Multiple Regression Type 2 Error Rate (Beta) Formulas

Multiple Regression Type 2 Error Rate (Beta) Formulas

Below you will find descriptions and details for the 9 formulas that are used to compute type 2 error rates (false negative rates) for multiple regression studies.

Beta function:

Cumulative distribution function (CDF) for the F-distribution:

Variable definitions:
d₁ and d₂ = the degrees of freedom
I = regularized lower incomplete beta function

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

Variable definitions:
d₁ and d₂ = the numerator and denominator degrees of freedom
λ = noncentrality parameter
F = Fisher F-value
I = regularized lower incomplete beta function

Cumulative distribution function (CDF) for the normal distribution:

Variable definitions:
µ = mean
σ = standard deviation
erf = error function

Effect size for an F-test (Cohen's f²):

Effect size for an F-test (Cohen's f<sup>2</sup>) formula

Variable definitions:
R² = squared multiple correlation

Error function:

Lower incomplete beta function:

Noncentrality parameter for the noncentral F-distribution:

Variable definitions:
f² = effect size
n = sample size

Regularized lower incomplete beta function:

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