Asymptotic Variance of the Partial Correlation Coefficient

Computes the large-sample variance of the sample partial correlation coefficient $r_{XY \cdot Z_1 \cdots Z_J}$ under multivariate normality. Two formulas are available: the classical asymptotic variance on the raw $r$ scale (the default), and the variance $1/(n - J - 3)$ of the Fisher's Z transformation (Fisher, 1921, 1924), which is the appropriate quantity for constructing a confidence interval by transformation and back-transformation.

Usage

var_partial_r(r, n, J = 1, fisher_z = FALSE)

Arguments

r: The sample partial correlation coefficient $r_{XY \cdot Z_1 \cdots Z_J}$. Must be in $[-1, 1]$.
n: Total sample size.
J: Number of variables partialled out (i.e., the count of $Z_1, \ldots, Z_J$); must be at least 1. Defaults to 1.
fisher_z: Logical. If FALSE (the default), the function returns the raw-scale asymptotic variance. If TRUE, it returns the variance of the Fisher's Z transformation $Z = \mathrm{arctanh}(r)$ under the Fisher (1924) reduction.

Value

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

Details

Raw-scale asymptotic variance. Under multivariate normality the partial correlation $\hat r_{XY \cdot Z}$ has the large-sample variance (Fisher, 1924, for the reduction; the simple-correlation building block is, e.g., Olkin & Finn, 1995, their Equation 3): $$\mathrm{Var}(\hat r_{XY \cdot Z}) \;\approx\; \frac{(1 - \rho^2_{XY \cdot Z})^2}{n - J - 1},$$ a direct generalization of the classical asymptotic variance $(1 - \rho^2)^2 / (n - 1)$ of the simple Pearson correlation (Fisher, 1915) with the degrees of freedom reduced by the number of partialled variables. The function evaluates this with $\hat r$ substituted for $\rho$.

Fisher's Z Transformation. Fisher (1921) showed that for a Pearson correlation, the transformation $Z = \tfrac{1}{2}\log\{(1+r)/(1-r)\} = \mathrm{arctanh}(r)$ is approximately normal with variance $1/(n - 3)$. Fisher (1924) showed that the partial correlation based on $n$ observations with $J$ variables partialled out is distributed as a simple correlation from a sample reduced in size by $J$; combined with the Fisher (1921) variance of $Z$, the Fisher's Z transformation of $\hat r_{XY \cdot Z_1 \cdots Z_J}$ is therefore approximately normal with variance $1/(n - J - 3)$. This is the standard ingredient for constructing a confidence interval on $\rho_{XY \cdot Z}$ by transforming, building a Wald interval on $Z$, and back-transforming with $\tanh$.

References

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. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10(4), 507–521.

Fisher, R. A. (1921). On the "probable error" of a coefficient of correlation deduced from a small sample. Metron, 1, 3–32.

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.)

Olkin, I., & Finn, J. D. (1995). Correlations redux. Psychological Bulletin, 118(1), 155–164. doi:10.1037/0033-2909.118.1.155

Author

Ken Kelley kkelley@nd.edu

Examples

# Olkin-Finn (1995) asymptotic variance for r_p = .35, n = 80,
#     J = 2 control variables.
var_partial_r(r = 0.35, n = 80, J = 2)
#>  term          value
#>  var_partial_r 0.01 

# Variance of Fisher's Z transformation (Fisher, 1921, 1924) for the
#     same setting, useful for building a CI on rho_p.
var_partial_r(r = 0.35, n = 80, J = 2, fisher_z = TRUE)
#>  term                   value 
#>  var_fisher_z_partial_r 0.0133