Study Type Details in BioequivalencePower

Standard (average) Bioequivalence

This section discusses the study types RatioTOST and DifferenceTOST which are used for standard (average) bioequivalence. The study type RatioTOST is the common type to use as it is for hypothesis tests formalized in terms of a ratio. The DifferenceTOST type is for the case of a difference. Unlike some other specialized study types, the RatioTOST and DifferenceTOST types can be combined with all the supported experimental designs. As all studies they have supporting methods of the samplesize and power function. They are also supported by power_analysis. These types also have methods supported by the pvalue and confint functions.

A subtle difference between the RatioTOST and DifferenceTOST types is that in the former the variability parameter is called CV for "coefficient of variation" while in the latter it is called SE for "standard error". This differs from say the PowerTOST the R package, where the parameter name "CV" is always used also in cases when it is not interpreted as a coefficient of variation (square root of variance divided by the mean).

Statistical background

The standard/basic (average) bioequivalence test uses a Two One Sided T-Test (TOST). In the package we explicitly differentiate between the log scale formulation, RatioTOST where the hypothesis test is:

\[\begin{array}{ll} H_0:& \frac{T}{R} < \theta_1 \quad \text{or} \quad \theta_2 < \frac{T}{R} \\[10pt] H_1:& \qquad \theta_1 \le \frac{T}{R} \le \theta_2. \end{array}\]

Here $T$ is the average test formulation and $R$ is the average for the reference formulation. The ratio $T/R$ (ideally at 1), is prescribed via the peθ₀ value. The default values for $\theta_1$ and $\theta_2$ are $0.8$ and $1.25$, respectively. The variability assumed in this test is captured via CV.

An alternative formulation that is less common is the non log scale, or difference formulation, with DifferenceTOST. In this case the hypothesis test is:

\[\begin{array}{ll} H_0:& T-R < \theta_1 \quad \text{or} \quad \theta_2 < T-R \\[10pt] H_1:& \qquad \theta_1 \le T-R \le \theta_2. \end{array}\]

In the case of a difference formulation, the difference $T-R$ is ideally at 0 and the default values for $\theta_1$ and $\theta_2$ are $-0.2$ and $0.2$ respectively. Note that the variability parameter for this case is called SE.

Basic examples

samplesize(RatioTOST, ED2x2Crossover; CV = 0.23)

Determined sample size: n = 24  achieved power = 80.665% 
	HypothesisTestMetaParams Settings:   α = 0.05  target_power = 0.8
	Study: RatioTOST{ED2x2Crossover}  CV = 0.23  θ₀ = 0.95  [θ₁, θ₂] = [0.8, 1.25]

samplesize(RatioTOST, ED2x2Crossover; CV = 0.23, target_power = 0.9)

Determined sample size: n = 32  achieved power = 90.443% 
	HypothesisTestMetaParams Settings:   α = 0.05  target_power = 0.9
	Study: RatioTOST{ED2x2Crossover}  CV = 0.23  θ₀ = 0.95  [θ₁, θ₂] = [0.8, 1.25]

samplesize(RatioTOST, ED2x2Crossover; CV = 0.23, target_power = 0.9, peθ₀ = 0.85)

Determined sample size: n = 242  achieved power = 90.044% 
	HypothesisTestMetaParams Settings:   α = 0.05  target_power = 0.9
	Study: RatioTOST{ED2x2Crossover}  CV = 0.23  θ₀ = 0.85  [θ₁, θ₂] = [0.8, 1.25]

power(RatioTOST, ED2x2Crossover; CV = 0.23, peθ₀ = 0.85, n = 242)

Computed achieved power: 90.044%
	HypothesisTestMetaParams Settings: n = 242  α = 0.05
	Study: RatioTOST{ED2x2Crossover}  CV = 0.23  θ₀ = 0.85  [θ₁, θ₂] = [0.8, 1.25]

power(RatioTOST, ED2x2Crossover; CV = 0.23, peθ₀ = 0.85, n = [110, 132])

Computed achieved power: 89.83%
	HypothesisTestMetaParams Settings: n = [110, 132]  α = 0.05
	Study: RatioTOST{ED2x2Crossover}  CV = 0.23  θ₀ = 0.85  [θ₁, θ₂] = [0.8, 1.25]

power_analysis(RatioTOST, ED2x2Crossover; CV = 0.23, peθ₀ = 0.85, target_power = 0.9)

Key docstrings

Types

BioequivalencePower.RatioTOST — Type

RatioTOST{D}(peθ₀::HTRealParam = 0.95,
             θ₁::HTRealParam = 0.8,
             θ₂::HTRealParam = 1/θ₁,
             CV::HTRealParam = nothing) where D <: ExperimentalDesign

A bioequivalence power study with a ratio formulation. This is the standard (average) bioequivalence study. As in all bioequivalence power studies, θ₁ and θ₂ are the limits, and peθ₀ is the postulated value of the ratio for which power (or sample size) is to be evaluated. CV is the coefficient of variation.

BioequivalencePower.DifferenceTOST — Type

DifferenceTOST{D}(  peθ₀::HTRealParam = 0.05,
                    θ₁::HTRealParam = -0.2,
                    θ₂::HTRealParam = -θ₁,
                    SE::HTRealParam = nothing) where D <: ExperimentalDesign

A bioequivalence power study with a difference formulation. As in all bioequivalence power studies, θ₁ and θ₂ are the limits, and peθ₀ is the postulated value of the difference for which power (or sample size) is to be evaluated. SE is the standard error.