Asymptotic Variance of the Semipartial (Part) Correlation Coefficient

Computes the large-sample variance of the sample semipartial (also known as the part) correlation coefficient $r_{Y(X \cdot Z_1 \cdots Z_J)}$ under multivariate normality. The function provides two variances: the asymptotic variance under the alternative (the partial-correlation form applied by analogy) and, when the full-model coefficient of multiple determination $R^2_{Y \cdot X Z_1 \cdots Z_J}$ is supplied, the exact null-hypothesis variance derived from the multiple-regression F-test for the unique contribution of $X$ (Cohen, Cohen, West, & Aiken, 2003).

Usage

var_semipartial_r(r_sp, n, J = 1, R2_full = NULL)

Arguments

r_sp: Sample semipartial correlation coefficient $r_{Y(X \cdot Z_1 \cdots Z_J)}$, with the controlled variables partialled out of $X$ only (not $Y$). Must be in $[-1, 1]$.
n: Total sample size.
J: Number of variables partialled out of $X$ (i.e., the count of $Z_1, \ldots, Z_J$); must be at least 1. Defaults to 1.
R2_full: Optional coefficient of multiple determination $R^2_{Y \cdot X Z_1 \cdots Z_J}$ from the full model that includes $X$ and all controls. When supplied, the null-hypothesis variance $(1 - R^2)/(n - J - 2)$ is returned instead of the alternative-side asymptotic variance. Must be in $[0, 1]$.

Value

A one-row data.frame with columns term (either "var_semipartial_r" or "var_semipartial_r_under_null") and value (the requested variance).

Details

Background. The squared semipartial $r^2_{Y(X \cdot Z)}$ equals the increase in $R^2$ when $X$ is added to a model already containing the controls $Z_1, \ldots, Z_J$, i.e., the unique variance in $Y$ attributable to $X$. Unlike the partial, the semipartial is on the original scale of $Y$ rather than on the partialled scale, which makes it the natural effect size companion to standardized regression coefficients in multiple-regression reports (Cohen et al., 2003).

Asymptotic variance (default). Under multivariate normality the semipartial admits the same large-sample form as the partial (Fisher, 1924, applied by analogy): $$\mathrm{Var}(\hat r_{Y(X \cdot Z)}) \;\approx\; \frac{(1 - \rho^2_{Y(X \cdot Z)})^2}{n - J - 1}.$$ The function evaluates this with $\hat r_{sp}$ substituted for $\rho_{sp}$. This is the appropriate quantity for Wald-style inference and for AIPE-style precision planning analogous to that of the partial correlation. Aloe and Becker (2012) develop the asymptotic variance of the semipartial as a function of the full population correlation structure, and Yuan and Chan (2011) give exact higher-order results for the closely related standardized regression coefficients; the present approximation matches the leading $1/n$ behavior.

Null-hypothesis variance (when R2_full is supplied). In multiple regression the unique contribution of $X$ is tested with $$F \;=\; \frac{r^2_{Y(X \cdot Z)}\,(n - J - 2)}{1 - R^2_{Y \cdot X Z}} \;\sim\; F(1,\, n - J - 2)$$ under $H_0\!: \rho_{Y(X \cdot Z)} = 0$ (Cohen et al., 2003, equation 3.7.3). Equivalently $t = \hat r_{sp}\,\sqrt{(n - J - 2)/(1 - R^2_{Y \cdot X Z})}$ is a t-statistic on $n - J - 2$ degrees of freedom, so the under-the-null variance of $\hat r_{sp}$ is $$\mathrm{Var}_0(\hat r_{sp}) \;=\; \frac{1 - R^2_{Y \cdot X Z}}{n - J - 2}.$$ Supplying R2_full returns this null variance, which is the standard ingredient for testing the significance of $X$'s unique contribution.

References

Aloe, A. M., & Becker, B. J. (2012). An effect size for regression predictors in meta-analysis. Journal of Educational and Behavioral Statistics, 37(2), 278–297. doi:10.3102/1076998610396901

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum.

Fisher, R. A. (1924). The distribution of the partial correlation coefficient. Metron, 3, 329–332.

Kelley, K., & Maxwell, S. E. (2003). Sample size for multiple regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8(3), 305–321. doi:10.1037/1082-989X.8.3.305

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 4 on contrasts, Chapter 5 on multiple comparisons, and Chapter 9 on ANCOVA.)

Yuan, K.-H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76(4), 670–690. doi:10.1007/s11336-011-9224-6

Author

Ken Kelley kkelley@nd.edu

Examples

# Olkin-Finn-style asymptotic variance of a semipartial r = .25 with
#     n = 100 and J = 3 controls in X.
var_semipartial_r(r_sp = 0.25, n = 100, J = 3)
#>  term              value  
#>  var_semipartial_r 0.00916

# With R^2 of the full model also supplied, the function returns the
#     null-hypothesis variance used in the F-test for X's unique
#     contribution.
var_semipartial_r(r_sp = 0.25, n = 100, J = 3, R2_full = 0.42)
#>  term                         value  
#>  var_semipartial_r_under_null 0.00611