C List of Symbols

 $$~$$ $$~$$ $$~$$ $$~$$ $$~$$ $$a,b,c,d$$ Events in the treatment group, non-events in the treatment group, events in the control group, non-events in the control group. $$\beta_0, \beta_1, \beta$$ Regression intercept, regression coefficient, Type II error rate. $$~$$ $$c_{\alpha}$$ Critical value assumed for the Type I error rate $$\alpha$$ (typically 1.96). $$\mathcal{HC}(x_0,s)$$ Half-Cauchy distribution with location parameter $$x_0$$ and scaling parameter $$s$$. $$~$$ $$\chi^2$$ Chi-squared statistic. $$\text{Cov}(x,y)$$ Covariance of $$x$$ and $$y$$. $$~$$ $$d$$ Cohen’s $$d$$ (standardized mean difference). $$D_g$$ Regression dummy. $$~$$ $$\delta$$ Non-centrality parameter (non-central $$t$$ distribution). d.f. Degrees of freedom. $$~$$ $$\epsilon$$ Sampling error. $$F$$ Snedecor’s $$F$$ statistic (used by the $$F$$-tests in ANOVAs). $$~$$ $$g$$ Small sample bias-corrected standardized mean difference (Hedges’ $$g$$). $$I^2$$ Higgins’ and Thompson’s $$I^2$$ measure of heterogeneity (percentage of variation not attributable to sampling error). $$~$$ $$\int f(x) dx$$ Integral of $$f(x)$$. $$k,K$$ Some study in a meta-analysis, total number of studies in a meta-analysis. $$~$$ $$\kappa$$ True effect of an effect size cluster. $$MD$$, $$SMD$$ (Standardized) mean difference (Cohen’s $$d$$). $$~$$ $$\bar{x}$$ Arithmetic mean (based on an observed sample), identical to $$m$$. $$\mu, m$$ (True) population mean, sample mean. $$~$$ $$n,N$$ (Total) sample size of a study. $$\mathcal{N}(\mu, \sigma^2)$$ Normal distribution with population mean $$\mu$$ and variance $$\sigma^2$$. $$~$$ $$\Phi(z)$$ Cumulative distribution function (CDF), where $$z$$ follows a standard normal distribution. $$\pi, p$$ True population proportion, proportion based on an observed sample. $$~$$ $$P(\text{X}|\text{Y})$$ Conditional probability of X given Y. $$\hat\psi$$ (Estimate of) Peto’s odds ratio, or some other binary effect size. $$~$$ $$Q$$ Cochran’s $$Q$$ measure of heterogeneity. $$RR$$, $$OR$$, $$IRR$$ Risk ratio, odds ratio, incidence rate ratio. $$~$$ $$\hat{R}$$ R-hat value in Bayesian modeling. $$R^2_*$$ $$R^2$$ (explained variance) analog for meta-regression models. $$~$$ $$\rho, r$$ True population correlation, observed correlation. $$SE$$ Standard error $$~$$ $$\sigma^2$$ (True) population variance. $$t$$ Student’s $$t$$ statistic. $$~$$ $$\tau^2, \tau$$ True heterogeneity variance and standard deviation. $$\theta$$ A true effect size, or the true value of an outcome measure. $$~$$ $$V$$, $$v$$, $$s^2$$, $$\widehat{\text{Var}}(x)$$ Sample variance (of $$x$$), where $$s$$ is the standard deviation. $$w$$, $$w^*$$, $$w(x)$$ (Inverse-variance) weight, random-effects weight of an effect size, function that assigns weights to $$x$$. $$~$$ $$z$$ Fisher’s $$z$$ or $$z$$-score. $$\zeta, u$$ Error’’ due to between-study heterogeneity, random effect in (meta-)regression models. $$~$$

Note. Vectors and matrices are written in bold. For example, we can denote all observed effect sizes in a meta-analysis with a vector $$\boldsymbol{\hat\theta} = (\hat\theta_1, \hat\theta_2, \dots, \hat\theta_K)^\top$$, where $$K$$ is the total number of studies. The $$\top$$ symbol indicates that the vector is transposed. This means that elements in the vector are arranged vertically instead of horizontally. This is sometimes necessary to do further operations with the vector, for example multiplying it with another matrix.