Decomposing Level-1 Predictors

Why Decompose a Level-1 Predictor?

A level-1 predictor varies across individuals within a cluster and may also vary between clusters. Those two sources of variation need not have the same association with an outcome. For example, an individual whose measured exposure is high relative to others in the same clinic may differ from an individual in a clinic with a high average exposure. Treating both comparisons as one association can obscure that distinction.

In multilevel imputation, this matters when a level-1 variable is used to predict another variable. Including a within-cluster term and a between-cluster term allows the imputation model to represent the two associations separately. The decomposition is an algebraic construction; it does not, by itself, account for clustering or guarantee that an imputation model is well specified.

The deviation \(x_{W,ij}\) describes how far an individual’s value is above or below their own cluster mean, in the original units of \(x\). The cluster mean \(x_{B,j}\) describes the cluster’s average level of the predictor. Within each cluster, the deviations sum to zero when the same individuals are used to compute the mean:

This identity is a useful coding check. It also makes clear that a deviation is not an individual’s raw value: it is a comparison with the individual’s own cluster.

Separate Associations in a Multilevel Model

A common linear specification for an outcome \(y_{ij}\) is

Here \(\beta_W\) is the within-cluster association: it describes the expected outcome difference associated with a one-unit difference in \(x\) between two individuals in the same cluster, holding the cluster mean fixed. The coefficient \(\beta_B\) is the between-cluster association: it describes the expected outcome difference associated with a one-unit difference in cluster means, comparing clusters at the same within-cluster deviation. These are model-based associations, not automatically causal effects.

In a random-intercept model, \(u_j\) represents cluster-specific variation in the outcome intercept, and \(\epsilon_{ij}\) represents individual-level residual variation. A usual Gaussian version assumes \(u_j\) has mean zero and variance \(\tau^2\), \(\epsilon_{ij}\) has mean zero and variance \(\sigma^2\), and the two are independent, with residuals independent across individuals conditional on the random intercept and predictors. The decomposition itself does not require normality; those distributional assumptions are relevant to this particular Gaussian model and imputation specification.

The earlier material on Within-Between Decomposition and Deriving The Contextual Effect established the interpretation of the coefficient difference. In this parameterization, the contextual association is \(\beta_B-\beta_W\): it compares clusters with different means while holding an individual’s raw \(x\) value fixed. The difference is meaningful only with the model and comparison stated; it is not a third independent predictor coefficient.

Why the Raw Predictor Alone Can Be Restrictive

Because \(x_{ij}=x_{W,ij}+x_{B,j}\), a model using only the raw predictor,

is equivalent to a decomposed model in which the within- and between-cluster coefficients are constrained to be equal. Substitution gives:

Thus, a raw-\(x\) model imposes \(\beta_W=\beta_B=\beta\). That equality may be suitable if it is substantively justified, but it is not guaranteed by the data structure. Including both terms removes this equality constraint. The random intercept still has a separate role: it represents residual cluster-level outcome variation, not the between-cluster association in \(x\).

Using the Terms in Multilevel Imputation

Suppose \(y\) is a level-1 variable to be imputed and \(x\) is a level-1 predictor. A conditional imputation model can include \(x_W\) and \(x_B\) as distinct predictors, along with other relevant variables and an appropriate representation of cluster dependence. For a continuous \(y\), one possible model is:

This model states the conditional distribution used for the illustration: the conditional mean is linear in the two predictor terms, residual variance is \(\sigma^2\), and cluster intercepts have a normal distribution with variance \(\tau^2\). The terms preserve the model’s distinct within- and between-cluster associations. Merely adding a cluster mean to a single-level imputation model would not, on its own, represent residual dependence among people in the same cluster.

In a MICE implementation, the decomposition should correspond to the data used in each completed data set. If \(x\) has missing values, its cluster means and deviations depend on the completed values; treating terms computed once from incomplete data as fixed can make the predictor columns inconsistent. Handling that missing-predictor situation requires additional care and is distinct from the basic construction shown here. The deterministic-identity principles from Passive Imputation apply when derived terms must be kept synchronized with their source variable.

Construction Checks, Alternatives, and Limitations

Before using the terms, verify that cluster membership is correct, that the mean is calculated over the intended individuals, and that the deviation is formed by subtracting that same cluster mean from each person’s \(x\). A simple numerical check is \(x_W+x_B=x\) for every row. Also check that within-cluster deviations sum to approximately zero, allowing for rounding.

Common warning signs include a deviation that does not sum to zero within clusters, a cluster mean that changes across rows within the same cluster, or a supposed within-cluster term that was centered around the overall sample mean instead. Grand-mean centering is a different transformation: it shifts the reference point but does not by itself isolate within-cluster variation. For a within-between decomposition, the centering reference for \(x_W\) must be the individual’s own cluster mean.

The main alternatives have different implications:

The decomposition is most directly useful when \(x\) varies within clusters and cluster means can be meaningfully defined. If a cluster has no variation in \(x\), its within-cluster deviations are all zero; that cluster provides no within-cluster contrast. Estimates may also be unstable when there are few clusters or little between-cluster variation. A mean calculated from a small cluster is an imprecise summary of its underlying cluster level; the decomposition does not remove that sampling uncertainty.

Finally, including both terms does not solve every modeling problem. The association may be nonlinear, vary across clusters, or interact with other predictors. The mean may be affected by missing values or the way cluster membership is observed. The imputation model must still reflect the target variables, dependence, and relevant structure. Decomposition clarifies which variation is represented; it does not certify that the complete imputation model is adequate.