# Methods and formulas for Power and Sample Size for 2-Sample Equivalence Test

Select the method or formula of your choice.

## Calculating power for test mean – reference mean (Difference)

This topic describes how power is calculated when you select Test mean - reference mean (Difference) in Hypothesis about.

### Power

Let tα,v be the upper α (one-sided) critical value for a t-distribution with v degrees of freedom. The power for the two-sided alternative hypothesis of Lower limit < test mean - reference mean < upper limit is given by:

For the alternative hypothesis of Test mean > reference mean or Test mean - reference mean > lower limit, the power is given by:

For the alternative hypothesis of Test mean < reference mean or Test mean - reference mean < upper limit, the power is given by:

where CDF( x; v, λ) is the cumulative distribution function, evaluated at x, for a noncentral t-distribution with noncentrality parameter, λ, and v degrees of freedom.

### Degrees of freedom

The degrees of freedom, v, is given by:

For power calculations, n is assumed to be the same for both groups.

### Noncentrality parameters

The noncentrality parameter that corresponds to the lower equivalence limit is denoted as λ1, and is given by:

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

The noncentrality parameter that corresponds to the upper equivalence limit is denoted as λ2, and is given by the following formula:

For the alternative hypothesis of Test mean < reference mean, δ2 = 0.

### Notation

TermDescription
αsignificance level for the test
Dmean of the test population minus the mean of the reference population
δ1lower equivalence limit
δ2upper equivalence limit
nsample size (For power calculations, n is assumed to be the same for both groups.)
σstandard deviation of the populations (For power calculations, σ is assumed to be the same for both populations.)

## Calculating power for test mean / reference mean (Ratio, by log transformation)

This topic describes how power is calculated when you select Test mean / reference mean (Ratio, by log transformation) in Hypothesis about.

### Power

Let tα,n be the upper α (one-sided) critical value for a t-distribution with v degrees of freedom. The power for the two-sided alternative hypothesis of Lower limit < test mean / reference mean < upper limit is given by:

For the alternative hypothesis of Test mean / reference mean > lower limit, the power is given by:

For the alternative hypothesis of Test mean / reference mean < upper limit, the power is given by:

where CDF(x; v , λ) is the cumulative distribution function, evaluated at x, for a noncentral t-distribution with noncentrality parameter, λ, and v degrees of freedom.

### Degrees of freedom

The degrees of freedom, v, is given by:

For power calculations, n is assumed to be the same for both groups.

### Noncentrality parameters

The noncentrality parameter that corresponds to the lower equivalence limit is denoted as λ1, and is given by:

The noncentrality parameter that corresponds to the upper equivalence limit is denoted as λ2 , and is given by:

### Sigma

The standard deviation, σ, is calculated using the coefficient of variation, CV, as follows:

### Notation

TermDescription
αsignificance level for the test
ρratio of the test population mean to the reference population mean
δ1lower equivalence limit
δ2upper equivalence limit
nsample size (For power calculations, n is assumed to be the same for both groups.)

## Calculating sample size

If you provide values for power and the difference (or ratio), Minitab calculates the sample size. Minitab uses the appropriate power formula and an iterative algorithm to identify the smallest sample size, n, for which the power is greater than or equal to the specified value. The actual power for n is likely to be greater than the specified power. This is because n must be a discrete integer value, and no value n is likely to yield exactly the specified power value.

## Calculating the difference

If you provide values for power and sample size, Minitab calculates values for the difference. Minitab uses the appropriate power formula and an iterative algorithm to identify the largest and/or smallest difference for which the power is greater than or equal to the specified value.

## Calculating the ratio

If you provide values for power and sample size, Minitab calculates values for the ratio. Minitab uses the appropriate power formula and an iterative algorithm to identify the largest and/or smallest ratio for which the power is greater than or equal to the specified value.
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