Computes the variance of the sample standardized mean difference (Cohen's d) under bivariate normality and homogeneous variances, using the exact noncentral t sampling distribution (Hedges, 1981) as the default and reporting the large-sample Hedges-Olkin (1985) approximation alongside for comparison. Optionally returns the variance of Hedges' g, the bias-corrected counterpart of d.
Arguments
- delta
Population standardized mean difference. Numeric scalar or vector.
- n_1
Sample size in group 1. Scalar or vector.
- n_2
Sample size in group 2. Scalar or vector. Defaults to
n_1(balanced design).- unbiased
Logical. If
TRUE, returns the variance of Hedges' g (the bias-corrected estimator); ifFALSE(the default), returns the variance of Cohen's d.
Value
A data.frame with rows for the exact (noncentral-
t) variance and the Hedges-Olkin large-sample approximation.
Columns are term ("var_smd_exact" or
"var_smd_approx") and value.
Details
var_smd() is a stand-alone variance utility: most R packages
return only the Hedges-Olkin large-sample approximation and conflate
the variances of d and g (Goulet-Pelletier & Cousineau,
2018). The drift between the exact and approximate forms becomes
non-trivial below about \(n = 30\) per group and matters whenever
var_smd() feeds into meta-analytic weighting, AIPE planning, or
a Wald-style standard-error report.
Exact noncentral t form. For
\(\hat d = (\bar Y_1 - \bar Y_2)/s_p\) with pooled \(s_p\), the
rescaled statistic \(t = \hat d \sqrt{n_1 n_2 / (n_1 + n_2)}\)
follows a noncentral t with \(\mathit{df} = n_1 + n_2 - 2\)
degrees of freedom and noncentrality parameter
\(\lambda = \delta \sqrt{n_1 n_2 / (n_1 + n_2)}\). The variance of a
noncentral t is (Johnson, Kotz, & Balakrishnan, 1995, Sec.\ 31.3)
$$\mathrm{Var}(t) \;=\;
\frac{\mathit{df}\,(1 + \lambda^2)}{\mathit{df} - 2} \,-\,
\lambda^2 \, c(\mathit{df})^{2},$$
where \(c(\mathit{df}) = \sqrt{\mathit{df}/2}\,
\Gamma((\mathit{df}-1)/2)\,/\,\Gamma(\mathit{df}/2)\); dividing by the
design factor \(n_1 n_2 / (n_1 + n_2)\) returns \(\mathrm{Var}(\hat d)\).
For Hedges' g, multiply the result by \(J(\mathit{df})^2\)
where \(J(\mathit{df}) = 1/c(\mathit{df})\) is the Hedges-Olkin
(1985) bias-correction factor (see expected_smd).
Hedges-Olkin large-sample approximation. The frequently quoted approximation (Hedges & Olkin, 1985, equation 8) is $$\mathrm{Var}(\hat d) \;\approx\; \frac{n_1 + n_2}{n_1 n_2} \;+\; \frac{\delta^2}{2(n_1 + n_2 - 2)}.$$ This approaches the exact form only as the degrees of freedom grow: even at \(\delta = 0\) it returns \(1/(n_1 n_2 / (n_1 + n_2))\) while the exact noncentral t variance is \([\mathit{df}/(\mathit{df} - 2)]/(n_1 n_2 / (n_1 + n_2))\), so the approximation is biased downward by a factor of \((\mathit{df} - 2)/\mathit{df}\), and the downward bias grows with \(\delta\) and small \(n\). Goulet-Pelletier & Cousineau (2018) document the drift and recommend the exact form for \(n < 30\) per group.
Companions. var_smd() is the variance partner of
expected_smd (mean) and ci_smd (CI). For
design-stage AIPE planning that solves for \(n\) given a target CI
width on d, see ss_aipe_smd.
References
Goulet-Pelletier, J.-C., & Cousineau, D. (2018). A review of effect sizes and their confidence intervals, Part I: The Cohen's d family. The Quantitative Methods for Psychology, 14(4), 242–265. doi:10.20982/tqmp.14.4.p242
Hedges, L. V. (1981). Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128.
Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Academic Press.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (2nd ed.). Wiley.
Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in parameter estimation via narrow confidence intervals. Psychological Methods, 11(4), 363–385. doi:10.1037/1082-989X.11.4.363
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 individual comparisons and Chapter 3 on one-way ANOVA.)
See also
smd, ci_smd,
expected_smd, ss_aipe_smd
Other variance utilities:
var_alpha(),
var_cv(),
var_ete(),
var_indirect_effect(),
var_omega_squared(),
var_r(),
var_smd_trimmed()
Author
Ken Kelley kkelley@nd.edu
Examples
# 1. Balanced design, delta = 0.5, n = 20 per group.
var_smd(delta = 0.5, n_1 = 20)
#> term value
#> var_smd_exact 0.109
#> var_smd_approx 0.103
# 2. Hedges-Olkin approximation drifts from exact when n is small.
var_smd(delta = 0.5, n_1 = 5)
#> term value
#> var_smd_exact 0.56
#> var_smd_approx 0.416
var_smd(delta = 0.5, n_1 = 50)
#> term value
#> var_smd_exact 0.0422
#> var_smd_approx 0.0413
# 3. Variance of Hedges' g (bias-corrected).
var_smd(delta = 0.5, n_1 = 20, unbiased = TRUE)
#> term value
#> var_smd_exact 0.105
#> var_smd_approx 0.0992