Estimability of Reference Variability in Partial Replicates

Why Replication Makes a Difference

A partial-replicate crossover gives each participant more than one administration of the reference formulation, but does not repeat every formulation. That repeated reference exposure has a specific statistical value: it lets us compare two reference responses from the same participant. The participant’s stable response level can then cancel from the comparison, leaving information about within-reference variation.

The key is not simply that the reference appears more than once somewhere in the study. It must appear more than once within the same participant. A group receiving reference once in one period and a different group receiving it once in another period does not provide the same within-participant contrast. The repeated administration is what makes a direct estimate possible.

Responses are commonly analysed on a logarithmic scale in bioavailability studies. In the examples below, \(Y_{ij}\) denotes participant \(i\)’s response in period \(j\), on that analysis scale. A difference is therefore in log units, and its variance is in squared log units. The same argument applies to another analysis scale if the model assumptions are appropriate for it.

Model and the Repeated-Reference Contrast

Consider the following model for a response in period \(j\):

Here \(\mu\) is an overall mean, \(d_{ij}\) identifies the formulation given to participant \(i\) in period \(j\), \(\tau_{d_{ij}}\) is its formulation effect, and \(\pi_j\) is the period effect. The participant effect \(b_i\) represents a stable difference in response between participants. The residual \(\varepsilon_{ij}\) represents within-participant variation not accounted for by those terms.

For direct estimation from repeated-reference contrasts, assume that \(b_i\) has mean zero and variance \(\sigma_b^2\); the residuals have mean zero and common reference-period variance \(\sigma_{eR}^2\) when the reference is given; and residuals from separate periods are independent of each other and of \(b_i\). The reference formulation effect is the same at both administrations. Period effects may differ, but their differences must be accounted for or held constant within the contrasts being analysed. These assumptions are important: if the residual variance differs by period, or the two residuals are correlated, the simple factor-of-two result need not hold.

Suppose participant \(i\) receives the reference in periods \(j\) and \(k\). Subtract the response in period \(k\) from the response in period \(j\):

The reference formulation effect and the participant effect cancel. The period difference remains, so the contrast is not necessarily centred at zero. But if \(\pi_j-\pi_k\) is common to participants with that same pair of periods, it shifts all their contrasts by the same amount and does not change their variance. Independence and equal residual variance give the following justification:

Consequently, the variance of repeated-reference contrasts divided by two estimates \(\sigma_{eR}^2\). Taking its square root estimates the within-reference residual standard deviation, \(\sigma_{eR}\). This estimate concerns variation on the response’s analysis scale; it is not the between-participant variance \(\sigma_b^2\).

Different Period Pairs Need Separate Treatment

Partial-replicate schedules can place the two reference administrations in different period pairs for different participants. For example, reference may occur in periods 1 and 2 for one sequence, in periods 1 and 3 for another, and in periods 2 and 3 for a third. Those contrasts can have different means because the corresponding period differences are different. Pooling their raw values as if they had a common mean can mistake period-related differences for within-reference variability.

A straightforward variance calculation first groups contrasts by their period pair (or otherwise adjusts for the relevant period effects). Within each group, subtract that group’s mean contrast. If there is a common residual variance and the assumptions above hold, the pooled within-group variance estimates \(2\sigma_{eR}^2\):

Here \(g\) indexes groups with the same reference-period pair, \(n_g\) is the number of participants in group \(g\), \(D_{ig}\) is a participant’s contrast, and \(\bar{D}_g\) is that group’s mean. This calculation requires enough participants within the groups to estimate their contrast variances. A fitted model can also adjust for period effects, but its estimate still relies on a defensible residual covariance structure.

What Replication Does—and Does Not—Establish

The defining design feature is a participant-level repeated reference administration. With just one reference response per participant, there is no reference-minus-reference contrast for that participant. The reference observations then combine stable participant differences and response variation. A model may sometimes separate variance components using other covariance information, but that inference depends on additional assumptions; it is not the direct repeated-reference estimate derived above.

For example, under the random-intercept model, after accounting for formulation and period means, the reference response is \(Y_{iR}=\mu+\tau_R+\pi_{j(i)}+b_i+\varepsilon_{iR}\), where \(j(i)\) is the period in which participant \(i\) receives reference. Its variance alone combines \(\sigma_b^2\) and \(\sigma_{eR}^2\). In a nonreplicated design, there is no second reference response for the same participant to subtract and cancel \(b_i\). This is why omitting the period term would be inappropriate: participants may receive reference in different periods, and their reference means can differ accordingly.

Diagnostics, Alternatives, and Limitations

Before interpreting a variance estimate, check whether the design and data support the assumptions used to obtain it. In particular:

• Confirm replication within participants. Count the reference administrations for each participant; repeated reference observations across different participants do not form paired contrasts.
• Check period-pair structure. Record which periods are being subtracted. Contrasts from different pairs may have different means and should be adjusted or grouped accordingly.
• Inspect contrast distributions by pair. Unusual outliers or markedly different spreads can signal data errors, unequal variances, or a poor common-variance assumption. With small groups, such checks have limited power.
• Consider dependence and missing records. Correlated period residuals change the contrast variance to \(\sigma_{eR,j}^2+\sigma_{eR,k}^2-2\operatorname{Cov}(\varepsilon_{ij},\varepsilon_{ik})\). Missing one of a participant’s reference responses removes that participant’s direct paired contrast.

A matched-contrast calculation is transparent and makes the cancellation of participant effects explicit. A mixed-effects model can instead estimate variance components while accounting for period and sequence structure, and can accommodate unbalanced records under its assumptions. The approaches are not interchangeable by default: the model-based result depends on its specified covariance structure, while a simple pooled contrast variance depends on comparable residual variances and appropriate handling of contrast means.

A partial replicate with repeated reference but a single test administration directly informs within-reference variability, not a test-minus-test within-participant variance. Nor does replication remove all design concerns: period effects, carryover, missingness, and departures from the assumed residual structure can affect interpretation. If period-specific reference variances differ, the pairwise contrast variance estimates a sum of period-specific variances rather than twice one common variance. If residuals are correlated, the covariance term matters. These cases require a model that represents the actual structure; the simple divide-by-two calculation is not justified automatically.

A common mistake is to say, “There are several reference observations in the study, so their overall variance estimates within-reference variability.” That reasoning ignores whether the observations are repeated within participants and whether period means differ. The correction is to identify same-participant reference pairs, form contrasts, account for the relevant period effects, and then use the stated variance assumptions to interpret the contrast variance.