Methods and formulas for Test mean - reference mean for 2-Sample Equivalence Test

Notation

Term	Description
D	Difference
	Test mean
	Reference mean

The mean of the reference sample, , is given by:

The standard deviation of the test sample, S₁, is given by:

The standard deviation of the reference sample, S₂, is given by:

Notation

Term	Description
X i	Observations from the test sample, with i = 1, ..., n1
Y i	Observations from the reference sample, with i = 1, ..., n2
n1	Number of observations in the test sample
n2	Number of observations in the reference sample

Do not assume equal variances (default)

Assume equal variances

If you select the option to assume equal variances, then Minitab calculates the pooled standard deviation, S_p, and the standard error of the difference, SE, using the following formulas:

Notation

Term	Description
S1	Standard deviation of the test sample
n1	Number of observations in the test sample
S2	Standard deviation of the reference sample
n2	Number of observations in the reference sample
Sp	Pooled standard deviation

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:

Do not assume equal variances (default)

By default, the degrees of freedom for the test, v, are given by the following formula:

Minitab displays v rounded down to the nearest integer.

Assume equal variances

If you select the option to assume equal variances, then Minitab calculates the degrees of freedom as follows:

Notation

Term	Description
S1	Standard deviation of the test sample
n1	Number of observations in the test sample
S2	Standard deviation of the reference sample
n2	Number of observations in the reference sample

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