Signal to Noise Estimators for the Squared Multiple Correlation Coefficient

Computes five estimators of the population signal to noise ratio \(\phi^2 = \rho^2 / (1 - \rho^2)\) associated with the squared multiple correlation coefficient \(\rho^2\). Two are functions of the unadjusted and the Wherry-adjusted sample \(R^2\); the other three are the Muirhead (1985) unique minimum variance unbiased estimators that improve substantially on the plug-in estimator at small N and modest numbers of predictors.

Usage

signal_to_noise_R2(R2, N, p)

Arguments

R2: The usual sample estimate of the squared multiple correlation coefficient (no degrees of freedom adjustment). Numeric scalar in \((0, 1)\).
N: Sample size.
p: Number of predictor variables.

Value

A data.frame with columns term and value and one row per estimator: phi2_hat, phi2_adj_hat, phi2_umvue, phi2_umvue_l, and phi2_umvue_nl.

Details

The signal to noise ratio \(\phi^2 = \rho^2 / (1 - \rho^2)\) is a natural reparameterization of \(\rho^2\) that is bounded only below (at zero) and so behaves more like a variance ratio than a proportion. It is also the noncentrality parameter (up to a factor of N) for the omnibus F-test of \(\rho^2 = 0\) under fixed predictors; see convert_R2_f.

The five estimators returned, in increasing order of bias-correction machinery, are:

phi2_hat: the plug-in estimator \(\hat\phi^2 = R^2 / (1 - R^2)\). Biased upward in small samples because the sample \(R^2\) is itself biased upward.
phi2_adj_hat: the plug-in estimator applied to the Wherry-adjusted \(R^2\). Removes the leading-order bias in \(R^2\) but is not itself unbiased for \(\phi^2\).
phi2_umvue: Muirhead's (1985) unique minimum variance unbiased estimator (his \(\theta_U\), their Eq. 4); equivalent to Stuart, Ord, and Arnold's (1999) equation 28.97. Requires \(N \ge p + 6\) (the gate on all three Muirhead estimators); for smaller \(N\) the value is NA.
phi2_umvue_l: Muirhead's (1985) linearly-improved unique minimum variance unbiased estimator (his \(\theta_L\)); equivalent to Stuart et al.\ (1999) equation 28.98. Dominates phi2_umvue in mean squared error.
phi2_umvue_nl: Muirhead's (1985) nonlinearly-improved estimator (his \(\theta_{NL}\)). Dominates the linear improvement in MSE but requires \(p \ge 5\); for smaller \(p\) the value is NA.

The nonlinear estimator dominates the linear one in risk when \(p \ge 5\), though Muirhead notes the two perform very similarly in practice; for smaller \(p\) the linear estimator is preferred over the plug-in and adjusted-\(R^2\) forms. All three Muirhead estimators are reported truncated at zero, so the returned value is on the same scale as \(\phi^2\) (the truncation introduces a negligible bias only when the population \(\phi^2\) is near zero).

As \(N\) grows with \(p\) fixed, the five estimators converge to a common value (the population \(\phi^2\)); the difference between them is the small-sample bias machinery in operation. The @examples block illustrates that convergence.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

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.)

Muirhead, R. J. (1985). Estimating a particular function of the multiple correlation coefficient. Journal of the American Statistical Association, 80, 923–925.

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

# 1. Fixed R^2 = 0.5 and p = 2, growing N: the five estimators agree
#    to within a small fraction once N is moderate.
signal_to_noise_R2(R2 = .5, N = 50,   p = 2)
#>  term          value
#>  phi2_hat      1    
#>  phi2_adj_hat  0.918
#>  phi2_umvue    0.878
#>  phi2_umvue_l  0.806
#>  phi2_umvue_nl <NA> 
signal_to_noise_R2(R2 = .5, N = 100,  p = 2)
#>  term          value
#>  phi2_hat      1    
#>  phi2_adj_hat  0.96 
#>  phi2_umvue    0.939
#>  phi2_umvue_l  0.901
#>  phi2_umvue_nl <NA> 
signal_to_noise_R2(R2 = .5, N = 500,  p = 2)
#>  term          value
#>  phi2_hat      1    
#>  phi2_adj_hat  0.992
#>  phi2_umvue    0.988
#>  phi2_umvue_l  0.98 
#>  phi2_umvue_nl <NA> 

# 2. With p = 5 the nonlinear estimator is available; it differs
#    most from the plug-in at small N.
signal_to_noise_R2(R2 = .5, N = 50,  p = 5)
#>  term          value
#>  phi2_hat      1    
#>  phi2_adj_hat  0.796
#>  phi2_umvue    0.755
#>  phi2_umvue_l  0.691
#>  phi2_umvue_nl 0.692
signal_to_noise_R2(R2 = .5, N = 500, p = 5)
#>  term          value
#>  phi2_hat      1    
#>  phi2_adj_hat  0.98 
#>  phi2_umvue    0.976
#>  phi2_umvue_l  0.968
#>  phi2_umvue_nl 0.968