Preserving Within-Person Change in Longitudinal Imputations

Why Preserve Change, Not Just Levels?

A time-varying covariate is not simply one value per participant. The sequence of values can describe both a participant’s typical level and how that participant’s level changes from occasion to occasion. Those are distinct sources of information. If an imputation procedure replaces every missing value with a participant’s mean, it may preserve a rough estimate of typical level while erasing the very fluctuations needed to study within-person change.

For example, a person’s blood pressure may generally be higher than another person’s, while also rising or falling over follow-up. A model of an outcome may distinguish the association with the person’s average blood pressure from the association with blood pressure being above that person’s own average at a particular time. Imputation should retain enough structure to estimate both associations. It should not collapse repeated observations into one summary value.

Write \(X_{ij}\) for a time-varying covariate measured for participant \(i\) at occasion \(j\), where \(i=1,\ldots,n\). Let \(t_{ij}\) denote the measurement time. In a completed imputed data set, define the participant mean and the occasion-specific deviation as

where \(m_i\) is the number of occasions represented for participant \(i\). The mean \(\bar{X}_i\) describes between-person level; \(X_{ij}^{W}\) describes within-person deviation. The deviations sum to zero across the occasions used to calculate the mean. When values are missing, these summaries must be formed from each completed data set, not treated as known constants.

Models That Represent Both Sources of Variation

A useful imputation model for a continuous repeated covariate is a multilevel model with participant-specific effects. A simple random-intercept and random-slope form is

Here, \(\beta_0+\beta_1t_{ij}\) describes the population-average time pattern; \(b_{0i}\) lets participant \(i\) have a different level; and \(b_{1i}\) lets that participant have a different time trend. A common model assumes the participant effects have a multivariate normal distribution with mean zero and covariance matrix \(G\), and that the residuals \(\epsilon_{ij}\) are independent normal errors with mean zero and variance \(\sigma^2\), conditional on the participant effects and included predictors. The covariance matrix \(G\) allows participant-specific levels and slopes to be associated.

These are modeling assumptions, not consequences of putting the data in long format. The time function should reflect plausible patterns; a straight-line trend may be inadequate if change is nonlinear. Additional predictors may be included when appropriate to the missingness and analysis models. If there is little information for estimating a participant’s slope, a random-slope model can be unstable; a random-intercept model or a simpler time structure may be more defensible.

In a MICE procedure, the conditional imputation model for \(X\) must account for repeated observations from the same participant. A multilevel conditional model can draw participant effects and residual variation as part of imputation. Drawing only fitted participant means, or inserting best linear predictions without uncertainty, understates variability. In addition, the conditional models used in chained equations need not automatically correspond to one coherent joint distribution. Their specifications should be checked for compatibility with the intended analysis and the repeated-measures structure.

A within-between outcome model often uses \(\bar{X}_i\) and \(X_{ij}^{W}\) as separate predictors:

In this model, \(\gamma_B\) describes the association between differences in participants’ average covariate levels and the outcome, while \(\gamma_W\) describes the association between an occasion-specific departure from a participant’s average and the outcome. These are associations, not necessarily causal effects. This decomposition is useful for understanding the analysis target, but it does not mean that \(\bar{X}_i\) and \(X_{ij}-\bar{X}_i\) should both be inserted as separate predictors when imputing \(X_{ij}\): they are algebraically constructed from \(X_{ij}\) itself. The imputation model must handle the repeated measurements coherently, for example through participant effects or carefully constructed summaries of other occasions.

A Conditional-Draw Derivation

A small random-intercept model shows why an imputation should combine a participant’s observed information with residual uncertainty. Suppose the time-adjusted covariate for a participant follows \(Z_{ij}=b_i+\epsilon_{ij}\), with \(b_i\sim N(0,\tau^2)\) and independent \(\epsilon_{ij}\sim N(0,\sigma^2)\). Assume these variances are known for this derivation. Given \(k\) observed adjusted values \(z_1,\ldots,z_k\), the conditional distribution of \(b_i\) is normal. Its precision is the prior precision plus the observed-data precision:

For a missing occasion with the same adjusted mean structure, the predictive distribution adds a new residual error. Thus its conditional mean is the posterior mean of \(b_i\), and its predictive variance is the posterior variance of \(b_i\) plus \(\sigma^2\). A draw from this distribution reflects the person’s observed level without pretending that the missing value is known exactly.

Separating Within-Person and Between-Person Information

The distinction matters because identical covariate values can have different meanings for different participants. An occasion value above one participant’s typical level may be below another participant’s typical level. A single population-wide regression term for \(X_{ij}\) can blend these two comparisons.

Practical Imputation and Diagnostics

A sound workflow keeps the repeated measurements in their participant-and-occasion structure and uses an imputation model that reflects that structure. The participant mean and deviations used in the substantive analysis should be recalculated within every completed data set. If they are used as imputation predictors, their construction needs special care: a summary that includes the target value can create circularity. A multilevel model with latent participant effects avoids treating a target-containing mean as an independent predictor; alternative summary-based methods must specify how summaries are constructed and what happens when little observed data are available for a participant.

Diagnostics should examine more than the overall distribution of the imputed covariate. Compare observed and imputed means, spreads, and ranges by time; inspect participant trajectories; and compare within-person variability and between-person variability. For continuous covariates, the intraclass correlation, the fraction of variation attributable to differences between participants under a specified model, can help reveal whether imputation has greatly changed the clustering structure. Also check whether fitted time trends and residual patterns are plausible. A narrow pile of imputed values, implausibly smooth trajectories, or almost no within-person variation can signal over-shrinkage or deterministic prediction.

Such diagnostics can reveal a poor model but cannot prove that missingness assumptions are true. As in other missing-data analyses, inference relies on assumptions about the relationship between missingness and observed or unobserved values. A common working assumption is that missingness is conditionally independent of the missing value given variables included in the imputation model. If that assumption is doubtful, sensitivity analysis may be needed; no trajectory diagnostic alone resolves the issue.

Common Pitfalls and Alternatives

A single-level imputation model that treats every row as independent ignores the dependence between measurements from the same participant. It can produce unrealistic combinations of within- and between-person variation, even if its overall mean looks reasonable. Conversely, imputing one participant mean and copying it into every missing occasion suppresses within-person change. Last observation carried forward is another deterministic substitution and does not represent the uncertainty or changing trajectories needed for valid inference.

Wide-format imputation, in which each occasion is a separate variable, can preserve occasion-specific distributions and associations, but it can become unwieldy with many occasions and may not represent participant-level trajectory structure well. A multilevel long-format model directly models clustering and time, but requires suitable model assumptions, adequate information, and implementation that draws uncertainty correctly. The choice is not simply between long and wide data: the conditional models must be appropriate to the analysis and the missingness structure.

• Do not equate prediction with imputation. A fitted mean or participant-specific best prediction omits residual uncertainty unless an appropriate stochastic draw is added.
• Do not use a target-containing summary as if it were external information. A person mean calculated from a value being imputed is mathematically dependent on that value.
• Do not assume that random slopes are always better. They may be weakly identified with few occasions or sparse observations; use a time structure supported by the data.
• Do not judge preservation from the pooled mean alone. Similar overall means can conceal collapsed within-person spread or distorted trajectories.

The central practical principle is that imputation should preserve the features the analysis needs. When the analysis separates participant averages from occasion-specific deviations, the imputation model must retain both sources of information and uncertainty. A model that only reproduces cross-sectional distributions can still be inadequate for longitudinal inference.