Why Unequal Visit Spacing Matters
Repeated measurements are often labelled by visit number: first visit, second visit, third visit. That label records order, but not how much time passed. A measurement at a third visit could be taken four days after baseline for one person and six weeks after baseline for another. If the outcome changes with elapsed time, or if nearby measurements tend to be more alike than distant ones, treating both records as equally spaced can distort an imputation model.
The key is to represent the timing information that is relevant to the outcome. This can mean including actual elapsed time in the conditional mean, specifying dependence as a function of the time gap, or doing both. The appropriate choice depends on the scientific process and on whether a proposed model gives a reasonable account of the observed data.
An origin such as enrollment or treatment initiation should be chosen to match the question; time can also be centered at a meaningful reference point. Units matter: a slope per day is not numerically the same as a slope per month. The observed time should be retained even when the associated outcome is missing. The discussion here concerns imputing a continuous outcome in a longitudinal chained-equation system.
Mean Structure and Time-Dependent Dependence
A basic conditional mean model for a continuous outcome is a regression on actual time and relevant covariates. A longitudinal version can also allow each subject to have a different starting level and rate of change. One possible model is
Here \(\mathbf{X}_{ij}\) denotes included covariates, \(\beta_1\) is the population-average linear time trend, and \(b_{0i}\) and \(b_{1i}\) are subject-specific deviations in starting level and slope. A common assumption is that the random effects have a mean-zero multivariate normal distribution, independent across subjects, and independent of the residual process conditional on the model covariates. These assumptions should be adapted if the study design or data suggest otherwise.
Unequal spacing also matters for residual dependence. One continuous-time option, conditional on the fixed and random effects, is an exponential correlation structure:
In this model, \(\sigma_{\varepsilon}^{2}\) is the residual variance and \(\phi\) is a decay rate measured in inverse units of time. Residual correlation is \(\exp(-\phi\lvert t_{ij}-t_{ik}\rvert)\): it decreases as the elapsed-time gap increases. This is a modeling assumption, not a consequence of unequal timing itself. It is reasonable only when the observed residual dependence is adequately described by that pattern. Random slopes and residual dependence capture different features: the former allows persistent subject-specific trajectory differences, while the latter describes remaining within-subject dependence over time.
In MICE, an outcome’s conditional imputation model should use actual time in a way that reflects the analysis model and the data. If the imputation model uses a time-dependent covariance structure, draws for multiple missing outcomes within a subject should respect their joint conditional distribution rather than be generated as unrelated residuals. Parameters and random effects also carry uncertainty; imputing only fitted conditional means generally understates uncertainty.
Worked Examples
Worked Example: Visit Order and Elapsed Time Give Different Predictions
Hypothetical data. For one subject, measurements are scheduled at days 0, 1, and 4. The observed outcomes at days 0 and 4 are 10 and 18 units; the day-1 outcome is missing. To isolate the timing issue, suppose these two observed values lie exactly on a straight line and use that line for prediction, without adding random error.
Using actual time, the slope is \((18-10)/(4-0)=2\) outcome units per day. The predicted day-1 value is \(10+2(1-0)=12\) units.
A line based only on visit index assigns indices 1, 2, and 3 to days 0, 1, and 4. Its slope between the observed endpoints is \((18-10)/(3-1)=4\) outcome units per visit. At the second visit it predicts \(10+4(2-1)=14\) units.
The difference is not an arithmetic error: the second visit occurs only one day after baseline, not halfway through the four-day elapsed interval. The visit-index calculation treats it as halfway between the observed visits. These are point predictions under two different mean specifications, not multiple imputations; a MICE draw must also reflect uncertainty. The example assumes a linear trajectory for illustration and does not establish that either model is suitable for real data.
Worked Example: Conditioning on Unequally Spaced Residuals
Hypothetical data. Suppose the mean model has already been fitted, leaving observed residuals \(2\) and \(-1\) units at days 0 and 4. The residual at day 1 is missing. For this calculation only, assume the exponential residual model above with known \(\sigma_{\varepsilon}^{2}=4\) units squared and \(\phi=0.5\) per day, and assume Gaussian residuals.
The residual correlations between day 1 and the observed times are \(\exp(-0.5\times1)=0.6065\) and \(\exp(-0.5\times3)=0.2231\). The correlation between the two observed residuals is \(\exp(-0.5\times4)=0.1353\). Write \(\mathbf{R}\) for the correlation matrix of the two observed residuals and \(\mathbf{r}\) for their correlations with the missing residual:
For jointly Gaussian residuals with mean zero, the conditional mean of the missing residual is \(\mathbf{r}^{T}\mathbf{R}^{-1}\) times the vector of observed residuals. Here the weights are
Thus the conditional residual mean is \(0.5871(2)+0.1437(-1)=1.0305\) units. Its conditional variance is
Under these assumed known parameters, a residual draw would be from a normal distribution with mean about 1.031 units and standard deviation \(\sqrt{2.447}=1.564\) units. Add that draw to the fitted mean at day 1 to obtain an imputed outcome. Using the conditional mean alone would omit residual uncertainty; in an actual MICE analysis, uncertainty in model parameters must also be represented.
Worked Example: Same Visit Number, Different Elapsed Time
Hypothetical model. Suppose two subjects have the same covariates and the same baseline level, and the fitted conditional mean is \(5+0.08t\), where \(t\) is days since baseline and the outcome is measured in points. Their records are both labelled visit 3, but the first measurement is at day 14 and the second at day 42.
For day 14, the fitted mean is \(5+0.08(14)=6.12\) points. For day 42, it is \(5+0.08(42)=8.36\) points. The difference is \(8.36-6.12=2.24\) points, which also equals \(0.08(42-14)\). A model that uses only visit number and otherwise identical predictors assigns the same visit-3 time contribution to both records.
This comparison assumes a linear time effect and equal other predictors; it illustrates what the model encodes, not a finding about any real population. A nonlinear time effect may be needed if the trajectory bends. For example, a prespecified spline in actual elapsed time can allow curvature without pretending that irregularly spaced visits are equally far apart.
Why the Conditional Gaussian Formula Works
The conditional calculation in the second example follows from the joint normal distribution. Let \(\boldsymbol{\varepsilon}_{o}\) be a vector of observed residuals and \(\varepsilon_m\) a missing residual, with covariance blocks \(\boldsymbol{\Sigma}_{oo}\), \(\boldsymbol{\Sigma}_{om}\), and \(\boldsymbol{\Sigma}_{mm}\). For a jointly Gaussian vector, the conditional distribution of \(\varepsilon_m\) given \(\boldsymbol{\varepsilon}_{o}\) has mean and variance
To see the mean, subtract the proposed linear prediction from the missing residual: \(\boldsymbol{\eta}=\varepsilon_m-\boldsymbol{\Sigma}_{mo}\boldsymbol{\Sigma}_{oo}^{-1}\boldsymbol{\varepsilon}_{o}\). Its covariance with \(\boldsymbol{\varepsilon}_{o}\) is \(\boldsymbol{\Sigma}_{mo}-\boldsymbol{\Sigma}_{mo}\boldsymbol{\Sigma}_{oo}^{-1}\boldsymbol{\Sigma}_{oo}=\mathbf{0}\). For jointly Gaussian quantities, zero covariance implies independence. Therefore, conditional on the observed residuals, the prediction error has mean zero; expanding its variance gives the stated variance formula. This argument also shows why actual gaps enter: they determine covariance blocks under the chosen continuous-time model. The formula assumes the covariance matrix is correctly specified and invertible.
Model Checks, Alternatives, and Limitations
Before relying on an elapsed-time imputation model, check the assumptions it needs. The time origin and units must be known, and recorded times must correspond to the occasions at which outcomes were measured. The conditional mean and covariance should be plausible over the observed follow-up range. As in other MICE models, the imputation assumptions also concern missingness: a missing-at-random assumption is more plausible when the imputation model includes observed variables that predict missingness or the outcome. It does not follow merely from including time.
Useful checks include plotting observed outcomes and fitted means against actual elapsed time, inspecting residuals against time and against the gap since the preceding measurement, and comparing residual dependence for short and long gaps. Check whether residual spread or trajectory shape changes over follow-up. In chained equations, also examine trace plots and compare observed and imputed outcome distributions across meaningful time ranges. These diagnostics can reveal a poor mean form, unmodeled time-varying variance, or dependence that does not decay as assumed; they cannot prove the model is correct.
A visit-order model can be a reasonable alternative when visits are genuinely scheduled at nearly equal intervals, or when the scientific estimand concerns visit stage rather than duration. If actual intervals vary materially and change the expected outcome or residual dependence, visit order alone discards relevant information. Categorizing elapsed time may be useful when the response differs by clinically defined period, but broad categories can also hide meaningful differences within a period. A model with both visit order and elapsed time is sometimes justified, but the terms must represent distinct scientific features rather than being added automatically.
Actual visit times can also be informative. For example, worsening health might trigger an earlier visit. If visit timing depends on unobserved outcomes even after conditioning on observed information, treating the times as fixed predictors may not address the resulting selection. The visit process may need explicit consideration, or the assumptions may need careful sensitivity analysis. Other limitations include sparse observations at some time ranges, extrapolation beyond observed follow-up, and misspecification of linear trends or exponential correlation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Three visits occur on days 0, 2, and 10. What are the elapsed gaps between consecutive visits, and why is it inaccurate to treat them as equal intervals?
- Under the exponential correlation model with \(\phi=0.2\) per day, calculate the residual correlation for observations 3 days apart. State its interpretation.
- A fitted mean increases by 0.5 outcome units per week. Two records have the same visit number but occur 6 weeks apart. Under a linear time effect and equal other predictors, what is the difference in fitted means?
- Why is substituting a conditional mean for every missing value not equivalent to drawing a proper multiple imputation?
Solutions. (1) The gaps are \(2-0=2\) days and \(10-2=8\) days. They differ, so the same increment in visit index does not represent the same elapsed time. (2) The correlation is \(\exp(-0.2\times3)=\exp(-0.6)\approx0.5488\). Under the assumed model, residuals 3 days apart have correlation about 0.549, conditional on the model’s mean structure. (3) The difference is \(0.5\times6=3\) outcome units; the later record has the larger fitted mean if the slope is positive. (4) A conditional mean omits the residual variation represented by the conditional predictive distribution and, if parameters are treated as fixed, omits parameter uncertainty too. Repeatedly inserting means therefore fails to represent the uncertainty that multiple imputation is intended to carry forward.