Effect sizes

Effect sizes measure how large a difference or association is, in raw or standardised units, independently of the sample size.

d = \frac{\xbar - \mu_0}{s}, \qquad d = \frac{\xbar_1 - \xbar_2}{s_p}, \qquad d_z = \frac{\bar d}{s_d}

Cohen's d: one sample, two independent samples (pooled SD), paired samples.

$s$: Sample standard deviation (n − 1).
$s_p$: Pooled SD $\sqrt{((n_1-1)s_1^2 + (n_2-1)s_2^2)/(n_1+n_2-2)}$ .
$\bar d, s_d$: Mean and SD of the paired differences.

g = J(m)\,d,\qquad J(m) = \frac{\Gamma(m/2)}{\sqrt{m/2}\,\Gamma((m-1)/2)} \approx 1 - \frac{3}{4m-1}

Hedges' g: d with a small-sample bias correction (the engine uses the exact J by default).

$m$: Degrees of freedom $n_1 + n_2 - 2$ .

\text{RD} = \phat_1 - \phat_2,\quad \text{RR} = \frac{\phat_1}{\phat_2},\quad \text{OR} = \frac{ad}{bc},\quad h = 2\arcsin\sqrt{\phat_1} - 2\arcsin\sqrt{\phat_2}

Two proportions: risk difference, relative risk, odds ratio, Cohen's h.

$\phat_1, \phat_2$: Proportions with the outcome in groups 1 and 2.
$a, b, c, d$: Cells of the 2×2 table [[a, b], [c, d]] (rows = groups, first column = outcome present).

\eta^2 = \frac{SS_{\text{effect}}}{SS_{\text{total}}},\qquad \omega^2 = \frac{SS_{\text{effect}} - df_{\text{effect}}\,MS_{\text{error}}}{SS_{\text{total}} + MS_{\text{error}}},\qquad V = \sqrt{\frac{X^2}{N(\min(r,c)-1)}}

ANOVA (η², ω²) and contingency tables (Cramér's V, using the uncorrected Pearson X²).

$SS, MS, df$: Sums of squares, mean squares and degrees of freedom from the ANOVA table.
$X^2, N, r, c$: Pearson chi-square statistic, total count, numbers of rows and columns.

r_{rb} = \frac{2U_1}{n_1 n_2} - 1,\qquad r_{rb}^{\text{paired}} = \frac{W^+ - W^-}{W^+ + W^-}

Rank-biserial correlations for the Mann–Whitney and Wilcoxon signed-rank tests.

$U_1$: Mann–Whitney U of sample 1.
$W^+, W^-$: Sums of ranks of positive and negative differences.

When to use

Always, alongside the p-value. Prefer raw effects in meaningful units (minutes, SAR, percentage points) for decisions; use standardised effects (d, g, h, η², V) to compare across studies or to plan sample sizes.

Assumptions

Standardised effects depend on the SD used (pooled vs control-group SD, raw vs paired), so always say which one. η² and ω² describe the sample design they came from.

How to interpret

Cohen's rough benchmarks for d (0.2 small, 0.5 medium, 0.8 large) and h (same) are a last resort; judge size in context.
Relative risk 1.25 = "25% higher risk"; risk difference 0.03 = "3 percentage points higher". Report both: a big relative change of a rare outcome can be a tiny absolute change.
Odds ratios approximate relative risks only when the outcome is rare.
η² is the share of total variation explained by groups in the sample; ω² is a less biased estimate for the population (it can be slightly negative).
Rank-biserial r ranges from −1 to 1: for Mann–Whitney it is $2\hat P(X > Y) - 1$ (ties counting ½).
Report a confidence interval for the effect when possible.

Common mistakes

Equating "significant" with "large".
Reporting only relative changes ("doubles the risk") without the baseline.
Comparing paired $d_z$ with independent-groups d: they use different SDs.
Calling an odds ratio a risk ratio when the outcome is common.

Example

Two groups (26.5 vs 22.0, SDs 2.43 and 2.16, n = 6 and 7): $d = 1.97$ , Hedges' $g = 1.83$ .
Conversion 150/1000 vs 120/1000: RD = 0.03 (3 points), RR = 1.25, OR = 1.29, $h = 0.088$ (tiny on Cohen's scale, yet possibly valuable commercially).
One-way ANOVA with SS between 8, SS total 14 (3 groups of 4): $\eta^2 = 0.571$ , $\omega^2 = 0.455$ .
A 2×3 table with $X^2 = 4$ , $N = 150$ : $V = 0.163$ .