Two-Level Normal Imputation By Hand

From a Random-Intercept Model to a Missing-Value Draw

A clustered outcome contains two sources of variation: differences between clusters and differences between individuals within the same cluster. An imputation that ignores the first source can make people in a cluster look more independent than they are. An imputation that uses only a cluster’s observed average can go too far in the other direction, treating that average as if it were known without error.

This tutorial works through the conditional calculation for one missing Gaussian outcome. It builds on the earlier material on Random-Intercept Imputation Models, but focuses on the arithmetic of conditioning on a cluster’s observed outcomes and then making a stochastic draw.

For this hand calculation, take \(\beta_0,\tau^2,\sigma^2\) as known. This simplifies the conditional distribution without claiming that real imputations should ignore parameter uncertainty. The model assumes a common intercept, normally distributed random intercepts and residuals, and constant residual variance. It also assumes that, given the cluster effect, individuals’ outcomes are conditionally independent. These assumptions should be assessed in the application: strong skew, outliers, changing within-cluster spread, or systematic cluster patterns not represented by the intercept can make the calculation a poor description of the data.

The calculation conditions on the observed outcomes in a cluster. It does not, by itself, establish that missingness is ignorable or that the imputation model is correctly specified for an analysis. Those conditions matter for valid inference; the conditional distribution below is the model-based draw once its inputs and assumptions are accepted.

Deriving the Cluster-Specific Conditional Distribution

Suppose cluster \(j\) has \(k\) observed outcomes \(y_{1j},\ldots,y_{kj}\). Write each observed outcome as its deviation from the population intercept, \(r_{ij}=y_{ij}-\beta_0\). Conditional on \(u_j\), these deviations have mean \(u_j\) and variance \(\sigma^2\). The prior distribution for \(u_j\) is normal with mean zero and variance \(\tau^2\).

The conditional density of \(u_j\) is proportional to the prior density times the likelihood of the observed outcomes. Collecting the terms involving \(u_j\) in the exponent gives a quadratic whose coefficient is the posterior precision and whose linear term determines the posterior mean:

Completing the square shows that the conditional distribution is normal. Define \(V_j\) as its variance and \(m_j\) as its mean:

Thus \(u_j\mid\text{observed outcomes}\sim N(m_j,V_j)\). For a missing outcome \(Y_{\mathrm{mis},j}\) in that cluster, the model gives \(Y_{\mathrm{mis},j}=\beta_0+u_j+\epsilon_{\mathrm{mis},j}\). The residual for that individual is independent of the observed outcomes and of \(u_j\). Adding the means and variances therefore gives the predictive distribution:

The \(V_j\) term represents uncertainty about the cluster’s contribution after seeing its observed outcomes. The \(\sigma^2\) term represents individual variation around that contribution. Both appear in the predictive variance; a conditional mean alone is not a stochastic imputation.

Worked Calculations

Applying the Calculation and Checking It

In an actual multiple-imputation procedure, the model parameters are generally not known. A proper procedure accounts for their uncertainty rather than treating estimates as fixed in every completed data set. The hand calculations above isolate the conditional outcome draw given specified parameter values; they do not include a parameter-draw step. Accordingly, they illustrate the within-model calculation, not a complete prescription for fitting and running a multilevel MICE algorithm.

When more than one outcome is missing in a cluster, the missing outcomes share the same cluster effect. A joint draw should use one draw of \(u_j\) for that cluster and a separate residual draw for each missing individual. Drawing a new cluster effect independently for every missing person would discard the dependence induced by their common cluster.

Several checks can reveal when the model or its implementation is going wrong:

• Verify the units. The means \(m_j\) and \(u_j\) are in outcome units; \(V_j,\tau^2,\sigma^2\) are in squared outcome units.
• Check the arithmetic in the precision \(1/\tau^2+k/\sigma^2\). More observed outcomes increase precision and reduce \(V_j\); they do not increase uncertainty about the cluster effect.
• Compare the distribution of observed cluster means and within-cluster residuals with the model’s assumptions. Marked skewness, influential observations, or visibly different residual spreads across clusters may indicate poor fit.
• Check completed-data draws for implausible values and inspect whether the cluster patterns are plausible. A normal model can generate values outside a bounded outcome’s possible range.
• Confirm that repeated missing outcomes in a cluster share one cluster-effect draw. Independent cluster-effect draws at the individual level produce the wrong dependence structure.

The alternatives are not interchangeable. A single-level model may be reasonable only if its assumptions match the data and estimand; a mean-only fill-in is a deterministic prediction, not a stochastic imputation. Conversely, a two-level normal model is not automatically adequate just because observations are clustered. It can be sensitive to the normality and variance assumptions, to poorly estimated model parameters, and to how the model represents the relevant predictors. A realistic draw under a misspecified model does not repair that misspecification.

Common Mistake