Methods and formulas for Power and Sample Size for 2x2 Crossover Design Equivalence Test

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

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 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 sequences.

Noncentrality parameters

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

where σ is the within-subject standard deviation.

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:

where δ₂is the upper equivalence limit.

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

Notation

Term	Description
α	significance level for the test
D	mean of the test population minus the mean of the reference population
δ1	lower equivalence limit
δ2	upper equivalence limit
n	number of participants in each sequence. (For power calculations, n is assumed to be the same for both sequences.)

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

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 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 sequences.

Noncentrality parameters

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

where σ is the within-subject standard deviation as described below.

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 within-subject coefficient of variation (CV) as follows:

Notation

Term	Description
α	significance level for the test
ρ	ratio of the test population mean to the reference population mean
δ1	lower equivalence limit
δ2	upper equivalence limit
n	number of participants in each sequence. (For power calculations, n is assumed to be the same for both sequences.)

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.

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.

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.