Crossover Bioequivalence with Differential Dropout
A two-period, two-sequence crossover pilot is run in 18 healthy volunteers to compare a test formulation (T) and a reference (R) on log-transformed AUC. Ten volunteers receive T then R; eight receive R then T. Four volunteers withdraw before period 2 - three in the T-then-R sequence and one in the R-then-T sequence. For the 14 completers, the mean within-subject log difference (T−R) is 0.07 with SD 0.18. The sponsor proposes an unpaired two-sample t-test using all 18 period-1 observations and all 14 period-2 observations, arguing that pairing loses the dropouts. A regulatory reviewer wants a 90% confidence interval for the geometric mean T/R ratio and a statement of the estimand. Period-1 means are 7.21 for T and 7.05 for R on the log scale; the sponsor notes these are close enough. No carryover test has been reported.
1. Is the sponsor's unpaired analysis defensible? Identify the design features that make it invalid and state what the unpaired comparison actually estimates.
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The unpaired analysis is not defensible. The design is a two-period crossover with repeated measurements on the same volunteer, so observations are correlated within subject and the two formulations are compared within subject. An unpaired comparison of all period-1 observations with all period-2 observations confounds formulation with period and with sequence; it also ignores the within-subject pairing and the intended estimand. Because the T and R observations come from different people in different periods, between-subject variance enters the standard error, and period effects are not removed. The four dropouts do not justify breaking the design. The unpaired estimate does not estimate the average within-subject log T/R difference; at best it estimates a mixture of period and formulation effects. The paired analysis on completers is valid under missing completely at random, but its estimand is the average treatment effect among completers.
2. Using the completer data, compute the 90% confidence interval for the geometric mean T/R ratio and state whether it meets the usual 80–125% bioequivalence limits.
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For the 14 completers, dbar=0.07, sd=0.18, n=14. SE=0.18/sqrt(14)=0.0481. With t_{0.05,13}=1.771, the 90% CI for the mean log difference is 0.07 ± 1.771(0.0481)=0.07±0.0852, i.e. (-0.0152, 0.1552). Exponentiating gives geometric mean ratio CI (exp(-0.0152), exp(0.1552)) = (0.985, 1.168). The point estimate is exp(0.07)=1.072. The entire interval lies inside 0.80 to 1.25, so on completer data the test formulation meets the usual average bioequivalence criterion. This does not repair the missing-data problem.
3. How should the four dropouts be handled, and what estimand should be pre-specified?
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Pre-specify the estimand as the average within-subject log T/R difference in the randomized population, not only completers. The four dropouts have only period-1 data, so a completer-only analysis can be biased if withdrawal is related to treatment or response. Use a mixed-effects model for the crossover with fixed period, sequence and treatment effects and a random subject effect, fitted to all available observations under a stated missing-at-random assumption. Report the completer analysis as a sensitivity analysis and state that it assumes MCAR. Because three dropouts are in the T-then-R sequence, a differential dropout pattern by sequence should be checked. Do not pool unpaired observations.