Inference on the Fuzzy Equivalence

Abstract

Under some clinical circumstances, statistical test for establishing equivalence with respect to the response rate between standard (S) and test (T) drugs specifying a maximum allowable value Δ₀ of the true difference Δ=ρ_S-ρ_T is not feasible, because of the difficulty or impracticality in specifying the allowable difference. Instead of defining a crisp difference, a fuzzy tolerable difference may be introduced by formulating a membership function, say, by asking opinions of clinicians and/or patients.
The membership function μ (Δ), taking values between 0 and 1, indicates the degree of reluctance of accepting a seemingly less effective but much less toxic or more convenient therapy; namely, at the lower boundary μ (Δ_L)=0 and at the upper booudary μ (Δ_U)=1. If Δ_L=Δ_U=Δ₀, then the function jumps up from 0 to 1 at the crisp tolerable difference Δ₀. Thus the crisp argument is a special case of the fuzzy one.
Combining the membership function with the distribution for the difference in response rate, a more flexible fuzzy equivalence test is formulated . The probability density function for the difference obtained by joining beta functions is replaced by a normal approximation without the continuity correction to give fairly good results.