Free Post-hoc Statistical Power Calculator for Multiple Regression

The following formulas are involved in the calculation of post-hoc statistical power values for multiple regression studies:

	Beta function:
	Cohen's f² effect size for an F-test: where R² is the squared multiple correlation.
	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 post-hoc statistical power calculator for multiple regression:

	Soper, D.S. (2012) "Post-hoc Statistical Power Calculator for Multiple Regression (Online Software)", http://www.danielsoper.com/statcalc3.
	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.

The following statistics calculators are related to this post-hoc statistical power calculator for multiple regression, and may be useful for your research:

	A-priori Sample Size Calculator for Multiple Regression
	Adjusted R-square Calculator (Population R-square)
	Beta (Type II Error Rate) Calculator for Multiple Regression
	Confidence Interval Calculator for an f-square Effect Size
	Critical F-value Calculator
	Effect Size Calculator for Multiple Regression
	F-Value and p-Value Calculator for Multiple Regression
	Post-hoc Statistical Power Calculator for a Student t-Test
	Post-hoc Statistical Power Calculator for Hierarchical Multiple Regression
	R-square Calculator (from an f-square Effect Size)
	R-square Confidence Interval Calculator
	Regression Coefficient Confidence Interval Calculator
	Regression Intercept Confidence Interval Calculator