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B Effect Size Formulas


Effect Size (Λ†ΞΈ) Standard Error (SE) Function
Arithmetic Mean (3.2.1) Λ‰x=βˆ‘ni=1xin SEΛ‰x=s√n mean
Proportion (3.2.2) p=kn SEp=√p(1βˆ’p)n
Proportion (3.2.2) plogit=loge(p1βˆ’p) SEplogit=√1np+1n(1βˆ’p)
Correlation
Product-Moment Correlation (3.2.3.1) rxy=Οƒ2xyΟƒxΟƒy SErxy=1βˆ’r2xy√nβˆ’2 cor
Product-Moment Correlation (3.2.3.1) z=0.5loge(1+r1βˆ’r) SEz=1√nβˆ’3
Point-Biserial Correlation1 (3.2.3.2) rpb=(Β―y1βˆ’Β―y2)√n1N(1βˆ’n1N)sy cor
(Standardized) Mean Difference
Between-Group Mean Difference (3.3.1.1) MD=Λ‰x1βˆ’Λ‰x2 SEMD=spooledβˆ—βˆš1n1+1n2
Between-Group Standardized Mean Difference (3.3.1.2) SMD=Λ‰x1βˆ’Λ‰x2spooledβˆ— SESMD=√n1+n2n1n2+SMD2between2(n1+n2) esc_mean_sd
Within-Group Mean Difference (3.3.1.3) MD=Λ‰xt2βˆ’Λ‰xt1 SEMD=√s2t2+s2t1βˆ’(2rt1t2st1st2)n
Within-Group Standardized Mean Difference (3.3.1.3) SMD=Λ‰xt2βˆ’Λ‰xt1st1 SESMD=√2(1βˆ’rt1t2)n+SMD2within2n
Binary Outcome Effect Size
Risk Ratio (3.3.2.1) petreat=antreat SElogRR=√1a+1cβˆ’1a+bβˆ’1c+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=SMDΓ—(1βˆ’34nβˆ’9) hedges_g
Unreliability (3.4.2) rxyc=rxy√rxx SEc=SE√rxx
Unreliability (3.4.2) rxyc=rxy√rxx√ryy SEc=SE√rxx√ryy
Unreliability (3.4.2) SMDc=SMD√rxx
Range Restriction (3.4.3) U=sunrestrictedsrestricted SErxyc=rxycrxySErxy
Range Restriction (3.4.3) rxyc=UΓ—rxy√(U2βˆ’1)r2xy+1
Range Restriction (3.4.3) SMDc=UΓ—SMD√(U2βˆ’1)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=√(n1βˆ’1)s21+(n2βˆ’1)s22(n1βˆ’1)+(n2βˆ’1).