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Asymptotic Variance of the Pearson Correlation Coefficient

Source: R/var_r.R
var_r.Rd

Computes the asymptotic (large-sample) variance of the sample Pearson product-moment correlation \(r\) under bivariate normality (Fisher, 1915) and, optionally, the Bonett-Wright (2000) kurtosis-corrected variance for non-normal margins. Stand-alone variance utility: surprisingly absent from CRAN despite being a building block for AIPE planning, meta-analytic weighting, and Wald-style inference on \(r\).

Usage

var_r(rho, n, kurtosis_x = NULL, kurtosis_y = NULL)

Arguments

rho

Population correlation coefficient. Numeric scalar or vector in \((-1, 1)\).

n

Total sample size on which the Pearson \(r\) would be computed. Scalar or vector.

kurtosis_x

Optional excess kurtosis of the marginal distribution of \(X\). When kurtosis_x and kurtosis_y are both supplied (along with rho_xy_2x2y if available), the Bonett-Wright (2000) corrected variance is returned in addition to the normal-theory variance.

kurtosis_y

Optional excess kurtosis of \(Y\); see kurtosis_x.

Value

A data.frame with the rows

  • var_r_normal, the normal-theory asymptotic variance \((1 - \rho^2)^2 / (n - 1)\) (Hotelling, 1953, Section 7),

  • var_r_bonett_wright (when kurtoses are supplied) the Bonett & Wright (2000) kurtosis-corrected variance,

  • var_fisher_z, the variance of the Fisher \(Z\) transform, \(1/(n - 3)\).

Details

Normal-theory variance (default). Under bivariate normality the large-sample variance is $$\mathrm{Var}(\hat r) \;\approx\; (1 - \rho^2)^2 / (n - 1),$$ the leading term of the exact moment expansion (Hotelling, 1953, Section 7; the exact density of \(r\) is Fisher's, 1915). This is the workhorse variance and is exact in the limit; it is also what Fisher's \(Z\) CI ci_r uses on the transformed scale.

Bonett-Wright kurtosis correction. When the marginals are not normal, the asymptotic variance picks up a kurtosis- dependent correction (Bonett & Wright, 2000): $$\mathrm{Var}(\hat r) \;\approx\; (1 - \rho^2)^2 / (n - 1) \cdot \bigl(1 + \rho^2 (\gamma_2^{(X)} + \gamma_2^{(Y)}) / 4\bigr),$$ where \(\gamma_2^{(X)}\), \(\gamma_2^{(Y)}\) are the excess kurtoses of the two marginals. The correction is exact when the joint distribution is elliptical; for non-elliptical joints it is a first-order approximation. When the kurtosis arguments are NULL, only the normal-theory variance is returned.

Connection to Fisher's Z transform. On the variance- stabilized scale \(Z = \tanh^{-1}(r)\) the asymptotic variance is \(1/(n-3)\) regardless of \(\rho\) (Fisher, 1921). This is reported alongside the raw-scale variance because it is the natural working scale for CI construction (ci_r).

References

Bonett, D. G., & Wright, T. A. (2000). Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrika, 65(1), 23–28. doi:10.1007/BF02294183

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.

Hotelling, H. (1953). New light on the correlation coefficient and its transforms. Journal of the Royal Statistical Society, Series B, 15(2), 193–232. (Section 7 gives the exact moments of \(r\).)

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.

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

See also

expected_r, ci_r, var_partial_r, var_semipartial_r

Other variance utilities: var_alpha(), var_cv(), var_ete(), var_indirect_effect(), var_omega_squared(), var_smd(), var_smd_trimmed()

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Normal-theory variance:
var_r(rho = 0.30, n = 50)
#>  term         value 
#>  var_r_normal 0.0169
#>  var_fisher_z 0.0213

# 2. With Bonett-Wright correction for leptokurtic margins
#        (excess kurtosis = 3 for each variable):
var_r(rho = 0.30, n = 50, kurtosis_x = 3, kurtosis_y = 3)
#>  term                value 
#>  var_r_normal        0.0169
#>  var_fisher_z        0.0213
#>  var_r_bonett_wright 0.0192