From expected value of R2 in Regression to expected value of partial eta squared in ANOVA

Question

In general, $R^2$, the estimator of "multiple correlation coefficient" in regression, is known to be positively biased. Given $K$ predictors, and $N$ total sample size, Johnson, Kotz, & Balakrishnan, 1995, Vol. 2, p. 621, provide the correct expected value of $R^2$ as follows (Equation 1):

where $P^2$ is the population value of $R^2$, and $H$ is the hypergeometric function.

The distribution of $R^2$ is also known to be a non-central $F$ distribution with $K = df1$, $N - K - 1 = df2$, and the non-centrality parameter ($ncp$) of $(N \times R^2) / (1 - R^2)$.

Question:

Partial eta-squared $(\eta_p^2)$ in ANOVA is exactly distributed like $R^2$ except that $K -1 = df1$, $N - K = df2$, and the non-centrality parameter ($ncp$) of $(N \times \eta_p^2) / (1 - \eta_p^2)$.

My question is, is it possible to re-express the Equation 1 for partial eta-Squared?

Below, I'm showing an R implementation of Equation 1 for $R^2$ which I hope to be able modify to work for partial eta-Squared (I appreciate a possible R implementation).

library(gsl) # used for hypergeometric function

expected.R2 <- function(P2, K, N) {
  Value <- 1 - ((N - K - 1)/(N - 1)) * (1 - P2) * 
  gsl::hyperg_2F1(1, 1, 0.5 * (N + 1), P2)
  Value <- max(0, Value)
  return(Value)
}
# Example of use:
expected.R2(P2 = .2, K = 2, N = 120)

Answer 1

Here you have a degrees of freedom issue:

we can rerwite:

$$ \begin{aligned} &1-\frac{N-K-1}{N-1}(1-p^2)H[1;1;\frac{N+1}{2};P^2]\\ &1-\frac{N-df1-1}{N-1}(1-\eta^2)H[1;1;\frac{N+1}{2};\eta^2]\quad\text{Changed } K \text{ to } df1\\ &1-\frac{N-(K-1)-1}{N-1}(1-\eta^2)H[1;1;\frac{N+1}{2};\eta^2]\\ &1-\frac{N-K}{N-1}(1-\eta^2)H[1;1;\frac{N+1}{2};\eta^2] \end{aligned}$$

library(gsl) # used for hypergeometric function

expected.R2 <- function(P2, K, N) {
  Value <- 1 - ((N - K - 1)/(N - 1)) * (1 - P2) * 
  gsl::hyperg_2F1(1, 1, 0.5 * (N + 1), P2)
  Value <- max(0, Value)
  return(Value)
}
# Example of use:
expected.R2(P2 = .2, K = 2+1, N = 120)

or:

library(gsl) # used for hypergeometric function

expected.R2 <- function(P2, K, N) {
  Value <- 1 - ((N - K)/(N - 1)) * (1 - P2) * 
  gsl::hyperg_2F1(1, 1, 0.5 * (N + 1), P2)
  Value <- max(0, Value)
  return(Value)
}
# Example of use:
expected.R2(P2 = .2, K = 2, N = 120)