Why an Unusual Exposure Profile Deserves Investigation
A subject with an unusually high AUC or Cmax is not automatically a subject who distorts the bioequivalence comparison. In a crossover study, the treatment effect is informed by how exposure differs within the same subject between test and reference. A subject whose exposure is high under both treatments may have an ordinary treatment contrast. Conversely, a subject with values that are not striking in either treatment alone may have an unusual test-to-reference contrast and strongly affect the estimated treatment effect.
The practical task is therefore twofold: identify profiles or treatment contrasts that merit review, then determine whether the reason is a documented data problem, a design or conduct issue, or genuine variability. Diagnostics flag questions; they do not answer them, and they do not by themselves justify removing a subject.
What the Crossover Model Uses
For AUC or Cmax, the usual analysis works on the logarithmic scale. Let \(Y_{ijk}\) be the log-transformed exposure for subject \(j\) in sequence \(i\), in period \(k\). In a 2×2 crossover, sequence determines which treatment a subject receives in each period. A conventional model can be written as
Here \(\text{treatment}_{ik}\) denotes the treatment assigned to sequence \(i\) in period \(k\); it is not determined by period alone. Sequence, period, and treatment are fixed effects, while subject nested within sequence is commonly modeled as a random effect. Under the usual model assumptions, subject effects and residual errors have mean zero, the subject effects are independent across subjects, residual errors are independent with a common variance, and the two sources of variation are independent. The model also assumes that the log-scale treatment effect is represented by a common effect and that the design and data support estimation of the specified effects. These assumptions are a working model, not proof that all individual profiles should look alike.
The treatment contrast is reported on the log scale and commonly exponentiated to obtain a geometric mean ratio. Subject-level review should therefore consider both the exposure levels and the treatment contrast. A useful plot displays each subject’s test and reference log exposure as a connected pair; a separate plot can show the paired log contrast by subject and sequence. Plotting concentration against time can reveal unusual profile shapes that a single AUC or Cmax value hides. These displays complement, rather than replace, the model.
Worked Example: High Exposures but an Ordinary Contrast
Suppose a hypothetical subject has AUC values of 1,520 mg·h/L for test and 1,460 mg·h/L for reference. Both values appear high compared with most subjects in the study. Calculate the subject’s test-to-reference ratio and log contrast.
Ratio. Use the unrounded AUC values:
Thus the test AUC is about 4.11% higher than the reference AUC for this subject. The corresponding log contrast is
The two absolute AUC values may merit a profile and data review, but their paired contrast is modest. If most subjects’ log contrasts are near 0.04, this subject may have little influence on the estimated treatment effect despite high exposure levels. If the other contrasts are near zero, the same subject could matter more. Influence is assessed relative to the whole pattern and fitted analysis, not from the raw values alone.
Influence Diagnostics and Their Interpretation
Begin with plots that preserve the crossover pairing. Plot each subject’s log AUC or log Cmax under both treatments and connect the observations. Then plot each subject’s test-minus-reference log contrast, retaining sequence labels. Review concentration-time profiles when a summary measure appears extreme or when its contrast differs sharply from the others. Sequence labels matter because an apparent contrast pattern could also reflect period structure; the earlier material on interpreting sequence and period patterns provides that context.
A useful model-based diagnostic is a subject-deletion comparison: fit the prespecified model to all evaluable subjects, refit it after removing one subject at a time, and record the change in treatment estimate, its standard error, and confidence interval. This is a diagnostic exercise, not a replacement primary analysis. In a balanced 2×2 study, deleting a subject also changes the information available and may unbalance the sequences, so changes in uncertainty and model estimates must be interpreted in light of the design.
In a simple analysis of paired contrasts, the effect of removing one contrast can be shown directly. If \(d_i\) is subject \(i\)’s test-minus-reference log contrast and \(\bar d\) is the mean of \(n\) contrasts, then
The first identity follows because the sum of all contrasts is \(n\bar d\), so the sum after removing \(d_i\) is \(n\bar d-d_i\). Dividing by the remaining \(n-1\) contrasts gives the leave-one-out mean. The second identity follows by subtracting the first expression from \(\bar d\). Thus, a contrast far from the mean has more potential to shift the estimate, especially when the number of subjects is small. In a fitted crossover model, the treatment estimate is not necessarily the unadjusted mean of the contrasts, so refitting the model is the relevant diagnostic.
Regression diagnostics such as studentized residuals, leverage, and Cook’s distance can be useful in models for which their definitions and assumptions apply. They should not be transferred mechanically to a repeated-measures crossover model: the two records from one subject are not independent subjects, and deleting one record is not the same as deleting that subject. For a mixed model, use a method appropriate to that model or directly compare subject-level deletion refits. Report what quantity was compared and how it was calculated; “influential” without a specified result or diagnostic criterion is incomplete.
Worked Example: A Contrast That Shifts the Estimate
Consider eight hypothetical subject-level log AUC contrasts, defined as test minus reference:
The first seven contrasts sum to 0.04, so the total is \(0.04+0.35=0.39\). The mean contrast using all eight subjects is
Exponentiating gives a geometric mean ratio of
The eighth contrast is far above the others. As a diagnostic, calculate the mean after omitting it:
The estimate changes substantially in this simple paired-contrast calculation. That indicates sensitivity to this subject and warrants checking the source data, profile, and study conduct. It does not establish that the subject should be excluded. The primary model should remain the prespecified analysis unless a documented, scientifically valid data-handling rule applies. For an actual crossover analysis, one would refit that model with and without the subject as a diagnostic and report changes in both the estimate and uncertainty.
Investigating a Flagged Subject
A flagged profile should prompt a structured review, not a search for a convenient reason to discard it. Check subject and treatment identifiers, period assignment, units, dose and sample times, concentration records, calculations of AUC or Cmax, and the audit trail. Review assay flags and any documented sampling or administration issues. Earlier material on data-integrity diagnostics and incomplete concentration profiles gives relevant context for verifying records; an unusual value by itself is not evidence that a record is wrong.
If the source data establish a transcription or processing error, correct it under documented procedures and recalculate the affected exposure measure. If the records are valid and the profile is biologically or pharmacokinetically unusual, retain it in the analysis specified by the protocol and statistical analysis plan unless a prespecified rule says otherwise. Distinguish this situation from a protocol deviation or an eligibility decision, which requires its own documented basis. An extreme response should not become an exclusion criterion after the results are known.
Worked Example: Verifying an Unusual Concentration
In a hypothetical profile, concentrations are recorded as 0 mg/L at time 0, 120 mg/L at 2 hours, 1,200 mg/L at 4 hours, and 80 mg/L at 8 hours. The 4-hour value produces a conspicuous peak. Suppose the source record and assay documentation are checked.
If the source record verifies 120 mg/L. The entered 1,200 mg/L is a documented transcription error. Using the linear trapezoidal rule for these adjacent intervals, the contribution from 2 to 8 hours with the entered value is
With the verified value of 120 mg/L, the same contribution is
The difference is \(3880-640=3240\) mg·h/L. The correction is justified by the source record, not by the size of its effect. The full AUC must be recalculated using the corrected input and the prespecified method.
If the source record instead verifies 1,200 mg/L and no data-quality issue is identified, the value should not be changed simply because it looks surprising or shifts the result. Retain the valid observation in the prespecified analysis and describe the diagnostic review and any sensitivity comparison transparently.
Comparing Approaches Without Automatic Exclusion
Different tools answer different questions. Subject-level plots reveal patterns and possible recording problems; they do not quantify the effect on the treatment estimate. Model-based subject-deletion refits quantify sensitivity of a specified result; they do not establish that the removed subject is invalid. A paired-contrast summary is intuitive in a balanced crossover but may not reproduce the adjusted estimate from the full model. Robust estimators can reduce the effect of extreme contrasts, but they target a different analysis and should not be substituted after seeing the results without a prespecified rationale.
A defensible report identifies the primary model, describes the flagged exposure or contrast, gives the data and conduct review outcome, and presents any planned diagnostic comparison with its effect on the estimate and interval. The primary and diagnostic analyses should be clearly distinguished. A changed conclusion after deletion is evidence that the result is sensitive to that deletion; it is not evidence that deletion is the correct analysis.
| Approach | What it helps assess | Important limitation |
|---|---|---|
| Paired exposure and contrast plots | Whether high values occur under both treatments or the treatment contrast is unusual | Visual flags depend on scale and do not establish influence or error |
| Subject-level model refits | How the fitted treatment estimate and uncertainty change when a subject is omitted diagnostically | Does not justify exclusion; deletion can alter sequence balance and precision |
| Prespecified primary model with all valid observations | The planned treatment comparison under the stated analysis assumptions | May be sensitive to genuine extreme responses or model misspecification |
| Robust alternative analysis | Whether conclusions depend on sensitivity to extreme contrasts under another estimator | May target a different quantity and should not be chosen post hoc as a replacement |
Common Pitfalls and Limits
- Equating a high AUC with influence. A high test value can be accompanied by a similarly high reference value. Inspect the paired contrast and its effect on the model estimate.
- Calling every large residual a data error. A residual is a model-relative discrepancy. Verify records and study conduct before drawing conclusions about its cause.
- Deleting the subject who changes the conclusion. This selects the analysis based on its result and can inflate the risk of a misleading conclusion. Use the prespecified analysis and label deletion refits as diagnostics unless a valid documented rule applies.
- Using an observation-level diagnostic for subject-level influence. Crossover records are clustered within subjects. The unit of deletion must match the question: remove the subject’s relevant records together when assessing subject influence.
Diagnostics also have limits. Small studies provide unstable estimates of how unusual a subject is, and plots cannot prove that a profile is erroneous or representative. A single influential subject can reveal genuine between-subject variability or a fragile estimate, but neither conclusion alone resolves the biological cause. The interpretation must combine the model, verified records, design, and prespecified analysis plan.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A subject has high AUC under both test and reference, but a test-to-reference ratio close to the group’s typical ratio. Does the high AUC alone establish that the subject is influential?
- In a paired-contrast analysis with \(n=10\), the mean log contrast is 0.02 and one subject’s contrast is 0.20. Calculate the mean after omitting that subject as a diagnostic.
- What is the appropriate next step if an unusual concentration is confirmed by the source record but its profile is still extreme?
- Why should deletion diagnostics remove a subject’s crossover records together rather than remove only the treatment observation that appears unusual?
Solutions. (1) No. The paired treatment contrast and its effect on the fitted result matter; similar proportional changes under both treatments may yield an ordinary contrast. (2) The total contrast is \(10(0.02)=0.20\). After removing 0.20, the remaining sum is zero, so the diagnostic mean is \(0/9=0\). This sensitivity does not itself justify exclusion. (3) Retain the valid value in the prespecified analysis, document the review, and report any planned diagnostic comparison without treating extremeness as proof of error. (4) Both records belong to one randomized subject and are correlated within that subject; removing only one breaks the paired comparison and no longer assesses subject-level influence.