Unbiased and Adjusted Estimators of the Population Squared Multiple Correlation

Estimates the population squared multiple correlation coefficient $\rho^2$ from an observed sample $R^2$, correcting the well-known positive (upward) bias of $R^2$. Two estimators are available: the (essentially) unbiased Olkin and Pratt (1958) estimator (the default), and the classic Ezekiel (1930) adjusted-$R^2$ shrinkage formula reported by summary.lm as adj.r.squared. This is the inverse direction of expected_R2, which gives the forward expectation $E[R^2 \mid \rho^2]$.

Usage

unbiased_R2(R2, N, p, method = c("olkin_pratt", "ezekiel"))

Arguments

R2: Observed sample squared multiple correlation coefficient, in $[0, 1]$.
N: Sample size.
p: Number of predictor variables.
method: Which estimator to compute: "olkin_pratt" (default) for the (essentially) unbiased Olkin-Pratt (1958) estimator, or "ezekiel" for the Ezekiel (1930) adjusted $R^2$ (the value summary.lm reports as adj.r.squared).

Value

A 1-row data.frame (class dmar_tbl) with columns term and value. The term is "unbiased_population_R2" when method = "olkin_pratt" and "adjusted_population_R2" when method = "ezekiel"; value is the corresponding estimate of $\rho^2$.

Details

The sample $R^2$ overestimates $\rho^2$; the bias is larger for smaller samples and for more predictors. Two corrections are offered.

The Ezekiel (1930) adjusted estimator is $$\hat\rho^2_{\mathrm{Ezekiel}} = 1 - \frac{N - 1}{N - p - 1}\,(1 - R^2).$$ It reduces the bias but is not unbiased; it is exactly the quantity summary(lm(...))$adj.r.squared reports.

The Olkin and Pratt (1958) estimator is (essentially) unbiased: $$\hat\rho^2_{\mathrm{OP}} = 1 - \frac{N - 3}{N - p - 1}\,(1 - R^2)\; {}_2F_1\!\left(1, 1; \frac{N - p + 1}{2}; 1 - R^2\right),$$ where ${}_2F_1$ is the Gaussian hypergeometric function (the same function expected_R2 uses for the forward direction; see Stuart, Ord, & Arnold, 1999, section 28). Both estimators can fall below 0 for very small $R^2$; that is expected behavior for a bias-corrected estimator and is not truncated here (matching adj.r.squared, which is also allowed to be negative).

References

Ezekiel, M. (1930). Methods of correlation analysis. Wiley.

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43(4), 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

# An observed R^2 = .50 with N = 50 and p = 5 predictors overstates rho^2.
# The Olkin-Pratt (essentially unbiased) estimate is the default:
unbiased_R2(R2 = .50, N = 50, p = 5)
#>  term                   value
#>  unbiased_population_R2 0.454

# The Ezekiel adjusted R^2 (what summary(lm) reports) over-shrinks slightly,
# so it typically sits a little below the Olkin-Pratt value:
unbiased_R2(R2 = .50, N = 50, p = 5, method = "ezekiel")
#>  term                   value
#>  adjusted_population_R2 0.443

# The Ezekiel option reproduces summary(lm)$adj.r.squared exactly.
set.seed(113)
d   <- as.data.frame(matrix(rnorm(50 * 6), 50, 6))
fit <- lm(V1 ~ ., data = d)
s   <- summary(fit)
unbiased_R2(R2 = s$r.squared, N = 50, p = 5, method = "ezekiel")$value
#> [1] -0.09483852
s$adj.r.squared
#> [1] -0.09483852

# The bias (and so the correction) shrinks as N grows for fixed R^2 and p.
unbiased_R2(.50, 50, 5)
#>  term                   value
#>  unbiased_population_R2 0.454
unbiased_R2(.50, 500, 5)
#>  term                   value
#>  unbiased_population_R2 0.496