Viechtbauer 2010 - Effect Size Measures and escalc

Summary

Before any model is fitted a meta-analyst needs, for every study, an effect size $y_i$ and its sampling variance $v_i$. escalc() computes these for 2x2 tables (RR, OR, RD, arcsine RD, Peto), raw/standardized mean differences, correlations (raw, unbiased, Fisher $z$) and proportions (raw, logit, arcsine, Freeman-Tukey), including zero-cell adjustments. Log or variance-stabilising transformations make the normality and known-variance assumptions of the model more plausible.

Overview

The models of Viechtbauer 2010 - Fixed, Random and Mixed-Effects Models take $(y_i,v_i)$ as data. Section 3.1 documents how these are produced. Signature:

escalc(measure, ai, bi, ci, di, n1i, n2i, m1i, m2i, sd1i, sd2i, xi, mi, ri, ni, data=NULL, add=1/2, to="only0", vtype="LS", append=FALSE)

with a second formula interface escalc(measure, formula, weights, data, add, to, vtype).

Main Content

2x2 table data

Each study gives cells $a_i,b_i$ (group 1: outcome 1, outcome 2), $c_i,d_i$ (group 2), with $n_{1i}=a_i+b_i$, $n_{2i}=c_i+d_i$. Designs: RCT (treatment/control), case-control, cross-sectional. Inputs: either $a_i,b_i,c_i,d_i$ or $a_i,c_i,n_{1i},n_{2i}$.

measure	Definition	Notes
"RR"	$\log \frac{a_i/n_{1i}}{c_i/n_{2i}}$	log relative risk
"OR"	$\log \frac{a_i{d}_i}{b_i{c}_i}$	log odds ratio
"RD"	$a_i/n_{1i}-c_i/n_{2i}$	risk difference
"AS"	$\arcsin \sqrt{a_i/n_{1i}}-\arcsin \sqrt{c_i/n_{2i}}$	arcsine-transformed RD (Rucker et al. 2009)
"PETO"	$\frac{a_i-s_i{n}_{1i}/n_i}{s_i{t}_i{n}_{1i}{n}_{2i}/(n_i^2(n_i-1))}$, $s_i=a_i+c_i$, $t_i=b_i+d_i$, $n_i=n_{1i}+n_{2i}$	Peto log OR; assumes true OR = 1 in all tables

Logs make RR and OR symmetric around 0 and closer to normal.

Zero cells

add (default 1/2) and to control continuity corrections: to="all" adds in all $k$ tables; "only0" (default) adds only in tables with a zero cell; "if0all" adds in all tables only if any table has a zero; "none" or add=0 means no adjustment (division by zero is recoded NA).

Mean differences

measure	Definition	Notes
"MD"	$m_{1i}-m_{2i}$	raw mean difference
"SMD"	$(m_{1i}-m_{2i})/s_{p,i}$	pooled SD $s_{p,i}$; automatically corrected for slight positive bias (Hedges and Olkin 1985); vtype="LS" large-sample variance, "UB" unbiased

Correlations

measure	Definition
"COR"	raw $r_i$; vtype LS or approximately unbiased UB (Hedges 1989)
"UCOR"	unbiased $r_i$, correcting slight negative bias (Olkin and Pratt 1958, eq. 2.7)
"ZCOR"	Fisher $z_i=\frac{1}{2}\log \frac{1+r_i}{1-r_i}$: variance-stabilising and normalising (Fisher 1921)

Proportions (single-group data, $x_i$ events of $n_i$)

measure	Definition
"PR"	$x_i/n_i$
"PLO"	$\log \frac{x_i}{n_i-x_i}$ (logit)
"PAS"	$\arcsin \sqrt{x_i/n_i}$
"PFT"	Freeman-Tukey double arcsine: $\frac{1}{2}\left[\arcsin \sqrt{\frac{x_i}{n_i+1}}+\arcsin \sqrt{\frac{x_i+1}{n_i+1}}\right]$

Zero-handling (add, to) applies to $x_i$ and $m_i=n_i-x_i$.

Standard error formulas used elsewhere in this folder

The paper does not print the variance formulas; two standard large-sample ones appear in the companion paper (Roever 2020 - Pediatric Liver Transplant Worked Example): log odds ratio $\text{SE}=\sqrt{1/a+1/b+1/c+1/d}$ and log relative risk $\text{SE}=\sqrt{1/a-1/(a+b)+1/c-1/(c+d)}$.

Formula interface

For 2x2 data in long format the formula is outcome ~ group | study with weights = cell frequencies (multipart formula via the Formula package). Long format required: one row per cell, columns study (k-level factor), grp (two-level factor), out (two-level factor) and freq. The advantage is generality for future multi-arm and multi-outcome structures; for ordinary one-row-per-study data the default interface is easier.

Examples

Log relative risk for BCG trial 1 (Aronson 1948)

Cells: treated TB+ $a=4$, TB- $b=119$ ($n_1=123$); control $c=11$, $d=128$ ($n_2=139$). $y_1=\log \frac{4/123}{11/139}=\log (0.03252/0.07914)=-0.8893$. Variance $v_1=\frac{1}{a}-\frac{1}{n_1}+\frac{1}{c}-\frac{1}{n_2}=0.25-0.00813+0.09091-0.00719=0.32558$, matching the printed yi=-0.88931133, vi=0.325584765. Negative values mean lower infection risk in the vaccinated group.

Connections

• In marketing measurement, the analogue of escalc is turning each experiment into a (lift estimate, standard error) pair on a scale where normality is credible: log relative lift, or iROAS with a delta-method SE. Meta-analysing raw ratios without a log transform is the same mistake as pooling raw odds ratios.
• The variance-stabilising transformations (Fisher $z$, arcsine) are what make “known $v_i$” a reasonable approximation, an assumption discussed in Viechtbauer 2010 - Technical Details, Validation and Package Comparison.

See Also

• Viechtbauer 2010 - BCG Vaccine Worked Example
• Viechtbauer 2010 - Conducting Meta-Analyses in R with the metafor Package - Overview