Table 1.

Default choices for τ_ω²

Test	Statistic	Standardized effect (ω)	τ ω 2
1-sample z	$\frac{\sqrt{n}\overline{x}}{\sigma }$	$\frac{\mu }{\sigma }$	$\frac{n{\omega }^2}{2}$
1-sample t	$\frac{\sqrt{n}\overline{x}}{s}$	$\frac{\mu }{\sigma }$	$\frac{n{\omega }^2}{2}$
2-sample z	$\frac{\sqrt{n_1{n}_2}({\overline{x}}_1-{\overline{x}}_2)}{\sigma \sqrt{n_1+n_2}}$	$\frac{{\mu }_1-{\mu }_2}{\sigma }$	$\frac{n_1{n}_2{\omega }^2}{2(n_1+n_2)}$
2-sample t	$\frac{\sqrt{n_1{n}_2}({\overline{x}}_1-{\overline{x}}_2)}{s\sqrt{n_1+n_2}}$	$\frac{{\mu }_1-{\mu }_2}{\sigma }$	$\frac{n_1{n}_2{\omega }^2}{2(n_1+n_2)}$
Multinomial/Poisson	${\chi }_{\nu }^2=\sum _{i=1}^k\frac{{(n_i-n{f}_i(\widehat{\mathit{\theta }}))}^2}{n{f}_i(\widehat{\mathit{\theta }})}$	${\left(\frac{p_i-f_i(\mathit{\theta })}{\sqrt{f_i(\mathit{\theta })}}\right)}_{k\times 1}$	$\frac{n\mathit{\omega }\prime \mathit{\omega }}{k}=n{\tilde{\omega }}^2$
Linear model	$F_{k,n-p}=\frac{(RS{S}_0-RS{S}_1)/k}{[(RS{S}_1)/(n-p)]}$	$\frac{{\mathbf{L}}^{-1}(\mathbf{A}\mathit{\beta }-\mathbf{a})}{\sigma }$	$\frac{n\mathit{\omega }\prime \mathit{\omega }}{2k}=\frac{n{\tilde{\omega }}^2}{2}$
Likelihood ratio	${\chi }_k^2=-2\log \left[\frac{l({\mathit{\theta }}_{r0},\widehat{{\mathit{\theta }}_s})}{l(\widehat{\mathit{\theta }})}\right]$	${\mathbf{L}}^{-1}({\mathit{\theta }}_r-{\mathit{\theta }}_{r0})$	$\frac{n\mathbf{\omega }\prime \mathbf{\omega }}{k}=n{\tilde{\omega }}^2$

For one-sample tests, x₁, …, x_n are assumed to be iid N(μ, σ²), where n refers to sample size. In two-sample tests, x_{j, 1}, …, x_{j, nj}, j = 1, 2, are assumed to be iid N(μ_j, σ²). Integers n₁ and n₂ refer to sample sizes in each group. A bar over a variable denotes the sample mean. The variance of normal observations is denoted by σ² and is assumed to be equal in both groups in two-sample tests. Standard deviations are denoted by s and are the pooled estimate in the two-sample t test. In multinomial/Poisson tests, $f(\mathit{\theta })$ maps an s × 1 vector $\mathit{\theta }$ into a k × 1 probability vector, where k denotes the number of cells. The degrees of freedom ν equals k − s − 1. The quantities p_i and n_i represent cell probabilities and counts, respectively, and n is the sum of all cell counts. In the linear model, the alternative hypothesis is $\mathbf{A}\mathit{\beta }=\mathbf{a}$, where A is a k × p matrix of rank k, $\mathit{\beta }$ is a p × 1 vector of regression coefficients, and a is a k × 1 vector. The quantities RSS₀ and RSS₁ denote the residual sum of squares under the null and alternative hypotheses, respectively. The quantity n is the number of observations, and σ² is the observational variance. In the likelihood ratio test, l(⋅) denotes the likelihood function for a parameter vector $\mathit{\theta }=({\mathit{\theta }}_r,{\mathit{\theta }}_s)$. The k × 1 subvector ${\mathit{\theta }}_r$ equals ${\mathit{\theta }}_{r0}$ under the null hypothesis. The maximum likelihood estimate of $\mathit{\theta }$ under the alternative hypothesis is $\widehat{\mathit{\theta }}$, and the maximum likelihood of ${\mathit{\theta }}_s$ under the null hypothesis is ${\widehat{\mathit{\theta }}}_s$. In the linear model and likelihood ratio tests, the matrix L⁻¹ represents the Cholesky decomposition of the covariance matrix for the tested parameters, scaled to a single observation. Further explanation of τ_ω² values appear in SI Appendix.