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B Effect Size Formulas


Effect Size (ˆθ) Standard Error (SE) Function
Arithmetic Mean (3.2.1) ˉx=ni=1xin SEˉx=sn mean
Proportion (3.2.2) p=kn SEp=p(1p)n
Proportion (3.2.2) plogit=loge(p1p) SEplogit=1np+1n(1p)
Correlation
Product-Moment Correlation (3.2.3.1) rxy=σ2xyσxσy SErxy=1r2xyn2 cor
Product-Moment Correlation (3.2.3.1) z=0.5loge(1+r1r) SEz=1n3
Point-Biserial Correlation1 (3.2.3.2) rpb=(¯y1¯y2)n1N(1n1N)sy cor
(Standardized) Mean Difference
Between-Group Mean Difference (3.3.1.1) MD=ˉx1ˉx2 SEMD=spooled1n1+1n2
Between-Group Standardized Mean Difference (3.3.1.2) SMD=ˉx1ˉx2spooled SESMD=n1+n2n1n2+SMD2between2(n1+n2) esc_mean_sd
Within-Group Mean Difference (3.3.1.3) MD=ˉxt2ˉxt1 SEMD=s2t2+s2t1(2rt1t2st1st2)n
Within-Group Standardized Mean Difference (3.3.1.3) SMD=ˉxt2ˉxt1st1 SESMD=2(1rt1t2)n+SMD2within2n
Binary Outcome Effect Size
Risk Ratio (3.3.2.1) petreat=antreat SElogRR=1a+1c1a+b1c+d
Risk Ratio (3.3.2.1) pecontrol=cncontrol
Risk Ratio (3.3.2.1) RR=petreatpecontrol
Risk Ratio (3.3.2.1) logRR=loge(RR)
Odds Ratio (3.3.2.2) Oddstreat=ab SElogOR=1a+1b+1c+1d esc_2x2
Odds Ratio (3.3.2.2) Oddscontrol=cd
Odds Ratio (3.3.2.2) OR=a/bc/d
Odds Ratio (3.3.2.2) logOR=loge(OR)
Incidence Rate Ratio (3.3.3) IRR=Etreat/TtreatEcontrol/Tcontrol SElogIRR=1Etreat+1Econtrol
Incidence Rate Ratio (3.3.3) logIRR=loge(IRR)
Effect Size Correction
Small Sample Bias (3.4.1) g=SMD×(134n9) hedges_g
Unreliability (3.4.2) rxyc=rxyrxx SEc=SErxx
Unreliability (3.4.2) rxyc=rxyrxxryy SEc=SErxxryy
Unreliability (3.4.2) SMDc=SMDrxx
Range Restriction (3.4.3) U=sunrestrictedsrestricted SErxyc=rxycrxySErxy
Range Restriction (3.4.3) rxyc=U×rxy(U21)r2xy+1
Range Restriction (3.4.3) SMDc=U×SMD(U21)SMD2+1 SESMDc=SMDcSMDSESMD
1 Point-biserial correlations may be converted to SMDs for meta-analysis (see Chapter 3.2.3.2).
* The pooled standard deviation is defined as spooled=(n11)s21+(n21)s22(n11)+(n21).