Below you will find descriptions and details for the 9 formulas that are used to compute sample sizes for hierarchical multiple regression studies.

Beta function:

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

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

Cumulative distribution function (CDF) for the normal distribution:

Effect size for hierarchical multiple regression studies:

R

R

Error function:

Lower incomplete beta function:

Noncentrality parameter for the noncentral F-distribution:

Regularized lower incomplete beta function:

the numerator = lower incomplete beta function

the denominator = beta function