Variance of the Squared Multiple Correlation Coefficient

Computes the sampling variance of the squared multiple correlation coefficient from the population value, the sample size, and the number of predictors, the quantity that governs how precisely \(R^2\) is estimated at a given design size.

Usage

var_R2(population_R2, N, p)

Arguments

population_R2: Population squared multiple correlation coefficient
N: Sample size
p: The number of predictor variables

Value

A 1-row data.frame with columns term and value. The term value is "var_R2" and value is the asymptotic variance of \(R^2\).

Details

Uses the hypergeometric function as discussed in and section 28 of Stuart, Ord, and Arnold (1999) in order to obtain the correct value for the variance of the squared multiple correlation coefficient.

Note

The Gauss hypergeometric function \({}_2F_1\) is computed in base R (see the internal .hyperg_2F1); no GSL system library is required.

References

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43, 524–555. doi:10.1080/00273170802490632

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3 on \(R^2\) as a model comparison effect size.)

Stuart, A., Ord, J. K., & Arnold, S. (1999). Kendall's advanced theory of statistics, volume 2A: Classical inference and the linear model (6th ed.). Arnold.

Author

Ken Kelley kkelley@nd.edu

Examples

var_R2(.5, 10, 5)
#>  term   value 
#>  var_R2 0.0268
var_R2(.5, 25, 5)
#>  term   value 
#>  var_R2 0.0169
var_R2(.5, 50, 5)
#>  term   value  
#>  var_R2 0.00926
var_R2(.5, 100, 5)
#>  term   value  
#>  var_R2 0.00482