Estimating Reference Variability in Full Replicate Designs

Why Full Replication Identifies Reference Variability

Reference-scaled bioequivalence uses variability in repeated reference administrations, not the spread of all observed responses. A full replicate design gives each subject repeated administrations of both the test and reference formulations. The repeated reference measurements make it possible to isolate how much a subject’s reference response changes from one occasion to another.

The response is usually analyzed on the log scale. Let \(Y\) denote a subject’s log-transformed pharmacokinetic endpoint, such as log AUC or log Cmax. The variance of interest, \(\sigma_{WR}^{2}\), is the within-subject reference variance: the variance of occasion-to-occasion reference deviations under the specified model. It is expressed in squared log units. Its square root, \(\sigma_{WR}\), is the within-subject reference standard deviation on the log scale.

The factor of two matters: a difference contains occasion error from both reference administrations. The sign of the contrast is arbitrary for variance estimation, provided the convention is used consistently. The earlier material on reference-replicate contrasts establishes why a repeated-reference contrast carries information about within-subject reference variability; here the focus is how to estimate that variability in a full replicate design.

Model Terms and the Reference Contrast

A useful model for a complete full replicate crossover separates systematic effects, subject-level effects, and occasion-level error. Write the log response for subject \(i\), in sequence \(s\), period \(p\), and formulation \(f\) as

Here \(\mu\) is an overall mean; \(\tau_f\) is the formulation effect; \(\pi_p\) is the fixed period effect; and \(q_s\) is the fixed sequence effect. The term \(b_{i(s)}\) is a subject effect for subject \(i\) nested within sequence \(s\), while \(u_{if}\) represents a subject-by-formulation deviation. The occasion error is \(\varepsilon_{isp}\). For the direct contrast estimator, assume reference occasion errors are independent across the subject’s two reference periods, normally distributed, and have common variance \(\sigma_{WR}^{2}\). Fixed period and sequence effects must be accounted for; the model also assumes that the variance is common across sequences and subjects.

For the two reference observations from one subject, subtract the second from the first. The subject effect \(b_{i(s)}\) cancels because the same subject contributes both measurements. The subject-by-formulation term \(u_{iR}\) also cancels because both measurements use the reference formulation. The formulation effect cancels for the same reason. The fixed period contribution need not cancel, since the two reference administrations occur in different periods.

The period contrast can differ between sequences because their reference administrations occur in different periods. Denote its sequence-specific value by \(\delta_s\). Then \(C_{is}=\delta_s+e_{is}\), where \(e_{is}\) is the difference between the two reference occasion errors. Under the independence and equal-variance assumptions, \(E(e_{is})=0\) and \(\operatorname{Var}(e_{is})=2\sigma_{WR}^{2}\).

Deriving the Pooled Contrast Estimator

In a complete design, let \(n_s\) be the number of subjects in sequence \(s\), and let \(\bar{C}_s\) be the mean reference contrast in that sequence. Centering contrasts within sequence removes the sequence-specific mean contrast, including the systematic period difference. Define the pooled within-sequence sum of squares \(Q\) and its degrees of freedom \(\nu\) as follows.

For each sequence, the centered contrasts fluctuate around their common sequence mean. Under the model, each such contrast has variance \(2\sigma_{WR}^{2}\). The expected sum of squared deviations within sequence \(s\) is therefore \((n_s-1)2\sigma_{WR}^{2}\). Summing across sequences gives \(E(Q)=2\nu\sigma_{WR}^{2}\). Dividing by \(2\nu\) consequently produces an unbiased method-of-moments estimate:

This estimator uses the within-sequence variation, not the variation between sequence means. Under normal errors, \(Q/(2\sigma_{WR}^{2})\) has a chi-square distribution with \(\nu\) degrees of freedom when the sequence means are estimated from the data. That result describes the uncertainty in the estimate; it does not make the estimate itself a measure of uncertainty in the treatment effect. Estimation of degrees of freedom and variance components in more general models requires additional methods.

Worked Contrasts and Period Adjustment

The sequence means in the estimator are not optional if systematic contrast means differ by sequence. For example, in a four-period design, one sequence might receive R in periods 1 and 3, while another receives R in periods 2 and 4. A period effect can therefore shift the average R contrast differently across sequences. Centering within sequence removes a constant sequence-specific shift, but it does not correct for a period pattern that varies across subjects or is otherwise misspecified.

What the Estimate Does and Does Not Measure

This distinction is the practical reason to use repeated reference administrations. A subject may have consistently higher responses than another subject, yet show little occasion-to-occasion fluctuation. Conversely, subjects with similar average responses may have appreciable within-subject variation. The contrast targets the latter, not the former.

Assumptions, Diagnostics, and Alternatives

The direct contrast estimator is clearest for a complete, balanced full replicate design with two reference administrations per subject. Its interpretation depends on several conditions:

• Both reference observations are available for each included subject, and their period assignments are known.
• Systematic sequence and period effects are handled in the contrast means or in an equivalent model.
• Reference occasion errors are independent across the two administrations, have mean zero after adjustment, and share a common variance across subjects and sequences.
• The log-scale error distribution is reasonably compatible with the normal model if normal-based inferential results are used.

A contrast plot by sequence is a useful first check. Look for a sequence shift in the contrast means, unusually large contrasts, or a pattern in which contrast magnitude changes with period or subject characteristics. A strong period pattern suggests that a simple sequence-centered contrast is insufficient; the fixed-effects structure should represent the period pattern. Repeated measurements may also be correlated. If the two reference occasion errors have variance \(\sigma_{WR}^{2}\) and correlation \(\rho\), then the contrast variance is \(2\sigma_{WR}^{2}(1-\rho)\), not \(2\sigma_{WR}^{2}\). In that case, dividing by two does not estimate the marginal occasion variance as defined under independence.

For complete balanced data, the contrast calculation is transparent and directly shows which observations identify reference variability. A model-based analysis instead represents the individual period, sequence, subject, subject-by-formulation, and residual terms together; it can be more flexible when the design is unbalanced or when the fixed-effects structure is more involved. The two approaches should target the same reference occasion variance under compatible assumptions, but a model can estimate variance components differently. Later material treats those broader variance-component estimation procedures; they are not needed to understand the contrast estimator here.

The estimate also has limitations. With few subjects, it can be unstable, and a single unusually large contrast can materially increase the squared-deviation sum. A near-zero estimate in a small sample is not proof that reference variability is absent. Missing reference periods prevent the simple paired calculation for the affected subject, and replacing a missing value with an imputed response would not preserve the estimator’s assumptions. Finally, reference variance is endpoint-specific and scale-specific: it should not be substituted with a variance from another endpoint or with the total variability of all observations.