## 7.5 Multilevel imputation by fully conditional specification

Another possibility is to iterate univariate multilevel imputation over the variables (Van Buuren 2011). For example, suppose there are missing values in lpo and sex in model (7.5). One way to draw imputations is to alternate the following two steps:

\begin{align} \dot{{\texttt{lpo}}}_{ic} & \sim N(\beta_0 + \beta_1 {{\texttt{den}}}_{c} + \beta_2 {{\texttt{sex}}}_{ic} + u_{0c}, \sigma_\epsilon^2)\tag{7.8}\\ \dot{{\texttt{sex}}}_{ic} & \sim N(\beta_0 + \beta_1 {{\texttt{den}}}_{c} + \beta_2 {{\texttt{lpo}}}_{ic} + u_{0c}, \sigma_\epsilon^2)\tag{7.9} \end{align}

where all parameters are re-estimated at every iteration. Since the first equation corresponds to the complete-data model, there are no issues with this step. The second equation simply alternates the roles of lpo and sex, and uses the inverted mixed model to draw imputations. The above steps illustrate the key idea of multilevel imputation using FCS. It is not yet clear when and how the idea will work.

Resche-Rigon and White (2018) studied the consequences of model inversion, and found that the conditional expectation of the level-1 predictor in a multivariate multilevel model with random intercepts depends on the cluster mean of the predictor, and on the size of the cluster. In addition, the conditional variance depends on cluster size. These results hold for the random intercept model. Of course, including random slopes as well will only complicate matters. The naive FCS procedure in Equation (7.8) does not account for the cluster means or for the effects of cluster size, and hence might not provide good imputations. From their derivation, Resche-Rigon and White (2018) therefore hypothesized that the imputation model (1) should incorporate the cluster means, and (2) be heteroscedastic if cluster sizes vary. We now discuss these points in turn.

### 7.5.1 Add cluster means of predictors

Simulations done by Resche-Rigon and White (2018) showed little impact of adding the cluster means of the level-1 predictors to the imputation model, but did not hurt either. However, several other studies found substantial impact. described how the inclusion of cluster means preserves contextual effects. Adding a group mean of level-1 variables allows us to estimate the difference between within-group and between-group regressions. The aggregates are generally called contextual variables. Including both the individual and contextual variables into the same model is useful to find out whether the contextual variable would improve prediction of the outcome after the differences at the individual level have been taken into account. proposed to add contextual variables more generally to univariate multilevel imputation, requiring a change in the algorithm. In particular, cluster means need to be dynamically calculated from the currently imputed predictors, and depending on the model the original predictor needs to be replaced by its group-centered version, as in Kreft, De Leeuw, and Aiken (1995). In random slope models, the two specifications have different meanings, so the decision as to whether or not to use group-mean centered predictors at level 1 should be made in accordance with the analysis model. In random intercept models, the two specifications are equivalent and can be transformed into one another, so imputations should yield equivalent results regardless of centering. Mistler and Enders (2017) present a thorough and detailed analysis on the effects of this adaptation, both for a joint model and for an FCS model. Their paper demonstrated that in both cases, inclusion of the means of the clusters markedly improved performance. present an extensive simulation study contrasting many state-of-the-art imputation techniques. In all scenarios involving multilevel FCS, including the cluster means in the imputation model was beneficial.

The difference between the results of Resche-Rigon and White (2018) on the one hand, and the other three studies is likely to be caused by a difference in complete-data models. The latter three studies used a contextual analysis model, which includes the cluster means into the substantive model, whereas Resche-Rigon and White (2018) were not interested in fitting such a model. Hence, it appears that these studies address separate issues. Resche-Rigon and White (2018) are interested in improving compatibility among the conditional models without reference to a particular analysis model. Their result indicates that trying to improve compatibility of conditionals seems to have little effect, a result that is in line with the existing literature on FCS. The other three studies address the problem of congeniality, a mismatch between the imputation model and the substantive model. Improving congeniality had a major effect, which is in line with the larger multiple imputation literature. Section 4.5.4 explains the confusion surrounding the term compatibility in some detail.

A problematic aspect of including cluster means is that the contextual variable may be an unreliable estimate in small clusters. It is known that the regression weight of the contextual variable is then biased (Lüdtke et al. 2008). A solution is to formulate the contextual variable as a latent variable, and use an estimator that essentially shrinks the weight towards zero. Most joint modeling approaches assume a multivariate mixed-effects model, where cluster means are latent.

It is not yet clear when the manifest cluster means can be regarded as “correct” in an FCS context. When clusters are large and of similar size, the manifest cluster means are likely to be valid and have little differential shrinkage. For smaller clusters or clusters of unequal size, including the cluster means in the imputation model also seems valid because proper imputation techniques will use draws from the posterior distribution of the group means rather than using the manifest means themselves. All in all, it appears preferable to include the cluster means into the imputation model.

### 7.5.2 Model cluster heterogeneity

Van Buuren (2011) considered the homoscedastic linear mixed model as invalid for imputing incomplete predictors, and investigated only the 2l.norm method, which allows for heterogeneous error variances by employing an intricate Gibbs sampler. The 2l.norm method is not designed to impute variables that are systematically missing for all cases in the cluster. Resche-Rigon and White (2018) developed a solution for this case using a heteroscedastic two-stage method, which also generalizes to binary and count data. compared several univariate multilevel imputation methods, and concluded that “heteroscedastic imputation methods perform better than homoscedastic methods, which should be reserved with few individuals only.” Apart from the last paper there is relatively little evidence on the benefits of allowing for heteroscedasticity. It could be very well that heteroscedasticity is a useful option to improve compatibility of the conditionals, but the last word has not yet been said.