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Below you will find descriptions and details for the 9 formulas that are used to compute beta (type II error rate) values for hierarchical multiple regression studies.

Beta function:

Effect size for hierarchical multiple regression:

where R

Error function:

F-distribution cumulative distribution function (CDF):

where

Lower incomplete beta function:

Noncentral F-distribution cumulative distribution function (CDF):

where

Noncentral F-distribution noncentrality parameter:

where

Normal distribution cumulative distribution function (CDF):

where

Regularized lower incomplete beta function:

where the numerator is the lower incomplete beta function, and the denominator is the beta function.

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