Expected Value of the Squared Multiple Correlation Coefficient

Computes the expected value of the observed squared multiple correlation coefficient given the population squared multiple correlation coefficient, the sample size, and the number of predictors. The sample \(R^2\) is a positively biased estimator of its population value, and the expected value quantifies how large that bias is for a particular design.

Usage

expected_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 "expected_value_population_R2" and value is the expected value of \(R^2\) under random sampling.

Details

Uses the hypergeometric function as discussed in section 28 of Stuart, Ord, and Arnold (1999) in order to obtain the correct value for the squared multiple correlation coefficient. Many times an exact value is given that ignores the hypergeometric function. This function yields the correct value.

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.)

Olkin, I., & Pratt, J. W. (1958). Unbiased estimation of certain correlation coefficients. The Annals of Mathematical Statistics, 29(1), 201–211.

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

expected_R2(.5, 10, 5)
#>  term                         value
#>  expected_value_population_R2 0.754
expected_R2(.5, 25, 5)
#>  term                         value
#>  expected_value_population_R2 0.588
expected_R2(.5, 50, 5)
#>  term                         value
#>  expected_value_population_R2 0.542
expected_R2(.5, 100, 5)
#>  term                         value
#>  expected_value_population_R2 0.52 
expected_R2(.5, 1000, 5)
#>  term                         value
#>  expected_value_population_R2 0.502
expected_R2(.5, 10000, 5)
#>  term                         value
#>  expected_value_population_R2 0.5