Why Compare Residuals by Dose and Regimen?
A model can fit the overall concentration range reasonably well yet systematically overpredict at one dose level and underpredict at another. It can also miss the shape of repeated-dose profiles: for example, residuals may drift over successive dosing intervals when accumulation is not represented adequately. Grouping residuals by dose and regimen can make these patterns visible.
The comparison requires care. Dose changes the expected response, so raw concentrations from different doses should not be expected to match. A correctly specified model predicts different concentrations when doses differ. The diagnostic question is whether residuals—observations relative to their model predictions—show a systematic pattern after accounting for the dose and dosing history used to generate those predictions. Even a clear group difference is a clue, not proof of misspecification: dose groups may also differ in sampling times, patient characteristics, study conduct, or other design features.
Choose the Residual and the Comparison Carefully
Let \(y_{ij}\) be the observed response for individual \(i\) at observation \(j\), and let \(f_{ij}\) denote the corresponding model prediction. The raw residual is \(r_{ij}=y_{ij}-f_{ij}\). Its size depends on the response scale and may also depend on the prediction. Weighted residuals, conditional weighted residuals (CWRES), and individual weighted residuals (IWRES) scale residual information using an error or prediction-variance model. As established in the earlier material on residual definitions and standardization, the choice of residual affects what a plot can show.
For comparing dose groups with different predicted concentration ranges, standardized residuals are often more interpretable than raw residuals, because raw residuals may naturally be larger where concentrations and prediction variance are larger. CWRES are commonly used to assess population-level fit; IWRES describe discrepancies relative to individual predictions and should be interpreted with awareness that those predictions use estimated individual effects. No residual type removes the need to examine the model, design, and sampling scheme.
A useful display plots residuals against dose amount or labels dose categories with separate distributions. For regimens, useful labels may include dosing interval, loading-dose status, or position within a dosing sequence. Accompany group summaries with individual-level information: a group with many observations from a few individuals should not appear to provide the same breadth of evidence as a group sampled across many individuals. When measurements within an individual are correlated, counting every residual as independent can exaggerate how much information a group contains.
What a Pattern Can—and Cannot—Tell You
A dose-related residual shift can indicate that predictions are systematically too high or too low at some doses. Possible explanations include an inadequate dose-response relationship, a dose-dependent parameter or error structure not represented in the model, or a dose variable entered or coded incorrectly. But a shift may instead reflect a design difference. For instance, if high-dose subjects are mostly sampled late after dosing while low-dose subjects are sampled near peak concentration, a dose-group comparison can mix dose with time since dose.
Regimen comparisons require particular attention to history. Repeated doses can produce accumulation, and the model prediction should reflect the actual doses and timing relevant to each observation. Residuals changing across dosing occasions might signal an inadequate account of accumulation, but might also arise because later observations occur at different times, because actual administration differs from the recorded regimen, or because the individuals contributing observations change over time. A regimen category is not itself an explanation for a residual pattern.
The pattern's direction and location help focus follow-up. A dose-group shift across the full profile differs from a pattern limited to a particular sampling phase. A trend over successive dosing intervals differs from a consistent offset at all occasions. Check predictions, residuals, dose records, and sampling times together before proposing a model change. The earlier material on residuals against time since dose is useful when a regimen-group pattern may actually reflect different time-since-dose coverage.
Worked Example: A Shift Across Dose Levels
Suppose a hypothetical analysis compares mean CWRES across two dose groups. For a compact illustration, assume the listed residuals are from distinct individuals, with one observation per individual in the comparison. The low-dose group has CWRES values \(-0.2,\ 0.1,\ 0.3,\ 0.2\); the high-dose group has \(0.8,\ 1.0,\ 0.6,\ 0.8\).
Low-dose mean: \(\bar{r}_{L}=(-0.2+0.1+0.3+0.2)/4=0.4/4=0.10\).
High-dose mean: \(\bar{r}_{H}=(0.8+1.0+0.6+0.8)/4=3.2/4=0.80\).
The descriptive difference is \(0.80-0.10=0.70\) standardized residual units. Because positive residuals represent observations above predictions under the usual observed-minus-predicted convention, this display suggests underprediction in the high-dose group relative to the low-dose group. It does not establish why the pattern arose or prove that the structural model is wrong.
Next check whether both groups include comparable times since dose and whether the high-dose group differs in regimen, patient characteristics, or study procedures. Inspect individual profiles and dose records. If the shift remains within comparable time and design strata, it is a stronger reason to investigate dose-related model structure or residual-error assumptions; it remains a diagnostic finding rather than a formal test of a particular explanation.
A Short Justification for Comparing Standardized Residuals
Suppose, for illustration, an observation follows \(Y_{ij}=f_{ij}+\varepsilon_{ij}\), where \(f_{ij}\) is the correct prediction and the residual error has conditional mean zero and variance \(V_{ij}\), given the modeled information. Define a standardized residual \(z_{ij}=(Y_{ij}-f_{ij})/\sqrt{V_{ij}}\). Then:
Thus, under these assumptions, a systematic nonzero conditional mean across a dose or regimen category is not the expected pattern. Standardizing also puts residuals with different prediction variances on a more comparable scale. This argument does not imply that observed group averages must equal zero, nor that all standardized residuals are independent or exactly normally distributed. Estimated predictions and variance parameters, dependence within individuals, and unequal sampling can all affect the observed plot. A group mean is therefore a descriptive diagnostic, not automatically a test statistic with a valid standard error.
Worked Example: Regimen and Accumulation
Consider a hypothetical subject receiving the same dose every 12 hours. The model includes the repeated dosing history and predicts concentrations at four comparable post-dose times across successive intervals. The subject's CWRES, in chronological order, are \(-0.1,\ 0.2,\ 0.5,\ 0.8\).
The first-to-last change is \(0.8-(-0.1)=0.9\) standardized residual units. The sequence is increasing, and the mean is \((-0.1+0.2+0.5+0.8)/4=1.4/4=0.35\). In this hypothetical display, later observations tend to lie increasingly above the predictions. A possible concern is that the model underpredicts later concentrations as the dosing history accumulates.
That interpretation is justified only provisionally. Comparable post-dose times reduce one obvious timing difference, and the stated model includes the dosing history, but four observations from one subject do not establish a population-level trend. Confirm that actual doses and times match the recorded history, check whether other subjects show a similar sequence, and examine whether later intervals have different sampling or missingness. If the pattern recurs across subjects under comparable conditions, investigate whether the model represents accumulation and the relevant time course adequately.
Separate Dose Effects from Design Confounding
Dose and regimen are often linked to other features of a study. A high dose may be used only in a later study phase; one regimen may have more intensive early sampling; a dose cohort may consist of different participants. In such cases, a residual difference between groups cannot by itself isolate dose or regimen as the source. This is a diagnostic version of confounding: the comparison bundles the feature of interest with other systematically differing features.
A practical response is to compare like with like where the data allow it. Stratify or facet by sampling phase, time since dose, study period, or relevant design feature. Check whether dose levels overlap within those strata. If they do not, the available data may not support separating a dose pattern from the associated design difference. Report the limitation rather than treating the apparent dose trend as a confirmed dose effect.
Worked Example: Dose and Sampling Time Are Confounded
In a hypothetical data set, the low-dose group has four CWRES at early post-dose times: \(-0.2,\ 0.0,\ 0.1,\ 0.1\). The high-dose group has four CWRES at late post-dose times: \(0.6,\ 0.8,\ 0.9,\ 0.9\).
The low-dose mean is \((-0.2+0+0.1+0.1)/4=0.0\). The high-dose mean is \((0.6+0.8+0.9+0.9)/4=3.2/4=0.80\). The overall group difference is \(0.80-0.00=0.80\) standardized residual units.
However, dose and sampling phase do not overlap: all low-dose observations are early and all high-dose observations are late. The observed difference is therefore compatible with a dose-related shift, a time-related pattern, or both. These data alone cannot distinguish those explanations. The appropriate conclusion is that residuals differ between the observed dose-and-time combinations, not that the high dose caused underprediction. A more informative comparison would use overlapping dose levels at comparable times or another design that separates these features.
Practical Checks, Alternatives, and Limitations
Before interpreting a dose or regimen plot, confirm that dose units, dose amounts, administration times, and regimen labels are correct. Verify that the prediction corresponds to the recorded dosing history and that residuals are defined consistently. Then check how observations are distributed across dose, regimen, time since dose, and individuals. A pattern driven by one influential subject or by a handful of observations deserves a different interpretation from a broad pattern recurring across many subjects.
Useful displays include residuals against continuous dose, boxplots or group summaries by dose category, and residuals against time or dosing occasion with regimen labels. Each answers a slightly different question. A continuous-dose display can reveal a gradient hidden by arbitrary categories; categories can make a nonlinear or threshold-like pattern easier to inspect. Neither display corrects for confounding automatically. Subject-level summaries can complement observation-level plots, especially when some individuals contribute many more measurements than others.
| Approach | Useful for | Main caution |
|---|---|---|
| Residuals by dose category | Quickly showing shifts across prespecified dose groups | Group boundaries can hide gradients, and groups may differ in design |
| Residuals against continuous dose | Looking for a gradual or nonlinear dose-related pattern | Overlapping observations from the same individuals are not independent evidence |
| Residuals by regimen or dosing occasion | Checking for structure related to dose interval or accumulation history | Time since dose and sampling schedules may differ across regimens |
| Stratified or faceted comparisons | Checking whether a pattern persists at comparable sampling phases or designs | Small or empty strata limit what can be concluded |
These plots are primarily diagnostic and exploratory. A pattern can motivate checking the structural model, dose history, or residual-error model, but the plot alone does not identify which component needs revision. Conversely, a lack of visible structure does not prove adequacy, particularly with sparse observations or few subjects per regimen. The earlier material on residual spread and autocorrelation also matters: dependence and unequal variance can make simple group summaries misleading.
Common Mistakes
- Comparing raw concentrations as if dose groups should look alike. The correction is to compare observations with the predictions appropriate to each dose, using residuals suited to the question.
- Calling every group shift a model failure. A shift may reflect sampling-time imbalance, different participants, or data-recording issues. Check these before attributing it to model structure.
- Ignoring regimen history. Repeated doses and dose timing affect predictions. Confirm that the modeled history matches the administered history before interpreting accumulation-related patterns.
- Counting every observation as independent. Repeated residuals from one individual share information and may be correlated. Show subject contributions and avoid interpreting a large observation count as an equally large number of independent subjects.
- Treating a flat plot as proof of fit. Sparse or unbalanced data may hide systematic structure. State what the design can and cannot reveal.
Check Your Understanding
Answer each question, then compare with the solutions.
- A high-dose group has a positive mean CWRES. What does the sign suggest, and what does it not establish?
- Why can a rising residual sequence across repeated-dose intervals be relevant to accumulation, and name one alternative explanation to check.
- In a data set, all low-dose observations are early after dose and all high-dose observations are late after dose. Can a dose-specific residual effect be identified from this comparison alone?
- Why might four observations from one individual provide less independent evidence than four observations from four individuals?
Solutions. (1) Positive observed-minus-predicted residuals suggest observations tend to exceed predictions, or underprediction, in that group. The pattern alone does not establish that dose caused the discrepancy or that a particular model component is wrong. (2) Later positive residuals may indicate increasing underprediction as concentrations accumulate, if predictions and dosing history are correct. Check, for example, whether actual dose times were recorded accurately or whether sampling times differ across intervals. (3) No. Dose and time since dose are confounded in that comparison, so their contributions cannot be separated from those data alone. (4) Repeated observations from one individual may be correlated and share the same individual-specific characteristics; they do not generally provide as much independent information as observations spread across individuals.