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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.
The degrees of freedom, v, is given by:
For power calculations, n is assumed to be the same for both groups.
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.
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 | sample size (For power calculations, n is assumed to be the same for both groups.) |
σ | standard deviation of the populations (For power calculations, σ is assumed to be the same for both populations.) |
This topic describes how power is calculated when you select Test mean / reference mean (Ratio, by log transformation) in Hypothesis about.
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.
The degrees of freedom, v, is given by:
For power calculations, n is assumed to be the same for both groups.
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:
The standard deviation, σ, is calculated using the coefficient of variation, CV, as follows:
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 | sample size (For power calculations, n is assumed to be the same for both groups.) |