Formula Reference

General power

$\text{Power} = 1 - \beta = P(\text{reject } H_0 \mid H_1)$

For many tests, power is computed from a noncentral distribution:

$\text{Power} = 1 - F_{H_1}(c) + \text{(one/two-tailed adjustment)}$

where $c$ is the critical value under $H_0$ and $F_{H_1}$ is the CDF under $H_1$ .

t tests

Cohen’s d (one sample / paired)

$d = \frac{\mu_1 - \mu_0}{\sigma}$

Noncentrality: $\delta = d\sqrt{n}$ (one sample, paired).

Two independent means

$d = \frac{\mu_1 - \mu_2}{\sigma}$

$\delta = d\sqrt{\frac{n_1 n_2}{n_1 + n_2}}$

Point-biserial correlation

Convert $\rho$ to $d$ : $d = \frac{2\rho}{\sqrt{1-\rho^2}}$ .

Linear regression slope

$\delta = \frac{(\beta_1 - \beta_0)\sqrt{n}}{\sigma_{\text{residual}}} \cdot \sqrt{\sigma_x^2}$

Generic t (direct NCP)

User supplies noncentrality parameter $\delta$ and $df$ directly.

Vignette: t Tests

F tests and ANOVA

Cohen’s f from $\eta^2$

$f = \sqrt{\frac{\eta^2}{1-\eta^2}}$

Noncentrality (omnibus)

$\lambda = f^2 \cdot N \quad \text{or} \quad \lambda = f^2 \cdot (N - k - 1)$

depending on the test (see registry method field).

Multiple regression $f^2$

$f^2 = \frac{R^2}{1-R^2}$

$R^2$ increase

$f^2 = \frac{R^2_{\text{full}} - R^2_{\text{reduced}}}{1 - R^2_{\text{full}}}$

Two variances

$F = \frac{\sigma_1^2}{\sigma_0^2}$

Power from noncentral F with numerator $df_1 = 1$ .

Vignette: ANOVA and Regression

Chi-square tests

One-sample variance

Test $\sigma^2$ against $\sigma_0^2$ via $\chi^2$ with $df = n-1$ .

Cohen’s w (Gof / contingency)

$w = \sqrt{\sum_i \frac{(p_{1i} - p_{0i})^2}{p_{0i}}}$

Noncentrality: $\lambda = N w^2$ .

Vignette: Exact and Proportions

Exact and proportion tests

Binomial / one proportion / sign test

$H_0: p = p_0 \quad \text{vs} \quad H_1: p = p_1$

Exact binomial enumeration or normal approximation for large $n$ .

Fisher exact (two proportions)

$H_0: p_1 = p_2$

Uses hypergeometric enumeration; normal Cohen’s $h$ fallback for large tables.

McNemar (approximation)

Discordant-pair binomial proxy on $n_{01}$ vs $n_{10}$ .

Vignette: Exact and Proportions

z tests

Independent correlations (Fisher Z)

$q = \text{FisherZ}(r_1) - \text{FisherZ}(r_2)$

$z = \frac{q}{\sqrt{\frac{1}{n_1-3}+\frac{1}{n_2-3}}}$

Dependent correlations (Steiger)

Uses covariance of Fisher-Z transformed correlations; see z_corr_dependent_* tests.

Logistic regression (Wald)

$z = \frac{\log(\text{OR})}{\text{SE}(\log(\text{OR}))}$

Poisson regression (Wald)

$z = \frac{\beta_1}{\text{SE}(\beta_1)}$

Tetrachoric (approximation)

Large-sample Fisher-Z on tetrachoric $\rho$ .

Vignette: Correlation and z Tests

Nonparametric tests

Asymptotic relative efficiency (ARE)

Wilcoxon tests map to equivalent $d$ via ARE:

$d_{\text{eff}} = d \cdot \sqrt{\text{ARE}}$

then reuse t-test noncentrality.

Vignette: Nonparametric Tests

Biomarker discovery

ROC AUC — Hanley-McNeil (one sample)

$\text{SE}(A) = \sqrt{\frac{A(1-A) + (n_1-1)(Q_1 - A^2) + (n_0-1)(Q_2 - A^2)}{n_1 n_0}}$

$z = \frac{A - A_0}{\text{SE}(A)}$

Two independent AUCs — DeLong-style

$z = \frac{A_1 - A_2}{\sqrt{\text{Var}(A_1) + \text{Var}(A_2)}}$

Diagnostic accuracy

Separate binomial tests for sensitivity and specificity; reported power is $\min(\text{Power}_{\text{sens}}, \text{Power}_{\text{spec}})$ .

Log-rank (Schoenfeld)

$z \approx \frac{|\log(\text{HR})|}{\sqrt{1/D}} \cdot \sqrt{D \cdot p(1-p)}$

Cox regression

$z \approx |\log(\text{HR})| \sqrt{E}$

where $E$ is expected events.

FDR screening

Benjamini-Hochberg at level $q$ ; power from independent two-sample $t$ tests with proportion $\pi_0$ true nulls.

Vignettes: ROC / AUC, Diagnostic accuracy

Clinical trials

Superiority (continuous)

One-sided two-sample $t$ : $\delta = d\sqrt{n_1 n_2/(n_1+n_2)}$ .

Superiority (binary)

One-sided Fisher / proportion test on $p_0$ vs $p_1$ .

Non-inferiority (continuous)

$H_0: \mu_T - \mu_C \le -\Delta \quad \text{vs} \quad H_1: \mu_T - \mu_C > -\Delta$

Shifted mean difference: $d_{\text{NI}} = d + \Delta/\sigma$ .

Non-inferiority (binary)

Normal approximation on $p_T - p_C + \Delta$ .

Equivalence — TOST

Two one-sided tests:

$\text{Power}_{\text{TOST}} \approx \text{Power}_{\text{upper}} \times \text{Power}_{\text{lower}}$

Simon two-stage

$P(\text{success}) = \sum_{k} P(\text{stage 1}) \cdot P(\text{stage 2} \mid \text{stage 1})$

under $H_0$ and $H_1$ response rates.

Cluster RCT design effect

$\text{DE} = 1 + (m - 1) \cdot \text{ICC}$

Effective $n_{\text{eff}} = n / \text{DE}$ ; then standard two-sample formulas.

Vignettes: Phase III superiority, Non-inferiority, Equivalence

Support matrix — all 48 tests with modes and parity
Effect size conversions
Analysis modes
Approximation catalog