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Formulas: Beta (Type II Error Rate) for Hierarchical Multiple Regression

Beta (Type II Error Rate) for Hierarchical Multiple Regression Formulas

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:
Beta function formula
Effect size for hierarchical multiple regression:
Effect size for hierarchical multiple regression formula
where R2A is the variance accounted for by a set of one or more independent variables A, and R2AB is the combined variance accounted for by A and another set of one or more independent variables B.
Error function:
Error function formula
F-distribution cumulative distribution function (CDF):
F-distribution cumulative distribution function (CDF) formula
where d1 and d2 are the degrees of freedom, and I is the regularized lower incomplete beta function.
Lower incomplete beta function:
Lower incomplete beta function formula
Noncentral F-distribution cumulative distribution function (CDF):
Noncentral F-distribution cumulative distribution function (CDF) formula
where d1 and d2 are the numerator and denominator degrees of freedom, ? is the noncentrality parameter, F is the Fisher F-value, and I is the regularized lower incomplete beta function.
Noncentral F-distribution noncentrality parameter:
Noncentral F-distribution noncentrality parameter formula
where f2 is the effect size and n is the sample size.
Normal distribution cumulative distribution function (CDF):
Normal distribution cumulative distribution function (CDF) formula
where ยต is the mean, σ is the standard deviation, and erf is the error function.
Regularized lower incomplete beta function:
Regularized lower incomplete beta function formula
where the numerator is the lower incomplete beta function, and the denominator is the beta function.
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