Subject-Level Residual Summaries For Repeated Measures

Why Summarize Residuals by Patient?

A residual plot can show whether a longitudinal model fits the data overall, but an overall display can conceal patient-specific departures. One patient’s measurements may repeatedly lie above the fitted trajectory, while another’s residuals may alternate around zero. Summaries computed separately for each patient help locate these patterns and direct a closer review of the fitted longitudinal component.

The goal is diagnostic, not to label a patient as abnormal based on a single number. A subject-level residual mean captures the direction of average departure; a residual variance summarizes the size of departures around that patient’s mean; and a sign-run count describes whether positive and negative residuals are interspersed or clustered in measurement-time order. These summaries complement the marginal and conditional residuals introduced in earlier material, but their interpretation depends on how the residuals were constructed.

Assumptions and Choice of Residual

For a Gaussian longitudinal component, a useful working model is \(Y_{ij}=m_i(t_{ij})+\epsilon_{ij}\), where \(m_i(t_{ij})\) is the subject-specific mean trajectory represented by the fitted model. The basic reference interpretation of residual means and variances supposes that, conditional on the modeled trajectory and covariates, the errors have mean zero and a common variance \(\sigma^2\). The simplest reference also supposes independent errors across visits. These are assumptions to assess, not properties guaranteed by computing residuals.

The fitted mean must match the diagnostic question. A conditional fitted trajectory accounts for the subject’s estimated random effects; its residuals ask whether the measurements depart from that subject-specific fitted trajectory. A marginal fitted trajectory does not condition on the estimated random effects and answers a different question. A subject with a large estimated random intercept can have large deviations from the population trajectory but small residuals from a conditional trajectory. State which residual is being summarized, and do not interpret these two comparisons as interchangeable.

When measurement-error scales differ by visit or by subject, raw residual variances are not directly comparable. If a relevant residual standard deviation estimate is available, standardized residuals can be formed as \(r_{ij}=e_{ij}/\widehat\sigma_{ij}\); their variance can then be compared with the approximate unit-scale reference. Standardization does not fix an incorrect mean model, dependence, or a poorly estimated scale. Residuals also use estimated model parameters, and conditional predictions may use the same subject’s measurements. Consequently, idealized expectations provide diagnostic benchmarks, not exact finite-sample guarantees for each patient.

Mean and Variance Summaries

Under a correctly specified mean trajectory and zero-mean errors, the expected residual mean is approximately zero. A positive \(\bar e_i\) means the observed measurements average above the chosen fitted trajectory; a negative value means they average below it. Under independent errors with variance \(\sigma^2\), the unadjusted mean of \(n_i\) errors has variance \(\sigma^2/n_i\). Thus a mean based on two visits is much less stable than one based on ten. In fitted models, this simple variance is only a reference because parameter estimation and prediction of random effects can change the residual behavior.

The sample residual variance measures spread around \(\bar e_i\), not average departure from the fitted trajectory. It can be large even when the residual mean is near zero, for example when positive and negative errors are both substantial. Conversely, a patient can have a clearly nonzero residual mean and little spread if all residuals are similarly shifted. When \(n_i=1\), the sample variance is undefined; when \(n_i\) is small, it is highly uncertain.

Sign Runs in Measurement-Time Order

The sign-run summary reduces each nonzero residual to positive or negative and counts consecutive blocks in time order. A long run of positive signs is consistent with repeated measurements lying above the fitted trajectory over a stretch of visits; a long negative run has the analogous interpretation. Runs therefore highlight an ordered pattern that a mean or variance alone may not reveal. They do not measure the size of the residuals: a run of tiny positive residuals counts the same as a run of large positive residuals.

A simple reference for the number of runs comes from a random ordering of a fixed set of signs. Suppose \(n_+\) residuals are positive, \(n_-\) are negative, and \(N=n_++n_-\). Conditional on these counts, if every ordering is equally likely, the expected number of runs is

This expectation is a benchmark, not a universal test of model adequacy. Its random-ordering assumption is not assured merely because a longitudinal model has been fitted. For a very small number of visits, the run count takes few possible values and is a coarse summary. Zero residuals have no sign; a consistent convention is to omit them when counting signs and report how many were omitted. If ties arise from rounding, inspect the unrounded residuals where possible.

A Practical Diagnostic Sequence

Calculate summaries only after fixing the residual definition and ordering each patient’s observations by time. The following sequence keeps the three summaries distinct and makes comparisons more defensible.

Comparisons, Failure Modes, and Limits

A subject-level mean is more interpretable for persistent direction than a pooled residual mean, which can be near zero because positive and negative patient-level departures cancel. A subject-level variance adds information about within-patient spread, while the runs summary adds information about sign ordering. A residual plot retains magnitudes and timing, so it is usually the best companion display. These summaries do not replace the earlier material on marginal and conditional residuals, and they do not quantify residual autocorrelation; that is a separate diagnostic question.

Several features of joint longitudinal and event data require particular caution. Patients have different numbers and timings of measurements, and observation may stop at an event or censoring time. A summary based on measurements available before that time does not describe unobserved later biomarker values. Conditional residuals also depend on estimated subject effects, which are informed by that patient’s measurements; this can make departures look smaller than they would under an independently specified prediction. Finally, an extreme subject-level summary can arise by chance when many patients are inspected. These summaries identify records for follow-up; they do not establish that an individual is an outlier, that a model component is wrong, or that a biomarker departure caused an event.

A common mistake is to declare that a patient has a mean-model failure because their residual mean is positive, or to call residual signs independent because the observed run count is near its reference expectation. The first ignores uncertainty and the fact that some nonzero means occur by chance; the second treats one coarse summary as proof of an assumption. Instead, consider visit count, residual scale, fitted trajectory, timing, and the pattern across patients. Then use suitable model diagnostics and substantive review to decide whether the fitted longitudinal component needs further investigation.