Constructing the FDA Upper Confidence Bound

Why an Upper Bound Is Needed

The FDA linearized reference-scaled criterion compares the squared log-scale treatment difference with a term based on within-subject reference variability. A calculation using only the estimated treatment difference and estimated reference variance gives a point estimate of that criterion. But both quantities are estimated from finite data. The upper confidence bound addresses this uncertainty: it asks whether the data support keeping the criterion below zero even after allowing for estimation error.

This tutorial focuses on constructing and interpreting that bound. It is not enough to calculate the plug-in criterion and note that it is negative. The bound also depends on how precisely the treatment difference and reference variance have been estimated.

For the conventional FDA scaling constant, \(\theta=[\ln(1.25)/\sigma_{W0}]^2\), with \(\sigma_{W0}=0.25\) on the log scale; thus \(\theta\) is approximately \(0.7967\). The criterion, its estimate, and its bound are dimensionless because they are formed from log-scale quantities and variances on the log scale.

The inputs must refer to the same estimand and analysis model: \(\hat D\) is the estimated test-minus-reference log mean difference, \(SE_D\) is its standard error, \(S_{WR}^2\) is the estimated within-subject reference variance, and \(df\) is the degrees of freedom associated with estimating that variance for this bound. Do not substitute a coefficient of variation or a standard deviation where a variance is required.

Deriving the Upper Confidence Bound

Write the estimated criterion as \(g(\hat D,S_{WR}^2)=\hat D^2-\theta S_{WR}^2\). To approximate its uncertainty, use a first-order Taylor expansion around the estimates. The partial derivative with respect to the treatment difference is \(2\hat D\); the partial derivative with respect to the reference variance is \(-\theta\). Under the working approximation that the two estimates are independent, the delta method gives the variance of the estimated criterion as the sum of the two derivative-weighted variance contributions.

For a variance estimate based approximately on a normal chi-square model, \(df\,S_{WR}^2/\sigma_{WR}^2\) is treated as approximately chi-square with \(df\) degrees of freedom. A chi-square variable with \(df\) degrees of freedom has variance \(2df\). Therefore,

Substituting this approximation and taking the square root gives the estimated standard error of the criterion. Using the one-sided \(t\) critical value \(t_{0.95,df}\), the FDA upper confidence bound is

The factor of 2 in the reference-variance contribution is essential. It comes from the variance of a chi-square-based variance estimate, not from an optional adjustment. Omitting it understates the estimated uncertainty in \(S_{WR}^2\) and can make the upper bound too low.

The decision rule is strict: if \(UCB<0\), the upper bound lies below zero and supports the linearized reference-scaled criterion. If \(UCB\geq 0\), this bound does not establish that the criterion is below zero. A negative plug-in estimate alone is not enough.

Worked Calculations

Reading the Result and Checking the Calculation

The sign of the upper bound, not the sign of the plug-in estimate, determines this part of the decision. A negative estimated criterion may have a positive upper bound when the uncertainty allowance is large. Conversely, a bound well below zero indicates that the estimated uncertainty, as represented by this method, does not extend the criterion to zero.

The bound is a one-sided confidence procedure. In repeated applications under its assumptions, this first-order approximate procedure is intended to have approximately 95% coverage of the target criterion. It does not mean there is a 95% probability that the fixed population criterion lies below the calculated bound.

A useful calculation sequence is to write down the inputs with their meanings and units, calculate the plug-in criterion, calculate each variance contribution separately, add them before taking the square root, apply the one-sided critical value, and then compare the resulting upper bound with zero. Keeping the two uncertainty contributions visible makes an omitted term or an incorrect scale easier to detect.

Diagnostics, Alternatives, and Limitations

Several checks help identify a calculation that is going wrong. Since \(S_{WR}^2\) is a variance, it must be nonnegative; its fourth power in the formula is \(S_{WR}^4=(S_{WR}^2)^2\), not the fourth power of a standard deviation entered as though it were a variance. The two contributions inside the square root must be nonnegative, and the square root and resulting standard error must be real. Verify that \(df\) is positive and that \(t_{0.95,df}\) is a one-sided 95th-percentile critical value, not a two-sided 95% critical value. Finally, the confidence margin must be added to the point estimate when constructing an upper bound.

A common mistake is to use \(\theta^2S_{WR}^4/df\) for the reference-variance contribution. That omits the factor of two in \(\operatorname{Var}(S_{WR}^2)\approx2S_{WR}^4/df\). The correction is to use \(2\theta^2S_{WR}^4/df\) before adding the treatment-difference contribution. Omitting the factor makes the standard error too small and can produce an upper bound that is too low, potentially changing the decision.

The plug-in criterion is simpler to calculate, but it does not account for estimation uncertainty and is not a substitute for the upper-bound decision. Conventional average bioequivalence procedures address a different criterion; they should not be treated as interchangeable with this reference-scaled bound. This bound also does not by itself resolve every requirement of a scaled bioequivalence assessment. Its accuracy depends on the specified analysis model and on how well the normal and chi-square approximations describe the estimates, especially when the reference-variance estimate has few degrees of freedom or is unstable. Model fit and variance estimation therefore matter to the validity of the calculation, not just to its arithmetic.