Methods and formulas for Power and Sample Size for 1-Sample Equivalence Test

Power

Let t_α,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 Lower limit < test mean - target < upper limit is given by:

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

For alternative hypotheses of Test mean < target or Test mean - target < 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:

Noncentrality parameters

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

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

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

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

Notation

Term	Description
α	significance level for the test
D	mean of the test population minus the target value
δ1	lower equivalence limit
δ2	upper equivalence limit
n	sample size
σ	standard deviation of the population

If you provide values for power and the difference, 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.

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.