Why the Estimand Comes First
In nested data, individuals belong to clusters: patients may belong to clinics, and students to schools. An imputation model should not be chosen solely from the list of variables that happen to be incomplete. It should also reflect the estimand—the precise population quantity the analysis is intended to estimate. A model that preserves an individual-level association may not preserve a cluster-level association, and the reverse can also occur.
For notation, let \(i\) index individuals and \(j\) index clusters. Write \(Y_{ij}\) for an individual outcome, \(X_{ij}\) for an individual-level covariate, and \(Z_j\) for a cluster-level variable. A cluster summary, such as a mean or proportion, can be written with a bar or described explicitly. The goal here is not to prescribe one universal multilevel imputation model. It is to determine what information and variation an imputation model needs to retain for a stated target.
For example, the average outcome across all people gives more influence to people in large clusters. The average of the cluster-specific outcome means gives every cluster equal influence. These are different targets when cluster sizes differ or cluster means vary. Similarly, an association between a person’s exposure and outcome is not automatically the same quantity as an association between cluster-average exposure and cluster-average outcome.
What the Imputation Model Must Preserve
In MICE, an incomplete variable is imputed from a conditional model using observed information and current values of other variables. For the imputed values to support a particular analysis, the conditional model should include the variables needed to preserve the relationships relevant to that analysis. This normally includes the outcome as a predictor when imputing an incomplete analysis covariate: otherwise the completed covariate may fail to retain its association with the outcome.
For nested data, the relevant information can occur at more than one level. An individual-level covariate may vary among people in the same cluster and also differ systematically across clusters. A cluster-level predictor may explain differences between clusters but cannot explain why two people in one cluster differ. The imputation model therefore needs to represent the levels and sources of variation that contribute to the target association. It should not collapse distinct levels into one undifferentiated relationship.
The usual missing-at-random condition is that, conditional on the variables included in the imputation procedure, missingness does not depend on the missing value itself. This condition is not guaranteed by including cluster information, and it cannot be verified from observed data alone. The conditional models must also be suitably specified for the variable types and relationships, and the observed data must provide support for the predictions being made. Even under these conditions, imputation does not rescue an analysis whose estimand or weighting is unclear.
Worked Example: Imputing a Covariate for an Individual-Level Association
Suppose a hypothetical study follows patients within clinics. The outcome \(Y_{ij}\) is a symptom score, \(X_{ij}\) is an individual exposure, and \(Z_j\) is clinic staffing level. The analysis targets the association of individual exposure with symptom score, adjusted for staffing. One patient’s \(X_{ij}\) is missing, but that patient’s outcome and clinic staffing are observed.
A conditional mean model for illustration is
where \(u_j\) represents a clinic-specific contribution. Assume, only for this numerical illustration, that the fitted coefficients are as displayed, the patient has \(Y_{ij}=3\), the clinic has \(Z_j=0\), and \(u_j=0.2\). The predicted conditional mean is
The value 1.9 is a conditional mean, not a complete stochastic imputation. A proper imputation also reflects residual variation and uncertainty in the fitted model; it does not simply replace every missing value with its predicted mean. Including \(Y_{ij}\) helps preserve the individual-level \(X\)-outcome association, while including \(Z_j\) and clinic-level structure allows the conditional distribution to differ across relevant cluster contexts. A model using only an overall mean for \(X\) would ignore these observed relationships.
This example assumes that the conditional model is adequate, the missingness mechanism is compatible with missing at random given included information, and comparable observed cases provide support for the patient’s predictor values. The equation is a teaching illustration, not a claim that this particular linear model is appropriate for every exposure. The conditional distribution and link must suit the variable being imputed.
Cluster-Level Targets and Their Weighting
When the estimand is defined over clusters, the imputation strategy must preserve the information used to construct cluster-level quantities and their relationships with cluster-level predictors. This may require retaining individual observations that contribute to a cluster summary, as well as variables that explain differences among clusters. It is not enough to match the marginal distribution of an incomplete individual variable if doing so disrupts the cluster summaries that enter the analysis.
A central choice is weighting. An individual-weighted summary gives a cluster influence proportional to its number of individuals; an equal-cluster summary gives each cluster the same influence. Neither is universally preferable. The scientific question determines which is intended. When cluster size is related to outcomes or predictors, they can differ substantially.
Worked Example: A Cluster-Level Slope Across Four Clusters
Consider a hypothetical analysis in which the target is the descriptive slope of cluster-average outcome \(\bar{Y}_j\) on a cluster-level staffing measure \(Z_j\), giving each of four clusters equal weight. The observed cluster values are:
| Cluster \(j\) | \(Z_j\) (staffing units) | \(\bar{Y}_j\) (outcome units) |
|---|---|---|
| 1 | 1 | 5 |
| 2 | 2 | 7 |
| 3 | 3 | 6 |
| 4 | 4 | 10 |
The means are \(\bar{z}=(1+2+3+4)/4=2.5\) staffing units and \(\bar{Y}=(5+7+6+10)/4=7\) outcome units. The equal-cluster least-squares slope is
The descriptive slope is 1.4 outcome units per staffing unit. This calculation targets a relationship across four cluster means, with each cluster contributing one point. If individual outcomes or covariates are incomplete and the analysis will reconstruct cluster summaries, imputations need to preserve the relevant cluster-level relationships as well as the individual data that form those summaries. The slope is descriptive: it does not, by itself, show that changing staffing would cause the outcome to change.
This calculation assumes the four cluster summaries are measured on comparable scales and that equal cluster weighting matches the target. It does not establish that a linear relationship is an adequate scientific model. If the intended target instead weights individuals equally, the analysis and imputation must be aligned with that different weighting.
Unequal Cluster Sizes Can Change the Target
Worked Example: Equal-Cluster Versus Individual-Weighted Averages
Suppose a hypothetical dataset contains three clinics with sizes and observed outcome proportions as follows. The target is either the average clinic proportion or the proportion among all patients; these are distinct estimands.
| Clinic | Patients | Outcome proportion | Outcome count |
|---|---|---|---|
| A | 10 | 0.20 | 2 |
| B | 10 | 0.40 | 4 |
| C | 40 | 0.80 | 32 |
For the average clinic proportion, each of the three clinics receives equal weight:
For the proportion among all 60 patients, weight each clinic’s proportion by its patient count. The total number of outcomes is \(2+4+32=38\), so
The equal-clinic average is about 0.467, while the individual-weighted proportion is about 0.633. The difference arises because the largest clinic has the highest outcome proportion. If incomplete individual outcomes are imputed, a model intended to support the average clinic proportion must retain information about clinic-specific outcome proportions; an analysis targeting all patients uses patient-level weighting instead. The imputation model alone does not choose the estimand: the analysis and its weighting must state it.
The arithmetic assumes the listed counts and proportions are exact for this hypothetical dataset. It also illustrates why checking only the overall patient-level distribution can miss a mismatch in a cluster-level target.
Choosing Among Broad Imputation Strategies
The main alternatives differ in how they represent the hierarchy. A single-level imputation model treats records as if the conditional distribution were common across individuals, perhaps using observed cluster-level predictors but not representing residual similarities within clusters. It may be adequate in some settings, but it can fail to preserve cluster-specific variation and dependence when those features matter for the estimand.
A model that explicitly represents clusters can allow relevant conditional distributions to vary across clusters or account for within-cluster dependence. This is often more aligned with nested analyses, but it requires enough information to estimate the added structure and a plausible specification. A separate imputation model within every cluster is another conceptual option, but can be unstable when clusters contain few observed values. These broad strategies are not interchangeable, and none is automatically correct because it is more complex.
The practical comparison is therefore based on the target: does the proposed model retain the predictors, level-specific variation, and weighting needed by the analysis? The earlier material on representing nested data for imputation provides the data-structure context; here the key decision is to connect that representation to the estimand.
Diagnostics, Failure Modes, and Limits
Diagnostics should examine both the imputed values and the quantities the analysis will use. Useful checks include comparing observed and imputed distributions by cluster and by important cluster-level predictors; inspecting whether imputed values reproduce plausible within-cluster spread; and comparing cluster summaries and key associations across completed datasets. For a cluster-level estimand, compare the distribution of the cluster-level summaries directly. For an individual-level estimand, check relevant individual-level associations as well as their cluster pattern.
- Omitting the outcome when imputing a covariate. The imputed covariate may then be insufficiently associated with the outcome. Include the outcome when it is observed and relevant to the analysis, using a model suitable for its role.
- Using only a pooled distribution. This can erase meaningful cluster differences or within-cluster dependence. Check whether the target depends on those features and whether the imputation model retains them.
- Confusing the unit of analysis with the target population. An analysis with one row per person does not necessarily target an individual-weighted quantity, and a cluster summary does not necessarily imply equal-cluster weighting. State the estimand explicitly.
- Imputing only to match marginal distributions. Similar overall means and variances do not ensure that cluster-specific summaries or predictor-outcome associations are preserved. Check these analysis-relevant features directly.
Diagnostics can reveal incompatibility or poor fit, but cannot prove the missing-at-random assumption. Sparse clusters, extreme predictor combinations, and very limited observed information can make cluster-specific features difficult to estimate. More elaborate models can also be unstable or misspecified. When conclusions depend strongly on untestable assumptions about missingness, sensitivity analysis is needed; matching an imputation model to an estimand does not resolve that uncertainty.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A study targets the mean outcome among all patients, and clinics differ in size. Which weighting gives the largest clinics more influence?
- An individual-level exposure is incomplete, while the outcome is observed. Why can excluding the outcome from the exposure-imputation model be a problem?
- A researcher targets the average of clinic-specific outcome means. Should each clinic’s mean ordinarily receive weight proportional to the number of patients if the target gives every clinic equal influence?
- Name two diagnostics that directly assess whether imputation supports a cluster-level estimand.
Solutions. (1) Individual weighting gives each patient equal influence, so larger clinics contribute more patients and therefore more influence. (2) The imputed exposure may fail to retain its association with the observed outcome, which can distort the completed-data analysis. (3) No. Equal influence means each clinic mean receives the same weight; weighting by clinic size would change the target toward an individual-weighted quantity. (4) Compare imputed and observed cluster-specific summaries, and inspect whether key cluster-level associations and the distribution of those summaries are plausible across completed datasets.