Within-Subject Occasion Diagnostics

Why Compare Occasions Within a Subject?

A model can fit the overall data reasonably well while describing one occasion poorly. For example, profiles from the same subject might be systematically underpredicted at a later visit, even if residuals pooled across all visits are centered near zero. Pooling can conceal this pattern because positive residuals from one occasion may cancel negative residuals from another. Comparing occasions within subjects helps reveal such shifts while holding constant stable characteristics of each subject.

An occasion is a prespecified, scientifically meaningful period or study segment within a subject, such as a dosing interval or a visit. The occasion definition should follow the design and analysis question; it should not be chosen after inspecting residuals to make a pattern appear stronger. Use residuals on a comparable scale, such as the same type of standardized residual at every occasion. A residual's sign indicates whether the observation is above or below its corresponding prediction, subject to the residual convention used.

The diagnostic is exploratory: a pattern can flag a model or data feature to investigate, but it does not by itself identify a cause. In particular, an occasion contrast is not automatically evidence for a biological period effect or a need for an occasion-specific random effect.

Organizing the Comparison

Keep subject identity visible. Useful displays include subject-specific residual-versus-time panels separated by occasion, paired summaries for each subject, and observed-versus-individual-prediction overlays for subjects with observations on multiple occasions. Within a subject, compare occasions over time regions represented on both occasions. Also inspect the number and timing of observations: an occasion with mostly early samples should not be treated as directly comparable to one with mostly late samples.

A simple descriptive decomposition clarifies what a paired comparison can and cannot remove. Let \(r_{ijk}\) denote a residual for subject \(i\), occasion \(k\), at observation \(j\), and let \(t_{ijk}\) be the relevant time. For interpretation, consider the schematic representation

Here \(a_i\) represents a stable subject-level residual tendency, \(c_{ik}\) an occasion-associated component, \(g(t)\) a shared time-related pattern, and \(\varepsilon_{ijk}\) remaining variation. This is a diagnostic bookkeeping device, not a claim that residuals necessarily follow this exact model or that the components can be identified from a plot. In particular, a stable tendency in residuals is not the same thing as a subject's pharmacokinetic random effect.

Write \(\bar r_{ik}\) for the mean residual among the observations available for subject \(i\) on occasion \(k\). Subtracting occasion means cancels a common \(a_i\), but the other terms need not cancel:

The bars on \(g(t)\) and \(\varepsilon\) denote averages over the observations in the corresponding occasion. Thus, even if the occasion-associated components are equal, unequal time support or chance variation can produce a nonzero observed contrast. Similar sampling times can make a time contribution more comparable, but do not guarantee equality. Matching or stratifying by time can reduce this concern; it cannot remove all uncertainty or unmeasured differences.

Worked Comparisons

Diagnostics, Alternatives, and Limitations

A practical review proceeds from the display to the comparability of the data, then to possible explanations. Inspect raw residual points as well as subject-level summaries; a mean can hide a few large residuals or a time-ordered pattern. For each subject, ask whether an occasion difference persists at comparable times and whether it appears in more than a few subjects. Check that the same residual definition and scale are used throughout. When individual prediction overlays are available, look for systematic under- or overprediction on one occasion rather than relying only on a single numerical summary.

• Time trend: residuals may vary with time, and occasions with different time distributions can have different means even without an occasion-specific shift.
• Unequal sampling: a mean from many observations can dominate a pooled summary, while a sparsely sampled occasion may be unstable. Show counts and consider subject-level paired summaries.
• Changing conditions: dose, regimen, assay, study procedures, or other recorded factors may differ across occasions. A visual association with occasion does not distinguish these explanations.
• Residual dependence and noise: multiple observations from one subject are not automatically independent. A cluster of residuals is not equivalent to the same number of independent subjects.
• Model and data issues: an apparent shift can reflect prediction errors, recording or timing problems, or a mismatch in the residual scale, not only structural-model misspecification.

Within-subject plots are complementary to residuals pooled by study, site, or assay and to residuals plotted against time. Grouped plots can reveal differences between groups but may mix those differences with subject composition. Time plots can show a common trend but may obscure whether it occurs consistently within subjects. Occasion diagnostics focus on repeated periods within subjects; they do not replace either comparison. Formal models for repeated observations can quantify uncertainty under explicit assumptions, but choosing and interpreting such models requires more than visual inspection.

The main limitation is incomplete overlap. If occasion 1 contains only early times and occasion 2 only late times, the data may not separate an occasion contrast from a time trend without additional assumptions. Likewise, if only a few subjects have observations on both occasions, paired patterns may not represent the full sample. State these limitations rather than treating a plot as a causal or definitive test.