope Can Differ

The distinction between observed slopes and underlying model slopes matters. In a regression of observed outcomes on time with a common slope, the fitted slope is determined by the observed within-group covariances, not by a formula involving only the true slopes. When groups have the same time schedule, the pooled slope is a weighted average of their observed within-group slopes, with weights given by their within-group time sums of squares.

To see this, let \(S_{tt,g}=\sum_k(t_{kg}-\bar t_g)^2\) and let \(\widehat{s}_g\) be the observed least-squares slope in group \(g\). If the groups have the same mean time, then the between-group contribution to the pooled time-outcome covariance is zero. The pooled slope is

This follows because the within-group time-outcome cross-product is \(S_{ty,g}=S_{tt,g}\widehat{s}_g\), and, with common group mean time, the pooled cross-product is \(\sum_g S_{ty,g}\). Dividing by the pooled time sum of squares \(\sum_g S_{tt,g}\) gives the expression. If group mean times differ, between-group terms also enter both pooled cross-products, so this weighted-average result need not hold.

Including the Interaction in Chained Equations

In MICE, the conditional model for each incomplete variable is specified in turn, using current values of the other variables. When the target variable is the repeated outcome, form or include the time-by-\(W_j\) product in its imputation model. If time is centered, use the centered time in the product as well. The product may be generated deterministically from time and \(W_j\); its inclusion does not mean it is an independent observed variable.

A practical sequence is to state the trajectory structure implied by the scientific analysis, choose a conditional model that can represent it, and then check whether there is adequate observed information for its terms. The following outline emphasizes the modeling decisions rather than prescribing a particular software implementation.

Centering, Alternatives, and Limitations

Centering time changes coefficient interpretation but does not remove the need for an interaction. For example, let \(t_c=t-2\). Then \(t=t_c+2\), and

The interaction coefficient remains \(\beta_3\), while the coefficient for \(W_j\) now describes its association at time 2 rather than time 0. Centering can make that reference point more useful, but it does not create information where the data have little overlap in time or \(W_j\).

There are alternatives to a pooled interaction model. Separate imputation models within levels of a categorical \(W_j\) can allow different trajectories, but may be unstable when some groups have few observed values and cannot directly express a continuous \(W_j\) effect. A pooled common-slope model is simpler, but imposes no time modification by \(W_j\). A model with an explicit interaction shares information across levels while allowing the slope to vary systematically. The choice should reflect the analysis model, the data structure, and the available information.

Including an interaction is not a remedy for every imputation problem. The conditional model can still be misspecified, random-effects or serial dependence can be inadequately handled, and sparse data can make the interaction poorly estimated. Under a missing-at-random framework, the imputation model must condition on variables needed to make that assumption plausible and represent the relevant outcome relationships. The interaction does not establish that missingness is at random, and it does not resolve missing-not-at-random mechanisms. Chained conditional models can also be incompatible with a single joint distribution; retaining important terms is necessary for the intended relationships but does not guarantee full compatibility.

Diagnostics and Common Pitfalls

Check whether observed data contain enough time variation within clusters and enough clusters across levels of \(W_j\) to estimate the interaction. Inspect fitted or imputed trajectories by time and \(W_j\), and compare distributions of observed and imputed outcomes over those groups. Implausibly flat trajectories, abrupt jumps, or imputed values concentrated in a narrow range can indicate that the model or its uncertainty representation is inadequate. A convergence trace alone cannot establish that the interaction is preserved or that the imputation model is scientifically appropriate.

• Omitting the product term: A model with time and \(W_j\) main effects but no \(tW_j\) term allows different levels but forces a common time slope.
• Using only fitted means: Deterministic mean substitution omits residual and parameter uncertainty and can make completed data too regular.
• Confusing fitted and true slopes: The pooled-slope identity concerns observed within-group fitted slopes under its stated conditions, not generally the underlying trajectory slopes.
• Extrapolating beyond support: A fitted interaction can produce predictions for combinations of time and \(W_j\) scarcely represented in observed data; those predictions warrant caution.