Why Repeat the Reference?
A conventional two-period crossover gives each participant one administration of the reference formulation and one of the test formulation. It can compare their average responses, but it does not provide two reference responses from the same participant. Consequently, that design alone cannot reveal how much a participant’s response to the reference varies from one administration to another. Replicate designs add repeated administrations so that within-participant variation can be separated from differences between participants.
The key information comes from a within-participant contrast: compare the same participant’s responses after two separate administrations of the reference. On the log scale, a model for a reference response makes clear what the contrast measures and what assumptions are required.
Model and Assumptions
For reference administrations, use the additive model
Here, \(\mu_R\) is the overall reference mean on the log scale, \(b_i\) is participant \(i\)’s stable deviation from that mean, and \(\pi_j\) is the fixed effect of period \(j\). The within-reference deviation \(\varepsilon_{ij}\) has mean zero and variance \(\sigma_{WR}^{2}\). For the simple contrast derivation, assume the within-reference deviations are independent across separate administrations and participants, and have the same variance. The participant effect is stable across periods, and period effects act additively.
The model also requires that the responses represent comparable reference administrations. In particular, relevant carryover must be absent or adequately controlled, and a treatment-by-period pattern must not make the reference response’s expected value depart from the additive model. The endpoint must be defined consistently, such as log AUC measured using the same analysis rules in each period. The variability captured by \(\sigma_{WR}^{2}\) includes real within-participant response variation and, when applicable, measurement error in the recorded endpoint.
A repeat must mean a separate administration of the reference in a different treatment period. Multiple blood samples after one dose help characterize that dose’s concentration-time profile; they are not multiple reference administrations and do not form the contrast used here.
From Two Administrations to a Variance Estimate
Suppose participant \(i\) receives the reference in periods \(a\) and \(b\). Subtract the two model responses, correcting for the period effects:
Substituting the model shows why this contrast is informative. Both the overall reference mean and the participant effect cancel:
Because the two deviations are independent, each with variance \(\sigma_{WR}^{2}\), the variance of their difference is
Thus the sample variance of the adjusted participant-level contrasts estimates twice the within-reference variance. If \(m\) participants each provide a usable pair, and \(s_D^2\) is the sample variance of their adjusted contrasts, the basic estimate is
When everyone’s two reference administrations occur in the same pair of periods, their period-effect difference is the same constant. A constant shifts every raw contrast by the same amount and does not change its sample variance, so \(s_D^2\) can be calculated from the unadjusted differences in that case. If participants have reference administrations in different period pairs, period effects may shift their contrasts by different amounts. Period adjustment or a model that accounts for period is then needed; simply pooling raw differences can mistake period variation for within-reference variation.
The calculation estimates variability on the log scale. Under a lognormal interpretation of the original-scale response, a corresponding within-participant coefficient of variation is
This conversion depends on the lognormal interpretation; it is not a new estimate of the arithmetic-scale variance. For small log-scale standard deviations, \(CV_{WR}\) is approximately \(100\sigma_{WR}\%\), but the exponential formula gives the exact conversion under that interpretation.
Worked Example: Estimating Variability From Six Contrasts
In a hypothetical replicate crossover study, six participants each receive the reference in the same two periods. Let \(D_i\) be the second log response minus the first. Because the period pair is the same for everyone, any common period difference shifts all six contrasts equally and does not affect their sample variance. Suppose the contrasts, in log units, are \(-0.08,\ 0.02,\ 0.05,\ -0.01,\ 0.04,\ -0.02\).
Their mean is zero. The sample variance of the contrasts is
Divide by two to estimate the within-reference variance, then take the square root for the within-reference standard deviation:
The estimated standard deviation is 0.03376 log units. Under the lognormal interpretation, the corresponding estimate of the within-reference coefficient of variation is
This is an estimate of within-participant variation between reference administrations, not the variation among all six participants’ reference responses. The contrast removes stable participant differences, which is why the two sources of variation are not interchangeable.
Period Adjustment and Model-Based Estimation
Consider a second hypothetical data set in which four participants receive the reference in periods 1 and 2. Suppose the estimated period effects are \(\pi_1=0\) and \(\pi_2=0.10\) log units, and the observed log responses are as follows.
| Participant | Period 1 response | Period 2 response | Adjusted contrast \(D_i=(Y_{i2}-\pi_2)-(Y_{i1}-\pi_1)\) |
|---|---|---|---|
| 1 | 1.00 | 1.12 | 0.02 |
| 2 | 0.90 | 1.02 | 0.02 |
| 3 | 1.10 | 1.18 | -0.02 |
| 4 | 1.00 | 1.11 | 0.01 |
For example, participant 1’s adjusted contrast is \((1.12-0.10)-(1.00-0)=0.02\) log units. The four adjusted contrasts have mean \(0.0075\). Their sample variance is
The contrast-based estimate is therefore \(\widehat{\sigma}_{WR}^{2}=0.0003583/2=0.0001792\) log-squared units, and \(\widehat{\sigma}_{WR}=\sqrt{0.0001792}=0.01339\) log units. The adjustment matters when period pairs differ across participants or the analysis otherwise uses contrasts with different period expectations. In an actual analysis, period effects are generally estimated from the study data. A mixed-effects or appropriate analysis-of-variance model can estimate the within-reference variance while accounting for the design; its uncertainty calculation must also account for the fitted model rather than treating estimated period effects as known.
What Replication Is Required?
To obtain a direct within-reference contrast for a participant, that participant must receive the reference at least twice, with an endpoint available for each administration. Having many participants each receive the reference only once does not replace this replication: differences between their responses combine stable between-participant differences with within-participant variation. Nor does a participant receiving the test once and the reference once provide two reference responses.
At the design level, there must be enough participants and sequence-period coverage to support the planned model, and the reference repeats must occur in periods that permit period effects to be distinguished from the within-reference contrast. The contrast calculation also relies on a reference response being observed twice for the participant; a withdrawal or missing endpoint can remove that participant’s direct pair. Some designs can estimate reference variability using a model that combines information across sequences, but the existence and interpretation of that estimate depend on the design’s estimability. The separate question of estimability in partial replicates requires its own design analysis.
Worked Example: Checking Whether a Schedule Provides Reference Replication
A hypothetical study assigns 12 participants equally to the three-period sequences R-T-R and T-R-R. Here R denotes reference and T denotes test. Each participant assigned to either sequence receives R twice.
There are 6 participants in each sequence, so the study plans \(12\times2=24\) reference administrations. More importantly, there are 12 participants with a pair of reference responses, and therefore up to 12 within-participant reference contrasts if all planned endpoints are observed. The sequences place the reference repeats in different period pairs, so a period-aware analysis is needed rather than treating every raw difference as if it had the same period expectation.
If instead 12 participants each received R once and T once, there would be 12 reference administrations but no participant would have two reference responses. That schedule could not provide the direct within-reference contrasts derived above. The count of reference administrations across the whole study is not enough; the allocation of repeats within participants is essential.
Diagnostics, Alternatives, and Limitations
The contrast is useful only if it behaves like a difference of two comparable within-reference deviations. Inspect the participant-level contrasts and the fitted model residuals for unusual values, strong skewness, or patterns by sequence and period. Investigate whether large contrasts coincide with missing or questionable endpoint measurements, predose concentrations that suggest carryover, or protocol deviations. A consistent shift in contrasts across period pairs can indicate inadequate period adjustment. Strongly different contrast spreads across sequences or period pairs can call the common-variance assumption into question.
A paired-difference calculation is transparent and useful when the pairing and period structure are simple. A model-based estimate is usually preferable when the design has several sequences, unequal numbers of observations, or period effects that must be estimated. Both approaches depend on the design and assumptions; using a more complex model does not make confounding, carryover, or inadequate replication disappear.
The estimate also has limits. With few complete pairs, the sample variance is imprecise and can be sensitive to a single contrast. If within-reference variation changes across periods or participants, one common \(\sigma_{WR}^{2}\) may summarize the data poorly. Further, a within-reference estimate describes repeatability under the study’s conditions and endpoint definition; it does not automatically describe every population, setting, or formulation.
- Mistake: Divide the variance of paired differences by one and call it the within-reference variance. Correction: Under independent, equal-variance administration deviations, the difference variance is \(2\sigma_{WR}^{2}\), so divide \(s_D^2\) by two.
- Mistake: Treat repeated blood samples after one dose as reference replication. Correction: The contrast requires separate reference administrations, not multiple measurements within one administration.
- Mistake: Attribute every difference between participants’ reference responses to within-reference variability. Correction: A between-participant comparison includes stable participant differences; repeated reference responses within the same participant are what remove that component in the contrast.
Check Your Understanding
Answer each question, then check the solutions below.
- Under the stated model, why does subtracting two period-adjusted reference responses remove the participant effect?
- If the sample variance of adjusted paired contrasts is \(0.0060\) log-squared units, what are the estimated within-reference variance and standard deviation?
- Can a study in which every participant receives the reference once and the test once directly estimate within-reference variability from a reference pair? Explain.
- Why might raw reference differences be misleading if participants’ reference administrations occur in different period pairs?
- Do three blood samples collected after one reference dose count as three reference administrations for this contrast? Explain.
Solutions. (1) The participant effect \(b_i\) appears in both responses and cancels in their difference. (2) The variance estimate is \(0.0060/2=0.0030\) log-squared units; the standard deviation estimate is \(\sqrt{0.0030}=0.05477\) log units. (3) No. Each participant has only one reference response, so there is no within-participant reference contrast. (4) Different period pairs can have different period-effect differences; unadjusted contrasts may then include period variation as well as within-reference variation. (5) No. Those are repeated measurements after one administration, not separate reference administrations in different periods.