On confidence bounds for the ratio of net differences in the "gold standard" design with reference, experimental, and placebo treatment

The three-arm clinical study including a placebo has been recommended to be the preferred study design for the comparison of an experimental treatment relative to a reference treatment. In a confirmatory three-arm study multiplicity issues arise that are not present in two-arm studies. In the past a successful demonstration of superiority of the reference over placebo has been regarded a prerequisite validation step for the demonstration of superiority of the experimental treatment over placebo. However, for an investigator this last comparison is the most critical one. In a clinical study the demonstration of superiority of the experimental treatment over placebo is a result of its own value and this should therefore not be made dependent on tests that are of higher priority in a hierarchical test procedure. This can be achieved through a symmetrical formulation of Fieller's method for constructing confidence intervals for ratio of the expected values from normally distributed variables. In the symmetrical formulation the different meanings of nominator and denominator disappear, and simultaneous statements for comparisons between the experimental treatment and placebo, reference treatment and placebo, and reference and experimental treatment can be made. This is accomplished by moving the discussion on confidence sets from the line of real numbers to the unit circle which allows representing confidence sectors always as connected sets, gives always simple geometrical interpretations, and is easy to be transformed back to the real numbers if the traditional calculations behave well. The proposed procedures provides additional insight in existing methods but does obviously not answer the clinical question whether or not the demonstration of superiority of the reference treatment over placebo is a necessary validation step for further comparisons.