Asymptotic Variance of the Coefficient of Variation

Computes the asymptotic variance of the sample coefficient of variation \(\hat\kappa = s / \bar Y\) under normality, using McKay's (1932) original noncentral t-based approximation and Vangel's (1996) refinement. Companion to ci_cv and ss_aipe_cv.

Usage

var_cv(cv, n)

Arguments

cv

Population coefficient of variation \(\kappa = \sigma / \mu\). Numeric scalar in \((0, \infty)\). When \(\kappa\) is very large (> 0.5), both McKay's and Vangel's approximations degrade and exact methods (ci_cv) should be preferred.

n

Sample size.

Value

A data.frame with rows for the McKay (1932) and Vangel (1996) approximations; columns are term and value.

Details

McKay (1932). The classical large-sample variance of the sample CV under normality is $$\mathrm{Var}(\hat\kappa) \;\approx\; \frac{\kappa^2}{n - 1} \cdot \left(\frac{1}{2} + \kappa^2\right).$$ This is exact up to \(O(1/n)\) and is what most planning tables use. It begins to drift when \(\kappa > 0.3\) or so.

Vangel (1996). Vangel showed that a small-sample correction that adjusts the McKay form for the noncentral t mean factor gives substantially better coverage of CIs derived from the variance: $$\mathrm{Var}_{\mathrm{Vangel}}(\hat\kappa) \;\approx\; \frac{\kappa^2}{n - 1} \cdot \left(\frac{1}{2} + \kappa^2 \cdot \frac{n + 1}{n - 1}\right).$$ The two forms coincide in the large-\(n\) limit. We report both so the user can see the magnitude of the small-sample correction.

References

Kelley, K. (2007). Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach. Behavior Research Methods, 39(4), 755–766. doi:10.3758/BF03192966

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3.)

McKay, A. T. (1932). Distribution of the coefficient of variation and the extended t distribution. Journal of the Royal Statistical Society, 95(4), 695–698.

Vangel, M. G. (1996). Confidence intervals for a normal coefficient of variation. The American Statistician, 50(1), 21–26. doi:10.1080/00031305.1996.10473537

See also

ci_cv, ss_aipe_cv, ss_aipe_cv_sensitivity

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

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. CV = 0.20 in a sample of 30:
var_cv(cv = 0.20, n = 30)
#>  term          value   
#>  var_cv_mckay  0.000745
#>  var_cv_vangel 0.000749

# 2. The Vangel correction grows with cv (becomes non-trivial
#        for kappa > 0.3):
var_cv(cv = 0.50, n = 30)
#>  term          value  
#>  var_cv_mckay  0.00647
#>  var_cv_vangel 0.00661