Residual Diagnostics by Study or Occasion

Why Examine Residuals by Group?

A pooled residual plot can look acceptable even when residual behavior differs meaningfully across studies or occasions. For example, one study may have residuals predominantly above zero while another has residuals predominantly below zero. When both are combined, their average can be near zero, concealing the group-specific offsets. Conversely, a pooled plot may show broad scatter because one group is much more variable than the others.

Grouping residuals by study, occasion, or dosing period is a targeted diagnostic: it asks whether the residual pattern is similar across those defined subsets. It does not replace the standard residual-versus-time, residual-versus-prediction, or distributional displays. Rather, it adds the group label to the diagnostic question. The earlier material on residuals versus time, residuals versus predictions, residual distributions, and residual trends across covariates provides the relevant general plotting context.

For a group comparison to be meaningful, observations must be assigned consistently, and the plotted residuals must be on a scale that supports comparison. If residuals are raw concentration differences and expected measurement variability differs greatly with concentration, a wider raw spread in one group may reflect that scale difference rather than worse model performance. Use the residual type chosen for the standard diagnostics and state its units or, for standardized residuals, that it is dimensionless and measured in standard-deviation units.

What Grouped Displays Can Reveal

Common displays include separate panels for each group or a single plot that uses distinct symbols or colors. A horizontal reference at zero helps assess residual location. When panels are compared, keep the vertical scale common: independently rescaled axes can make very different residual spreads appear similar. For a small number of groups, overlaid points can support direct comparison; for many groups, separate panels can reduce visual clutter. In either layout, show the number of observations when practical and avoid interpreting a few points as a stable group pattern.

Inspect at least three features within each group: whether residuals are centered near zero, whether their spread is broadly comparable, and whether there is a structure such as a sequence or cluster of positive or negative values. A grouped plot against time within an occasion can reveal a pattern that a single group mean cannot. Study labels can also identify differences in data collection or measurement context that merit investigation. These displays flag patterns; they do not, by themselves, establish their cause.

Why Pooling Can Conceal a Group Offset

Suppose the residuals are divided into groups indexed by \(g\). Let \(n_g\) be the number of residuals in group \(g\), \(N=\sum_g n_g\), and \(\bar{r}_g\) the group mean. The pooled mean is a weighted average of group means, with weights equal to the groups’ proportions of observations.

This identity explains why a pooled center near zero does not guarantee that every group is centered near zero: positive and negative group means can cancel. There is a related decomposition for empirical variance. Define \(s_g^2\) here as the within-group variance using divisor \(n_g\), and define the pooled variance using divisor \(N\). Then:

To justify the decomposition, write each deviation as \(r_{ig}-\bar{r}=(r_{ig}-\bar{r}_g)+(\bar{r}_g-\bar{r})\), square it, and sum within group. The cross-product sums to zero because \(\sum_i(r_{ig}-\bar{r}_g)=0\). The remaining terms are the weighted within-group variance and the weighted variation among group means. Thus, pooled spread combines within-group spread with between-group differences in location. A pooled spread alone cannot tell which component is responsible.

Worked Comparisons of Group Spread and Occasion Pattern

Assumptions, Diagnostics, and Alternatives

Grouped residual plots are descriptive diagnostics, not tests that require independent observations or normally distributed residuals. Their interpretation nevertheless depends on several practical conditions:

• Correct group assignment. Study, occasion, and dosing-period labels must refer to the intended observations. Verify coding and boundary rules, especially when occasions or periods are constructed from times.
• Comparable residual meaning. Use a consistent definition across groups. If the error scale or transformation differs, explain how the residuals were made comparable; raw residual magnitudes may not be comparable across different units or scales.
• Visible sample sizes and scales. Small groups give unstable impressions, and different vertical axis limits can disguise spread differences. Retain a shared scale for direct visual comparison.
• Dependence is acknowledged. Repeated observations from the same individual are not necessarily independent. A sequence of positive residuals within one individual or occasion may carry different information from the same number of positive residuals scattered across independent individuals.

A useful diagnostic workflow is to first examine the pooled plot, then stratify by the label relevant to the question, compare centers and spreads on common axes, and finally investigate any persistent pattern in context. If the concern is a location shift, compare group centers; if it is variability, compare spreads. If the pattern is temporal, inspect the residuals in their time order within each group rather than relying only on a group summary.

The main alternative is to retain a pooled plot, which is simpler and can reveal broad overall problems but may hide group differences. Separate panels make group patterns easier to see, although many panels can become hard to compare. Overlaid groups facilitate direct comparison when there are few groups but can obscure points when they overlap. A numerical summary can complement either plot, but a summary alone loses ordering and local structure. Formal modeling or testing may be appropriate for a specific follow-up question, but a visual pattern is not itself a significance test and does not imply a causal group effect.

Common failures include drawing separate panels with independently scaled axes, interpreting an average near zero as evidence that all groups are well centered, and treating every visible group difference as proof of model failure. Another pitfall is to group by a label without checking how it was defined: a misclassified occasion can create an artificial pattern. Examine counts, units, residual type, and data assignments before interpreting the plot. Grouped diagnostics are also limited by sparse groups, influential observations, and correlations among repeated measurements; they cannot repair a misspecified model or establish why a pattern occurred.