Longitudinal Records As Two-Level Data

Why Repeated Records Need a Two-Level View

A longitudinal dataset often contains several records for each person: one row per person-visit, person-month, or person-assessment. Those rows are not independent people. They are repeated observations from the same individual, and they may share stable characteristics or unmeasured influences. An imputation model that treats every row as an unrelated unit can therefore misrepresent the variation and dependence in the data.

The basic task here is to identify the two levels before specifying an imputation model. Level 2 is the person; level 1 is a repeated record within that person. This description does not require that everyone have the same number of records or that observations occur at equally spaced times. It does require a clear record-to-person mapping and a decision about which variables belong to which level.

For example, blood pressure measured at each visit is a level-1 variable, while a person's treatment assignment at baseline is typically level 2. Age requires care: age recorded at each visit changes across records and can be treated as level 1; age measured only at study entry is a single baseline characteristic and is level 2. The variable's meaning and how it was recorded determine its level, not its name.

This indexing clarifies the data structure, but does not by itself prescribe a particular model for time, serial dependence, or changing covariate effects. Those are additional modeling choices. Here the central question is how to preserve the fact that multiple records belong to one person when constructing conditional imputation models.

From Records to Variables and Imputation Targets

Before imputing anything, identify the person identifier, the repeated-record index, each variable's level, and the unit at which the variable is genuinely observed. The person identifier links records; it is not usually an explanatory variable whose numeric value should enter the model. A level-2 value copied onto several rows remains one person-level value, not several independent measurements.

In chained equations, each incomplete variable is modeled conditionally on other variables. For a repeated-record target, the conditional model should generally represent both the association among records from the same person and relevant information at both levels. For a person-level target, the conditional model must be fitted at the person level, or otherwise be constructed so that repeated copies do not give a person with more records disproportionate influence.

One basic continuous-outcome model illustrates the structure:

Here \(Y_{ij}\) is the incomplete or modeled level-1 variable, \(X_{ij}\) is a level-1 predictor, \(Z_i\) is a level-2 predictor, \(u_i\) is a person-specific random intercept, and \(\varepsilon_{ij}\) is record-specific residual variation. The usual random-intercept formulation assumes that the \(u_i\) are independent across people, the residuals are independent across records conditional on the random intercept and predictors, and both components have mean zero with the stated variances. It also assumes the conditional mean is adequately represented by the specified predictors and that the random intercept is independent of those predictors, unless the model is extended to address dependence.

In an imputation setting, this equation is a model for the conditional distribution of a missing \(Y_{ij}\) given observed and currently imputed information. It is not enough to substitute a predicted mean and fill in the missing cell: an imputation procedure must also represent appropriate residual and model uncertainty. The precise fitting and drawing steps depend on the chosen multilevel imputation method.

Why the Random Intercept Represents Dependence

A short derivation shows why the person term matters. Consider two distinct records \(j\ne k\) from the same person, with predictors treated as fixed. Their random parts are \(u_i+\varepsilon_{ij}\) and \(u_i+\varepsilon_{ik}\). If the record-specific residuals are independent of each other and of \(u_i\), then:

For a single record, the corresponding conditional variance is \(\tau^2+\sigma^2\). Thus the within-person correlation under this simple model is \(\tau^2/(\tau^2+\sigma^2)\). This is a consequence of the random-intercept assumptions, not a universal description of all longitudinal dependence. If those assumptions are inappropriate, a random intercept alone may not capture the dependence in the records.

Assumptions, Diagnostics, and Alternatives

A two-level data layout is a structural description; it does not guarantee that a chosen imputation model is correct. The analyst still needs a plausible conditional model for each incomplete variable, compatible predictor sets across levels, and an appropriate treatment of the outcome type. A normal random-intercept model, for instance, can be unsuitable for a strongly skewed or bounded variable without adaptation. For binary, ordinal, nominal, or count targets, the conditional distribution must match the outcome scale.

As with other imputation under a missing-at-random assumption, the observed information in the model must be sufficiently rich: conditional on included observed data, the probability that a value is missing should not depend on its unobserved value. If missingness still depends on the missing value after conditioning, a missing-not-at-random model or sensitivity analysis may be needed. The longitudinal structure alone does not establish this assumption.

Useful structural checks include confirming that each record maps to exactly one person, that person-level variables are constant within person where they should be, and that record counts and missingness patterns are plausible. After imputation, inspect whether within-person and between-person variability are both represented, whether imputed values stay within meaningful ranges, and whether people with many records dominate a model for a person-level target. Implausibly narrow within-person variation can signal that the model has not retained record-level residual variation; implausible between-person distributions can indicate a poorly specified person-level component.

A rowwise single-level model is simpler and may be reasonable when clustering is negligible for the target and analysis, but it assumes away residual dependence if it treats all records as independent. Aggregating to one summary per person avoids duplicate person-level contributions, but discards record-specific information and cannot represent variation across repeated outcomes. A two-level model retains the record-level observations while accounting for shared person-level structure. Its cost is greater modeling complexity and dependence on assumptions about the random effects and conditional distributions.

A common mistake is to say, “There are eight rows, so there are eight independent units for every variable.” The correction is to distinguish the number of records from the number of people: a level-1 analysis may use all eight records, but records from the same person are clustered; a level-2 value still has only three person-level realizations. Another mistake is to assume that adding a person identifier as an ordinary numeric predictor solves clustering. An identifier is a label, not a meaningful quantitative covariate; dependence must be represented through an appropriate multilevel structure or another justified model.