Free Beta (Type II Error Rate) Calculator for Hierarchical Multiple Regression

The following formulas are involved in the calculation of beta (type II error rate) values for hierarchical multiple regression studies:

	Beta function:
	Effect size for hierarchical multiple regression: where R²_A is the variance accounted for by a set of one or more independent variables A, and R²_AB is the combined variance accounted for by A and another set of one or more independent variables B.
	Error function:
	F-distribution cumulative distribution function (CDF): where d₁ and d₂ are the degrees of freedom, and I is the regularized lower incomplete beta function.
	Lower incomplete beta function:
	Noncentral F-distribution cumulative distribution function (CDF): where d₁ and d₂ are the numerator and denominator degrees of freedom, λ is the noncentrality parameter, F is the Fisher F-value, and I is the lower incomplete beta function.
	Noncentral F-distribution noncentrality parameter: where f² is the effect size and n is the sample size.
	Normal distribution cumulative distribution function (CDF): where µ is the mean, σ is the standard deviation, and erf is the error function.
	Regularized lower incomplete beta function: where the numerator is the lower incomplete beta function, and the denominator is the beta function.

The following references can be used to cite this beta (type ii error rate) calculator for hierarchical multiple regression:

	Soper, D.S. (2012) "Beta (Type II Error Rate) Calculator for Hierarchical Multiple Regression (Online Software)", http://www.danielsoper.com/statcalc3.
	Abramowitz, M. and Stegun, I.A., eds. (1965), "Handbook of Mathematical Functions", Dover, New York, NY.
	Cohen, J. (1988) "Statistical Power Analysis for the Behavioral Sciences (2nd Edition)", Lawrence Earlbaum Associates, Hillsdale, NJ.
	Cohen, J., Cohen, P., West, S.G., and Aiken, L.S. (2003) "Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd edition)", Lawrence Earlbaum Associates, Mahwah, NJ.
	Raudenbush, S. and Bryk, A. (2002) "Hierarchical Linear Models (Second Edition)", Thousand Oaks: Sage Publications.

The following statistics calculators are related to this beta (type II error rate) calculator for hierarchical multiple regression, and may be useful for your research:

	A-priori Sample Size Calculator for Hierarchical Multiple Regression
	Effect Size Calculator for Hierarchical Multiple Regression
	F-Value Calculator for Hierarchical Multiple Regression
	Post-hoc Statistical Power Calculator for Hierarchical Multiple Regression