Treatment-Specific Variability Questions and Design Choice

Start With the Variability Question

A replicate design is useful when the question concerns more than the average difference between test and reference. It can provide repeated observations that help distinguish variability from one administration to another from stable differences among participants. But repeating a formulation is informative only for questions involving that formulation’s variability. The design should therefore follow the target quantity, not simply the number of periods that seems convenient.

The phrase between-formulation variability can refer to different quantities. It might mean the variability of each participant’s test-minus-reference response, or it might be used loosely to describe a difference between the test and reference within-subject variances. These are not interchangeable. State the target explicitly before choosing a schedule.

For clarity, suppose the analyzed response is on a suitable scale, such as log exposure, and write a simplified model for participant \(i\) at occasion \(j\):

Here \(\mu\) is an overall level, \(\pi_j\) is a period effect, \(b_i\) is a participant effect common to formulations, and \(u_{iF_{ij}}\) is a stable participant-by-formulation deviation for the formulation \(F_{ij}\), either test \(T\) or reference \(R\). The occasion error \(\varepsilon_{ij}\) represents variation specific to that administration. A replicate schedule supplies repeated occasions on which the same participant receives the same formulation.

For the variance decompositions below, assume period effects are absent or appropriately modeled, there is no relevant carryover, occasion errors from distinct administrations are independent with mean zero, and their variances are \(\sigma_{WT}^2\) for test and \(\sigma_{WR}^2\) for reference. Also assume the occasion errors are independent of the participant effects and formulation-specific deviations. The pair \((u_{iT},u_{iR})\) may be correlated; do not assume its covariance is zero unless the scientific model justifies that restriction. These assumptions are modeling conditions, not automatic properties of a crossover schedule.

Map the Target to Repeated Administrations

To estimate within-test variability from replication, some participants must receive test more than once. To estimate within-reference variability, some participants must receive reference more than once. A schedule that repeats only one formulation does not, by that fact alone, identify the other formulation’s within-subject variability. A full replicate schedule gives each participant repeated test and reference administrations; a partial replicate schedule repeats one formulation in at least part of the sequence set.

A two-period \(TR/RT\) crossover gives each participant one observation under each formulation. It can support a paired comparison of average responses, subject to the model and design assumptions, but it has no within-formulation repeat for either formulation. A three-period partial replicate can supply replication for one formulation; a four-period full replicate, such as \(TRTR\) and \(RTRT\), supplies replication for both. Sequence membership and coverage matter: the sequence set must actually include repeated administrations of the target formulation, and the analysis must account for period and other modeled effects.

Why Replication Separates Occasion Variation

The basic justification is that a within-participant difference between repeated administrations cancels components that are stable for that participant and formulation. Consider two reference administrations for the same participant, after period effects have been removed:

Subtracting the second observation from the first removes \(\mu\), \(b_i\), and \(u_{iR}\):

By independence of the two occasion errors, each with variance \(\sigma_{WR}^2\), the variance of this difference is \(2\sigma_{WR}^2\). Therefore, under these assumptions, half the variance of the adjusted repeated-reference differences estimates the within-reference variance. The same reasoning applies to repeated test administrations and \(\sigma_{WT}^2\). If period effects remain unaccounted for, the difference can include a period contrast as well as occasion error, so the simple interpretation fails.

Individual Contrasts and the Assumptions Behind Their Variance

With \(r_T\) test and \(r_R\) reference administrations for a participant, consider the contrast between that participant’s formulation averages. After removing modeled period effects, write \(\bar{Y}_{iT}-\bar{Y}_{iR}=(u_{iT}-u_{iR})+\bar{\varepsilon}_{iT}-\bar{\varepsilon}_{iR}\). The common participant effect cancels. Under independent occasion errors and their independence from the formulation-specific deviations, the variance is:

The first term is the variance across participants in stable test-minus-reference deviations. The remaining terms are the occasion-error contributions to the contrast of formulation averages. The formulation-specific deviations may be correlated, and their covariance is included in \(\operatorname{Var}(u_{iT}-u_{iR})=\operatorname{Var}(u_{iT})+\operatorname{Var}(u_{iR})-2\operatorname{Cov}(u_{iT},u_{iR})\). The decomposition does not follow merely from independent occasion errors: independence, or at least zero covariance, between occasion errors and the formulation-specific deviations is also needed.

Design Checks, Alternatives, and Limitations

Before choosing a schedule, state the target in words and identify the repeated administrations that would produce information about it. Then check that the proposed sequences include those administrations, that both formulations are observed where individual contrasts are needed, and that period effects and other design features can be handled by the intended model. Earlier material on covariance between repeated formulation responses is relevant when interpreting the stable contrast component: covariance between \(u_{iT}\) and \(u_{iR}\) affects the variance of their difference.

A full replicate is the direct choice when the question requires within-test and within-reference variability from repeated administrations of both formulations. A partial replicate is a more targeted alternative when the primary variability question concerns just one formulation; it does not automatically provide the other formulation’s within-subject variance. A nonreplicate two-period crossover is simpler for paired average comparisons, but cannot use within-formulation repeat differences to separate occasion error from stable participant components. These are trade-offs in information and design burden, not guarantees that one schedule is universally preferable.

• Missing repeat: If the relevant formulation occurs only once for every participant, the schedule lacks a within-participant repeat for estimating its occasion variability by the difference argument.
• Unmodeled period effects: A repeated-treatment difference may include a systematic period contrast; do not attribute all of its variance to occasion error without checking the model.
• Dependence or carryover: Correlated occasion errors or residual effects from earlier administrations can invalidate the simple variance calculation.
• Overinterpreting a contrast: The variance of individual test-minus-reference responses is not the same as the difference between \(\sigma_{WT}^2\) and \(\sigma_{WR}^2\), nor is either quantity the mean formulation difference.

Replicate schedules also have limitations. Repetition increases the number of administrations and can increase participant burden; the variance estimates still depend on the assumed model and the observed data. A schedule can supply the necessary repeated administrations without ensuring that estimates will be precise, unbiased under model misspecification, or applicable beyond the study conditions. In particular, the variance decomposition above should not be used if its independence conditions are unsupported.