Why the Scale of Analysis Matters
AUC and Cmax are positive pharmacokinetic measurements that can vary substantially between subjects. In many plausible settings, variability is more naturally described proportionally: a subject's exposure may differ from a typical value by, for example, about 10%, rather than by a fixed number of concentration-time units. Such data can be right-skewed, with a few large values. On the raw scale, those large observations can dominate averages and treatment contrasts.
The scale determines what a treatment effect means. A raw-scale analysis describes an arithmetic difference, such as a change of 20 AUC units. A log-scale analysis describes a difference in mean log exposure, which can be expressed as a ratio of geometric means. Neither estimand is automatically correct for every scientific question. The concern is that raw-scale ANOVA can be a poor fit when the treatment effect and variability are primarily multiplicative.
What Raw-Scale ANOVA Assumes
In a balanced two-treatment crossover, a simplified raw-scale model for subject \(i\) and treatment \(j\) is
where \(Y_{ij}\) is the observed AUC or Cmax, \(\mu\) is an overall mean, \(S_i\) represents a subject effect, \(\tau_j\) is the treatment effect, and \(\varepsilon_{ij}\) is the residual. For the usual ANOVA interpretation and its standard tests to be appropriate, the model must describe the relevant mean structure, and residuals should be independent across subjects, approximately normal for the inferential procedure, and have a common variance across the modeled treatment conditions. The crossover design also requires that the specified model adequately account for the design features included in the analysis.
The treatment effect in this model is additive: \(\tau_T-\tau_R\) is a difference in response units. The constant-variance assumption is also on the original scale. If the typical residual size grows with the fitted exposure, the raw-scale residuals may instead fan out as fitted values increase. If a small number of subjects have very large exposures, raw arithmetic means and differences can be especially sensitive to them.
A proportional model describes a different structure. For positive responses, one possible representation is
where \(M_{ij}\) is the systematic level of exposure. If log-scale residuals are approximately independent, normal, and of common variance, the log model represents multiplicative variation as additive residual variation. This is a modeling possibility to assess, not a guarantee that logging will fix every departure from assumptions.
Worked Example: A Raw Difference Can Reverse the Ratio Story
Consider six hypothetical subjects in a crossover study. The values below are hypothetical AUC measurements in the same units for both treatments. For each subject, the treatment-to-reference ratio is also shown.
| Subject | Reference AUC | Treatment AUC | Treatment/reference ratio |
|---|---|---|---|
| 1 | 100 | 110 | 1.10 |
| 2 | 200 | 180 | 0.90 |
| 3 | 400 | 480 | 1.20 |
| 4 | 800 | 800 | 1.00 |
| 5 | 1,600 | 2,080 | 1.30 |
| 6 | 3,200 | 2,560 | 0.80 |
The arithmetic treatment mean is \(6,210/6=1,035\) AUC units, while the reference mean is \(6,300/6=1,050\) AUC units. Thus the raw treatment-minus-reference contrast is
Yet the arithmetic mean of the six within-subject ratios is \((1.10+0.90+1.20+1.00+1.30+0.80)/6=1.05\), or 5% above 1. The geometric mean ratio, corresponding to the mean paired log ratio, is
The raw mean difference is negative, while both summaries of the ratios exceed 1. These are not contradictory calculations of the same estimand: one is an arithmetic difference and the others summarize ratios. The high-exposure sixth subject contributes a difference of \(-640\) AUC units, which strongly influences the raw contrast. The example illustrates why the scientific question and the plausible error structure must guide scale choice.
Why Multiplicative Variation Creates Raw-Scale Patterns
A short derivation shows why a constant spread on the log scale need not produce a constant spread on the raw scale. Suppose \(Y=M\exp(\varepsilon)\), where \(\varepsilon\sim N(0,\sigma^2)\). The lognormal mean and variance are
For fixed \(\sigma^2\), the raw-scale variance is proportional to \(M^2\). Therefore, as the typical exposure \(M\) increases, raw-scale residual scatter increases even though the log-scale residual variance stays constant. This produces a funnel-like residual pattern on the raw scale. Log transformation removes the multiplicative factor in this model because \(\log(Y)=\log(M)+\varepsilon\); it does not remove other sources of misspecification or guarantee normal residuals.
Worked Example: Comparing Residual Spread at Two Exposure Levels
Suppose two hypothetical subjects each have a reference AUC and a treatment AUC that is 10% higher. Their values are 100 and 110 for the first subject, and 1,000 and 1,100 for the second. Consider the within-subject midpoint as the fitted level for each pair to illustrate residual size around that level.
On the raw scale, the first pair has midpoint 105 and deviations \(-5\) and \(+5\) AUC units. The second has midpoint 1,050 and deviations \(-50\) and \(+50\) AUC units. The second pair's residual magnitudes are ten times larger, although the relative difference is the same.
On the log scale, the two observations in either pair lie symmetrically around the mean of their logs. The deviation magnitude for either observation is
This value is the same for both pairs because multiplying both measurements by 10 adds \(\log(10)\) to each log observation without changing their centered deviations. In this constructed illustration the log-scale spread is constant while the raw-scale spread grows with exposure. The calculation supports checking residuals on both scales; it does not establish that real data follow a lognormal model.
Comparing the Two Analyses in a Crossover
In a two-period crossover with one observation per treatment per subject, the treatment comparison can be expressed through each subject's paired difference or paired log ratio. For subject \(i\), define
A raw-scale analysis summarizes the \(D_i\) values in response units; a log-scale analysis summarizes the \(L_i\) values and exponentiates their mean to obtain a geometric mean ratio. The paired structure matters: treatment comparisons are made within subjects, rather than treating the two treatment groups as independent. More complex crossover designs require a model appropriate to their periods, sequences, and covariance structure.
Worked Example: Translate a Mean Log Ratio to a Relative Effect
Suppose, hypothetically, the mean of the paired log AUC ratios in a small crossover dataset is \(0.1823\). The corresponding geometric mean ratio is
Thus the estimated geometric mean AUC under treatment is 1.200 times the reference geometric mean, or 20.0% higher. This is a relative effect, not a 20.0-unit increase. To recover the implied log-scale contrast, \(\log(1.200)\approx0.1823\). This interpretation concerns the fitted treatment contrast; it does not by itself establish that the model assumptions hold or answer a separate equivalence question.
Diagnostics, Alternatives, and Limitations
A decision about scale should be supported by the data and the estimand. Examine residuals against fitted values and treatment, and inspect their distribution, using the principles of residual diagnostics for log-scale crossover models and assessment of residual normality. On the raw scale, increasing residual spread with fitted exposure, pronounced right skew, or a few observations dominating the treatment contrast are warnings that an additive, constant-variance model may be inadequate. Then inspect the corresponding log-scale residuals: a more even spread and less skew support, but do not prove, the proportional model.
The main alternatives answer different questions. Raw-scale ANOVA is appropriate when an additive difference in original units is the target and its assumptions are credible. Log-scale ANOVA is often more coherent when positive responses vary proportionally and a ratio is the target. Other models may be needed when neither model captures the mean and variance structure. A transformation is not a device for making an inconvenient result disappear: the chosen scale, response, and model should be justified before interpreting the treatment effect.
Log analysis also has limits. It requires positive observations, and a log-scale normal model can still have influential subjects, unequal residual variances, or other misspecification. Further, exponentiating a mean log response gives a geometric mean, not generally the arithmetic mean on the raw scale. If the scientific target is an arithmetic mean difference, a log-scale contrast does not directly estimate that target.
A Practical Scale-Choice Workflow
Decide whether the scientific question concerns a difference in AUC or Cmax units or a relative treatment effect.
Check positivity, skew, the influence of large observations, and whether spread increases with the fitted raw response.
Compare residual behavior on the candidate scale while accounting for the crossover design; do not rely on the appearance of the raw data alone.
Give an additive contrast in response units or exponentiate a log contrast and report a ratio, with interpretation appropriate to the model.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A raw-scale treatment contrast is 25 AUC units. What type of effect does it describe?
- Under a multiplicative model with constant log-scale variance, why might raw residuals fan out as fitted exposure increases?
- For a mean paired log ratio of \(-0.1054\), calculate and interpret the geometric mean ratio.
- If raw-scale residual spread increases with fitted values but log-scale residual spread is more even, what does that suggest, and what does it not prove?
Solutions. (1) It is an arithmetic treatment difference of 25 AUC units, not a ratio. (2) The raw-scale variance is proportional to the square of the typical exposure under the stated lognormal model, so larger fitted exposures have greater raw-scale scatter. (3) \(\exp(-0.1054)=0.9000\); the treatment geometric mean is 0.9000 times the reference geometric mean, or 10.0% lower. (4) The pattern supports considering a multiplicative model because proportional variability may be more plausible; it does not prove lognormality, validate every model assumption, or establish that the log scale matches the scientific target.