Evaluating Variance Estimates for Sample Size Planning

Why the Variance Estimate Deserves Scrutiny

A sample-size calculation can be executed correctly and still produce a poor plan if its variance input is not relevant to the study being planned. In average bioequivalence planning, within-subject variability on the log scale affects the precision of the treatment comparison and therefore the chance that the confidence interval will meet the equivalence limits. A variance estimate is not automatically suitable simply because it came from a pharmacokinetic study or was reported as a coefficient of variation.

Evaluation has three parts. First, consider the source: how much information supports the estimate, and how closely does its study population resemble the planned population? Second, consider its precision: how uncertain is the estimate as a consequence of finite degrees of freedom? Third, consider design compatibility: does the estimate represent the same within-subject variation that the planned design and analysis will use? The earlier material on average bioequivalence sample-size inputs and replicate crossover planning provides the calculation context; here the emphasis is on judging the variance input.

The word “within-subject” does not identify one universal quantity independent of design. The relevant variance depends on which observations are repeated, which fixed effects are adjusted for, and which treatment contrast is used. Thus, before transferring an estimate to a new calculation, identify the exact variance component or residual variance that the planned analysis requires.

Source and Design Compatibility

Potential sources include a previous study of the same drug and endpoint, a pilot study, a published study, or a variance component estimated from a replicate design. A source is more persuasive when the formulation, dose, endpoint, sampling and assay procedures, study conditions, participant characteristics, and analysis model are relevant to the planned trial. These features need not be identical in every respect, but differences should be examined rather than ignored.

The endpoint matters: within-subject variability for AUC need not match that for \(C_{\max}\). The design and model matter too. For example, a residual variance estimated after adjusting for period and sequence effects is not necessarily the same as the variance of unadjusted subject-level treatment contrasts. Nor should a variance component from a design with repeated administrations be assumed to equal the variance of a contrast in a different design without checking how that contrast is constructed.

If a source reports a coefficient of variation (CV) rather than a log-scale variance, confirm that it is the within-subject CV for the relevant endpoint and population. Under the usual lognormal conversion, a CV expressed as a proportion \(CV_w\) corresponds to log-scale variance

The CV must be entered as a proportion: for example, 20% is \(0.20\), not \(20\). This conversion does not make a mismatched estimate compatible; it only expresses a compatible CV on the log scale.

Precision of an Estimated Variance

A variance estimate based on few independent residual degrees of freedom can be quite uncertain, even when the reported value has several decimal places. Under a normal-error model with correctly specified fixed effects, independent residuals, and a common residual variance \(\sigma_w^2\), the residual sum of squares divided by its degrees of freedom gives \(s_w^2\). If that residual degrees of freedom is \(\nu\), then

This result gives a confidence interval for the variance. Let \(\chi^2_{q,\nu}\) denote the \(q\)-quantile of a chi-square distribution with \(\nu\) degrees of freedom. A two-sided \(100(1-\alpha)\%\) confidence interval is

The order of the quantiles in the denominators is intentional: the larger chi-square quantile gives the lower variance limit. The interval describes sampling uncertainty under the stated model; it does not account for a mismatch between the source study and the planned study. A narrow interval around an irrelevant estimate does not make that estimate relevant.

Why Raw Paired Contrasts Can Mislead

A useful check is to ask whether the statistic being called a within-subject variance actually estimates the model residual variance required for planning. Consider a two-period crossover in which the two period residuals for a subject are independent, have mean zero, and each have variance \(\sigma_w^2\). Their difference then has residual variance \(2\sigma_w^2\). That fact alone does not mean that the sample variance of raw paired treatment contrasts estimates \(2\sigma_w^2\): the contrasts may have different means across sequences because of a period contribution.

For a balanced two-sequence design, suppose the raw contrast means are \(+\delta\) and \(-\delta\), with within-sequence contrast variance \(2\sigma_w^2\). The variance across all raw contrasts is the average within-sequence variance plus the variance of the sequence-specific means:

Dividing that pooled variance by two would yield \(\sigma_w^2+\delta^2/2\), not \(\sigma_w^2\). Dividing a contrast variance by two is justified only when the contrast variance has been appropriately residual-adjusted, or when additional conditions ensure a common contrast mean. Otherwise, use the model-based residual variance appropriate to the planned analysis.

Practical Checks and Planning Consequences

Before using a variance estimate, record its endpoint, scale, estimator, degrees of freedom, source design, and model adjustments. Then check whether its uncertainty and provenance are credible for the planned study. Useful diagnostics include reviewing residual plots for strong departures from the assumed error structure, checking for influential observations, verifying the degrees of freedom from the fitted model, and confirming that the estimate was not calculated from a mixture of quantities with different meanings. Such checks can reveal concerns; they cannot prove that a historical estimate will transport perfectly.

A simple sensitivity check helps show the practical importance of the choice. Under a normal approximation with all other planning inputs held fixed, required sample size is approximately proportional to the variance. Exact sample size calculations need not follow that ratio exactly because degrees of freedom and the power calculation also change; they should be recalculated using each plausible input.

The main alternatives are not interchangeable procedures for “fixing” a weak estimate. A well-matched historical estimate can be useful when the new study is sufficiently comparable; a pilot estimate is direct but may have few degrees of freedom; and a published CV may be convenient but can conceal differences in endpoint, design, or estimator. A model-based residual estimate is preferable to a raw contrast summary when the planned model adjusts for effects that influence those contrasts. Each approach has limitations: historical data may not transport, pilots may be imprecise, and model-based estimates depend on model assumptions.

A common mistake is to treat the reported point estimate as exact because it is printed to several decimal places, or to choose the smallest estimate to obtain the smallest feasible study. The correction is to separate three questions: is the estimate relevant, how precise is it, and what happens to the plan across plausible values? The confidence interval addresses sampling precision under a model; the source review addresses relevance; and recalculation across plausible values shows sensitivity. None alone resolves all three.