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

The following methods and formulas are used for testing the ratio between the test mean and the reference mean.

In This Topic

Ratio
Means and standard deviations
Equivalence limits
Degrees of freedom (DF)
Pooled standard deviation
Confidence interval
T-values
P-values

Ratio

Notation

Term	Description
ρ	Ratio
	Test mean
	Reference mean

Means and standard deviations

The mean of the test sample,

, is given by:

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

Equivalence limits

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:

Degrees of freedom (DF)

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

Pooled standard deviation

Notation

Term	Description
S_p	Pooled standard deviation
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

Confidence interval

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

Do not assume equal variances (default)

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

CI = [min(C, ρ_L), max(C, ρ_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:
CI = [ρ_L, ρ_U]

Assume equal variances

If you select the option to assume equal variances, then the CI is calculated as follows.

Minitab cannot calculate the CI if any of the following three conditions are not met:

100(1 -α)% CI
Minitab calculates the 100(1 - α)% CI as follows:

CI = [min(C, ρ_L, max(C, ρ_U)]
Where:
100(1 - 2 α)% CI
If you select the option to use the 100(1 - 2 α)% 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 test sample
	Mean of the reference sample
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
δ₁	Lower equivalence limit
δ₂	Upper equivalence limit
S_ρ	Pooled standard deviation
v	Degrees of freedom
α	Significance level for the test
t_1-α,v	Upper 1 - α critical value for a t-distribution with v degrees of freedom

T-values

Do not assume equal variances (default)

Let t₁ be the t-value for the hypothesis,

, and let t₂ be the t-value for the hypothesis,

, where Λ is the ratio of the mean of the test population to the mean of the reference population. By default, the t-values are calculated as follows:

Assume equal variances

If you select the option to assume equal variances, then the t-values are calculated as follows:

Notation

Term	Description
	Mean of the test sample
	Mean of the reference sample
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
S_ρ	Pooled standard deviation
δ₁	Lower equivalence limit
δ₂	Upper equivalence limit

P-values

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

If , then:

H₀	P-Value

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

Note

For information on how the t-values are calculated, see the section on t-values.