Methods and formulas for Test mean / reference mean for Equivalence Test for a 2x2 Crossover Design

The ratio, ρ, is equal to the test mean, , divided by the reference mean, , as shown below:

where

The pooled variance for the reference periods, S²_RR, is given by the following:

The pooled variance for the test periods, S²_TT, is given by the following:

Let S_TR be defined as follows:

Notation

Term	Description
Yijk	Response for participant k during period j in sequence i (for more information, go to Methods and formulas for common concepts used in Equivalence Test for a 2x2 Crossover Design.)
ni	Number of participants in sequence i

100(1-α)% CI

Minitab cannot calculate the confidence interval (CI) if either of the following two conditions are not met:

where S²_RR represents the pooled variance for the reference periods, and S²_TT represents the pooled variance for the test periods, and both S²_RR and S²_TTare calculated as described in the Pooled variance section, and W is given by the following:

By default, Minitab calculates the 100(1 - α)% CI for the ratio as follows:

CI = [min(C, ρ_L), max(C, ρ_U)]

where:

where t = t_1-α,v, v = n₁ + n₂ – 2, and S_TR is calculated as described in the Pooled variance section.

100(1-2α)% CI

If you select the option to use the 100(1 – 2α)% CI, then the CI is given by the following:

CI = [ρ_L, ρ_U]

One-sided intervals

For a hypothesis of Test mean / reference mean > lower limit, the 100(1 – α)% lower bound is equal to ρ_L.

For a hypothesis of Test mean / reference mean < upper limit, the 100(1 – α)% upper bound is equal to ρ_U.

Notation

Term	Description
	Mean of the reference periods (for more information, see the section on the Ratio)
	Mean of the test periods (for more information, see the section on the Ratio)
ni	Number of participants in sequence i
v	Degrees of freedom
α	Significance level for the test (alpha)
t1-α,v	Upper 1 – α critical value for a t-distribution with v degrees of freedom

Let t₁ be the t-value for the hypothesis, , and let t₂ be the t-value for the hypothesis, , where is the ratio of the test mean to the reference mean for the populations. The t-values are calculated as follows:

where S²_RR represents the pooled variance for the reference periods, S²_TT represents the pooled variance for the test periods, and S_TR represents the overall standard deviation. For more information, see the section on the Pooled variance.

Notation

Term	Description
	Mean of the reference periods (for more information, see the section on the Ratio)
	Mean of the test periods (for more information, see the section on the Ratio)
δ1	Lower equivalence limit
δ2	Upper equivalence limit

The probability, P_H0, for each null hypothesis is given by the following:

If , then:

Notation

Term	Description
Λ	Unknown ratio of the mean of the test population to the mean of the reference population
δ1	Lower equivalence limit
δ2	Upper equivalence limit
v	Degrees of freedom
T	t-distribution with v degrees of freedom
t1	t-value for the hypothesis
t2	t-value for the hypothesis