t-Test Type 2 Error Rate (Beta) Formulas

Below you will find descriptions and details for the 7 formulas that are used to compute type 2 error rates (false negative rates) for t-tests.

Beta function:

Cumulative distribution function (CDF) for the noncentral t-distribution:

Variable definitions:
μ = noncentrality parameter
v = degrees of freedom
Γ = gamma function
I = regularized lower incomplete beta function

Effect size for a t-test (Cohen's d):

Variable definitions:
x₁ and x₂ = the means of group 1 and group 2
σ₁² and σ₂² = the variances of group 1 and group 2

Gamma function:

Lower incomplete beta function:

Noncentrality parameter for the noncentral t-distribution:

Variable definitions:
d = Cohen's d effect size
n₁ and n₂ = the sample sizes for group 1 and group 2

Regularized lower incomplete beta function:

Variable definitions:
the numerator = lower incomplete beta function
the denominator = beta function