Methods and formulas for Test mean - reference mean for Equivalence Test with Paired Data

Notation

Term	Description
D	Difference
	Test mean
	Reference mean

Minitab uses the following formula to calculate the standard error (SE) of the difference:

where S is the standard deviation of the differences as defined below.

Notation

Term	Description
SE	Standard error of the difference
S	Standard deviation of the differences
n	Number of pairs of observations
di	Pairwise differences (X i - Yi), i = 1, ..., n
	Average of the pairwise differences

Let k₁ be the value that you specify for the lower limit and k₂ be the value that you specify for the upper limit. By default, the lower equivalence limit, δ₁, is given by:

and the upper equivalence limit, δ₂, is given by:

Notation

Term	Description
v	Degrees of freedom
n	Number of pairs of observations

100(1-α)% CI

By default, Minitab uses the following formula to calculate the 100(1 – α)% confidence interval (CI) for equivalence:

CI = [min(C, D_l), max(C, D_u)]

where:

100(1-2α)% CI

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

CI = [D_l, D_u]

One-sided intervals

For a hypotheses of Test mean > reference mean or Test mean - reference mean > lower limit, the 100(1 – α)% lower bound is equal to D_L.

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

Notation

Term	Description
D	Difference between the test mean and the reference mean
SE	Standard error
δ1	Lower equivalence limit
δ2	Upper equivalence limit
v	Degrees of freedom
α	The 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 difference between the mean of the test population and the mean of the reference population. By default, the t-values are calculated as follows:

For a hypothesis of Test mean > reference mean, δ₁ = 0.

For a hypothesis of Test mean < reference mean, δ ₂ = 0.

Notation

Term	Description
D	Difference between the sample test mean and the sample reference mean
SE	Standard error of the difference
δ1	Lower equivalence limit
δ2	Upper equivalence limit

The probability, P_H0, for each null hypothesis (H₀) is given by the following:

Notation

Term	Description
	Unknown difference between the mean of the test population and 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