Estimating Within-Subject Variance From Replicate Data

Why Replication Matters for Variance Estimation

In a crossover bioequivalence study, observations from the same participant can be more alike than observations from different participants. Replicate administration of a formulation provides information about the variability of that participant’s response across administrations. The key is to separate this within-subject variation from differences among participants and from systematic effects such as period.

For a pharmacokinetic endpoint such as AUC or \(C_{\max}\), analysis commonly uses the natural logarithm of the measured value. Let \(Y_{ifj}\) denote the log endpoint for subject \(i\), formulation \(f\), and replicate occasion \(j\), where \(f\) is test (T) or reference (R). The within-subject variance \(\sigma_{W,f}^2\) describes the variability of repeated log outcomes for formulation \(f\), under a specified model and its assumptions. It is not the total variance across all observations: that total can also include subject, period, and other components.

The estimates depend on both the design and the model. Replicating reference but not test can estimate reference within-subject variance, but not test within-subject variance by the same direct comparison. Replicating both formulations can estimate both, provided the model distinguishes within-formulation replication from other sources of variation.

Estimating Variance From a Replicate Pair

Suppose each of \(m\) subjects receives formulation \(f\) twice. A simple model for the two log observations is \(Y_{if1}=\mu_f+b_i+\varepsilon_{if1}\) and \(Y_{if2}=\mu_f+b_i+\varepsilon_{if2}\). Here \(\mu_f\) is a common formulation mean, \(b_i\) is a subject effect shared by the pair, and the occasion errors \(\varepsilon_{if1}\) and \(\varepsilon_{if2}\) are independent with mean zero and variance \(\sigma_{W,f}^2\). For this direct calculation, assume no unmodeled occasion effect shifts one member of the pair relative to the other.

Taking the within-subject difference \(D_{if}=Y_{if1}-Y_{if2}\) removes both the common formulation mean and the shared subject effect. Under the assumptions, \(D_{if}=\varepsilon_{if1}-\varepsilon_{if2}\), so its variance is \(2\sigma_{W,f}^2\). If the differences have an unknown common mean, estimate their variance with the sample variance \(s_{D,f}^2\), then divide by two.

This estimator is unbiased under the stated model. If the mean difference is known to be exactly zero, an uncentered alternative is available: \(\widehat{\sigma}_{W,f}^2=\sum_i D_{if}^2/(2m)\). That alternative uses the known-zero assumption rather than estimating the mean difference from the data. Centering is generally important when a common nonzero difference could arise, for example, from a systematic period contrast.

Why the Divisor and Degrees of Freedom Matter

For normally distributed occasion errors, a replicate difference is normal with variance \(2\sigma_{W,f}^2\). When its mean is unknown and estimated by \(\bar{D}_f\), the sample variance of the \(m\) differences has \(m-1\) degrees of freedom. Consequently,

If instead the mean difference is known to be zero and the uncentered estimator is used, then each \(D_{if}^2/(2\sigma_{W,f}^2)\) has a \(\chi^2_1\) distribution. Summing the \(m\) independent terms gives

The factor of two in the denominator is essential: the variance of a difference of two independent observations, each with variance \(\sigma_{W,f}^2\), is \(2\sigma_{W,f}^2\). Using the wrong scaling changes the estimated variance and the reference distribution. The chi-square results also rely on normality and independence across subjects; they do not make a non-normal or dependent design behave as if those assumptions held.

How Design Contrasts Determine What Is Estimable

The same logic extends beyond a single pair. In a crossover model, a contrast is a weighted sum of observations whose coefficients sum to zero. That condition cancels any additive effect common to all observations from a subject. Under independent occasion errors, the variance of a contrast is the sum of each error variance multiplied by the square of its coefficient. This gives a direct way to see which variance components the design can reveal.

The formula assumes the errors at the occasions are independent, with variance determined by the formulation at that occasion. A difference between two replicates of the same formulation has coefficients \(1\) and \(-1\), so its variance is \(2\sigma_{W,f}^2\). A contrast involving different formulations can instead combine their variances. Such a contrast does not, by itself, separate them.

Model Adjustment, Diagnostics, and Failure Modes

Direct differences are transparent when the two observations in each replicate pair are comparable. In a multi-period study, however, a difference may also reflect a systematic period contrast. If one replicate is typically given later, a raw difference can combine within-subject error with a period effect. A suitable linear model can adjust for period and sequence, and its residual or contrast-based estimates can then target the within-subject component. The model must preserve the replication information: treating all observations as unrelated discards the within-subject structure, while treating every difference as if its mean were known to be zero can misstate uncertainty.

Useful checks include examining replicate differences by formulation, sequence, and period; looking for outlying differences and unequal spread; and confirming that the planned replicate observations are actually present. A systematic nonzero difference may signal an unmodeled period effect or a data issue, rather than a change in random variability. Dependence between occasions also matters: if the two occasion errors have correlation \(\rho_f\), then \(\operatorname{Var}(Y_{if1}-Y_{if2})=2\sigma_{W,f}^2(1-\rho_f)\). Dividing this contrast variance by two estimates \(\sigma_{W,f}^2(1-\rho_f)\), not the marginal variance \(\sigma_{W,f}^2\), unless \(\rho_f=0\) or the model accounts for the dependence.

A direct sample variance of differences is easiest to interpret and calculate when pairs are complete and the design is simple. A model-based contrast approach is more appropriate when period effects, unequal replication, or other planned covariates must be handled. Neither approach repairs a design that contains no information to identify a desired component: for example, an unreplicated formulation’s within-subject variance cannot be recovered from a single observation per subject without additional assumptions or external information.

These variance estimates describe variability on the log scale. If a coefficient of variation is useful for interpretation, the usual lognormal conversion is \(\mathrm{CV}_{W,f}=\sqrt{\exp(\sigma_{W,f}^2)-1}\), expressed as a proportion (multiply by 100 for a percentage). This conversion does not change which variance components the design identifies. Variance estimation should therefore be reported with the endpoint, scale, model, replicate contrasts, and assumptions made explicit.