Why Conditional Models Need a Multilevel Design
In multilevel chained equations, a list of plausible regressions is not enough. Each conditional model must reflect which level the target variable belongs to, how observations are clustered, and which relationships the final analysis is intended to estimate. A model that treats every row as independent can understate uncertainty and fail to preserve cluster patterns. A model that treats a cluster-level variable as if it were observed independently on every row can give that variable excessive influence.
The task here is to design the conditional models and assemble them into a multilevel fully conditional specification (FCS). FCS imputes each incomplete variable in turn from a model conditional on the other variables, cycling through the targets. The focus is model design: choosing the target unit, predictors, functional form, and cluster structure before running the cycle.
A specification is conditional model by conditional model. For a continuous level-1 target, a possible model has a cluster-specific intercept and separates within-cluster from between-cluster predictor information. A level-2 target instead has one record per cluster and may use cluster-level predictors and summaries of level-1 variables. Binary, ordinal, nominal, and count targets require conditional distributions suited to those outcomes; a normal linear model is not a generic substitute.
Design Principles and Assumptions
Before specifying a model, identify the target’s level and how many distinct target values exist. A level-2 value repeated down the rows of a person-level data file is still one value per cluster. Its imputation model should operate on the cluster-level information, not treat repeated copies as independent observations. Conversely, a level-1 target requires a model at the unit level that can represent within-cluster dependence when that dependence matters.
For each incomplete target, include predictors needed to preserve its relationships with the substantive analysis variables. This generally means including variables in the analysis model that predict the target or are associated with its missingness, along with useful auxiliary variables. If the analysis includes an interaction or nonlinear effect, the imputation specification should preserve the corresponding structure rather than impose an incompatible simple linear relationship. The earlier material on cross-level interactions and nonlinear effects develops those structures; here they are design checks for each conditional model.
For a level-1 predictor \(X_{ij}\), a within-between decomposition can help distinguish its individual-level association from its cluster-level association. Let \(\bar X_j\) denote the cluster mean. A conditional mean can include both \(X_{ij}-\bar X_j\) and \(\bar X_j\), rather than forcing one coefficient to represent two different comparisons.
Here \(b_j\) represents cluster-level variation in the target, and the conditional distribution also includes a level-1 residual. The within-cluster coefficient \(\beta_W\) describes how the expected target differs for two units in the same cluster whose \(X\) values differ by one unit. The coefficient \(\beta_B\) describes the association with the cluster mean, holding the individual’s deviation from that mean fixed. Whether this exact model is suitable depends on the analysis and the data; the decomposition is not an automatic requirement for every predictor.
The imputation models also rely on statistical and data-collection assumptions. The missingness mechanism must be adequately addressed by the observed information used in the imputation models; under a missing-at-random analysis, this means conditioning on relevant observed predictors of missingness and the incomplete values. The chosen conditional distributions must be plausible, the observed data must support estimation, and the cluster identifiers and variable levels must be correctly recorded. Adequate overlap is important: a model cannot reliably impute combinations of predictors that are absent from the observed data. These conditions do not make missingness at random empirically verifiable.
Assembling the Conditional Models
A practical specification begins with a target-by-target inventory. For every incomplete variable, record its level, type, analysis role, important predictors, and the dependence structure to preserve. Then select a conditional distribution and construct its predictors. The collection should be reviewed as a system: a useful predictor in one conditional model may itself be incomplete and therefore must be represented in the cycle.
Record whether it is level 1 or level 2, and whether it is continuous, binary, ordinal, nominal, or a count. Confirm whether the target is defined once per unit or once per cluster.
List the target’s role in the substantive model, including relevant within-between distinctions, cluster-level predictors, interactions, and nonlinear terms.
Use a distribution suited to the target and include a cluster structure appropriate to its level and observed dependence. Do not use a single-level model merely because it is computationally convenient.
Verify that important analysis variables and auxiliary predictors appear where needed, cluster-level targets are not duplicated as independent units, and generated summaries are defined consistently across iterations.
FCS does not require the conditional models to be derived from one specified joint distribution. That flexibility is useful, but it is also a limitation: independently chosen conditional models may be incompatible, meaning they do not correspond to the conditionals of a single joint distribution. A cycling algorithm can still produce imputations, but convergence alone does not establish that the resulting distribution is appropriate for the scientific analysis.
Worked Example: A Continuous Level-1 Target
Suppose a hypothetical study records a continuous symptom score \(Y_{ij}\) for patients within clinics. The incomplete target is \(Y_{ij}\); treatment \(X_{ij}\) varies among patients, clinic deprivation \(W_j\) is constant within a clinic, and the analysis needs separate within-clinic and between-clinic associations with treatment. For illustration, assume the fitted conditional mean has coefficients \(\beta_0=10\), \(\beta_W=2.0\), \(\beta_B=1.5\), and \(\beta_Z=-0.8\), with no additional interaction in this example.
For one patient, suppose \(X_{ij}=3\), that patient’s clinic mean is \(\bar X_j=4\), and clinic deprivation is \(W_j=1\). The fixed part of the conditional mean is
Thus the model’s fixed-part mean is 13.2 symptom-score points for this predictor profile. The within-clinic contribution is \(-2\) points because this patient’s treatment value is one unit below the clinic mean; the between-clinic contribution is \(6\) points. This calculation illustrates why one undifferentiated treatment coefficient could obscure the intended comparison. The value 13.2 is not itself a completed imputation: an imputation procedure must account for the model’s uncertainty and the target’s residual variability.
The model is defensible only if a continuous conditional distribution is reasonable for the score, the cluster structure is relevant, and the within-between terms match the analysis. If treatment is randomized and its mean is not scientifically meaningful, the between-clinic term may need a different justification; variables should not be included mechanically.
Worked Example: A Continuous Level-2 Target
Consider a hypothetical study of schools in which funding \(V_j\), measured in thousands of dollars per pupil, is incomplete and is constant for every student in a school. The target is one funding value per school. Suppose a cluster-level conditional model uses a school-level predictor \(Z_j\) and the school mean of a level-1 reading score, \(\bar R_j\). For this illustration, the fitted conditional mean is
For a school with \(Z_j=2\) units and \(\bar R_j=4\) score points, the model gives
The conditional mean is 11.0 thousand dollars per pupil. The target is imputed once for that school, not once for each student row. If the analysis needs to preserve the relationship between school funding and the distribution of student outcomes, a school mean may be insufficient on its own: additional summaries or other predictors may be required to represent the relevant association. The appropriate summaries depend on the scientific model and the amount of information available in each cluster.
Worked Example: A Binary Level-1 Target With a Cross-Level Predictor
Suppose a hypothetical study records whether a patient experiences a side effect, \(Y_{ij}\), where \(Y_{ij}=1\) indicates yes. A clinic-level exposure \(Z_j\) and a patient-level indicator \(X_{ij}\) are included, with a cross-level interaction. Consider the illustrative logistic conditional model
For \(X_{ij}=1\), \(Z_j=2\), and \(b_j=0\) as a reference value for this arithmetic illustration, the linear predictor is
The corresponding probability at that reference value is
So the model gives an approximately 0.711 conditional probability, or 71.1%, at the stated predictor values and reference cluster effect. The interaction is part of the conditional specification because the analysis requires it. The calculation is not a population-average probability across clinics, and setting \(b_j=0\) here is only a way to show the arithmetic; it is not a general method for handling cluster effects. The binary target calls for a binary-response model rather than a normal linear regression.
Derivation: Why the Within-Between Terms Matter
The within-between form follows from an identity, but its interpretation depends on including both pieces. Write each individual predictor as its deviation from the cluster mean plus the cluster mean:
A model with a single coefficient \(\beta X_{ij}\) imposes the same coefficient on the deviation and the cluster mean. Replacing that term by two terms gives
If \(\beta_W=\beta_B=\beta\), then substitution yields
Thus the conventional single-slope model is a special case that constrains the within- and between-cluster associations to be equal. Allowing distinct coefficients removes that constraint and makes the two comparisons explicit. This is useful when the analysis requires the distinction, but it also uses more information and may be unstable when clusters are few or the data offer little variation.
Diagnostics, Alternatives, and Limitations
Review the specification and the generated imputations, not just whether the algorithm reports that it ran. Warning signs include implausible cluster-level variation, imputed values outside possible ranges, overly concentrated values within clusters, unstable estimates, and imputed associations that differ sharply from the observed-data patterns without a scientific explanation. For categorical targets, check whether predicted probabilities are extreme or whether rare categories disappear from fitted models. Also check that a purported level-2 variable is constant within each cluster after imputation and that a level-1 variable retains plausible within-cluster variation.
A useful review asks whether each conditional model has enough observed information for its predictors, whether its functional form is suitable, and whether predictor summaries are calculated from the current completed data in a consistent way. Sparse categories, few clusters, and highly unbalanced cluster sizes can make otherwise reasonable models difficult to estimate. Simplifying a model may be necessary, but omitting a key analysis interaction or cluster structure can alter the relationships the imputations are intended to preserve.
The main alternatives have different trade-offs. A joint multilevel model specifies a multivariate distribution directly and can offer a coherent joint framework, but choosing and fitting that distribution may be difficult for mixed variable types. A single-level FCS uses separate conditional models but ignores clustering unless the dependence is represented another way; it is not a sound default when cluster dependence matters. Multilevel FCS offers flexible, target-specific conditional models, but its flexibility does not guarantee compatibility among those models or remove the need to check their consequences.
No specification can recover information absent from the observed data without assumptions. In particular, a well-designed set of conditional models does not by itself resolve departures from the assumed missingness mechanism, sparse-data problems, or incompatibility among conditionals. The design should be judged by whether it represents the scientific analysis and the data structure well enough for the intended estimand, followed by diagnostics and sensitivity analysis appropriate to the application.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A clinic-level exposure is copied onto every patient row. At what level should its missing values be imputed, and why?
- In a level-1 conditional model, what do \(X_{ij}-\bar X_j\) and \(\bar X_j\) represent?
- A binary level-1 target is imputed with a normal linear regression. What is the central specification problem?
- The analysis includes an interaction between a patient-level and a clinic-level predictor. What should be checked in the conditional imputation model for an incomplete outcome?
- Why does successful cycling not prove that the FCS conditionals form a coherent joint model?
Solutions. (1) Impute once per clinic because the exposure is a single cluster-level value; repeating it across patient rows does not create independent target observations. (2) The first term is the patient’s deviation from the clinic mean, representing within-clinic variation; the second is the clinic mean, representing between-clinic variation. (3) A normal model is not matched to a binary outcome and can yield invalid fitted values; use a binary-response conditional distribution. (4) Check that the corresponding cross-level interaction is represented when it is needed to preserve the analysis relationship, and that the model has enough information to estimate it. (5) FCS specifies conditionals separately, and those models may be incompatible: they need not be the conditional distributions of any one joint distribution. Successful cycling alone does not establish otherwise.