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Tutorials!in!Quantitative!Methods!for!Psychology! 2007,!Vol.!3(2),!p.! 51 "59.!
A!short!tutorial!of! GPower!
! Susanne!Mayr! Edgar!Erdfelder! ! Heinrich"Heine"Universität,!Düsseldorf,!Germany! Universität!Mannheim,!Mannheim,!Germany! ! ! Axel!Buchner! Franz!Faul! ! Heinrich"Heine"Universität,!Düsseldorf,!Germany! Christian"Albrechts"Universität,!Kiel,!Germany! ! The!purpose!of!this!paper!is!to!promote!statistical!power!analysis!in!the!behavioral!sciences!by!introducing! the! easy! to! use! GPower! software.! GPower! is! a! free! general! power! analysis! program! available! in! two! essentially! equivalent! versions,! one! designed! for! Macintosh! OS/OS! X! and! the! other! for! MS"DOS/Windows! platforms.! Psychological! research! examples! are! presented! to! illustrate! the! various! features! of! the! GPower! software.! In! particular,! a! priori,! post"hoc,! and! compromise! power! analyses! for! t "tests,! F "tests,! and!!^2 "tests! will!be!demonstrated.!For!all!examples,!the!underlying!statistical!concepts!as!well!as!the!implementation!in! GPower !will!be!described.! ! " (^) In!the!behavioral!sciences,!we!routinely!apply!statistical!
tests,! but! control! of! statistical! power! cannot! be! taken! for! granted.! However,! neglecting! statistical! power—the! probability! of! rejecting! false! null! hypotheses—can! have! severe! consequences.! For! example,! without! control! of! statistical! power! it! is! very! difficult! to! interpret! nonsignificant! results.! Statistical! tests! can! produce! nonsignificant! results! because! (a)! the! null! hypothesis! ( H 0 )! holds! and! is! retained! correctly! or! (b)! the! alternative! hypothesis! ( H 1 )! holds! but! the! test! has! not! been! powerful! enough!to!detect!the!deviations!from! H 0.!Obviously,!there!is! no!reasonable!way!to!decide!between!interpretations!(a)!and!
!
"! Susanne!Mayr,!Department!of!Experimental!Psychology,!Heinrich"
Heine"University,! Düsseldorf,! Germany;! Edgar! Erdfelder,! Department! of! Psychology,! Mannheim! University,! Mannheim,! Germany;! Axel! Buchner,! Department! of! Experimental! Psychology,! Heinrich"Heine"University,! Düsseldorf,! Germany;! Franz! Faul,! Department! of! Psychology,! Christian"Albrechts"University,! Kiel,! Germany.! Correspondence! concerning! this! article! should! be! addressed!to!Susanne!Mayr,!Institut!für!Experimentelle!Psychologie,! Heinrich"Heine"Universität,! D" 40225! Düsseldorf,! Germany.! Electronic! mail! may! be! sent! to! susanne.mayr@uni"duesseldorf.de.! This! work! is! based! on! the! German! language! tutorials! by! Buchner,! Erdfelder,!and!Faul!(1996)!and!Erdfelder,!Buchner,!Faul,!and!Brandt! (2004).!
(b)! when! the! power! of! the! test! is! unknown.! As! a! result! of! neglecting! statistical! power! analyses,! null! results! are! published! only! rarely.! Thus,! the! publication! of! research! findings! is! biased! in! favor! of! H 1! hypotheses! (Bredenkamp,! 1972,!1980).!! The!omission!of!power!control!is!frequently!justified!by! the! argument! that! power! analyses! are! too! complex! to! perform.!The! GPower !software_!_^1 !(Erdfelder,!Faul,!&!Buchner,!
- 1 "^! presented! in! this! article! should! largely! render! this! argument! obsolete.! GPower! is! an! easy! to! use! program! for! performing!various!types!of!power!analysis.!This!paper!tries! to! familiarize! readers! with! the! concept! of! statistical! power! analysis!in!general!and!with! GPower !in!particular.!
Types!of!power!analyses! Different! types! of! power! analysis! can! be! distinguished! with!respect!to!their!intended!purposes.!We!want!to!present! the! two! most! common! types—a! priori! and! post"hoc! power! analysis—as! well! as! a! third! variant,! compromise! power!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!
1! GPower! is! free! and! may! be! downloaded! from!
http://www.psycho.uni"duesseldorf.de/aap/projects/gpower.! Note! that!this!tutorial!refers!to! GPower !Version!2.!By!now,!Version! 3 !(Faul,! Erdfelder,!Lang,!&!Buchner,!2007)!is!already!available!via!the!same! weblink.! Version! 3! comprises! an! extended! functionality! which!
might!be!worthwhile!for!the!interested!reader.!
!
analysis.! All! three! types! can! be! accomplished! with! the! GPower !software.!! An!a!priori!analysis!is!done!before!a!study!takes!place.!It! is!the!ideal!type!of!power!analysis!because!it!provides!users!
with!a!method!to!control!both!the!type" 1 !error!probability! #!
(i.e.,!the!probability!of!incorrectly!rejecting! H 0 !when!is!in!fact!
true)!and!the!type" 2 !error!probability! $!(i.e.,!the!probability!
of! incorrectly! retaining! H 0! when! it! is! in! fact! false).! By! implication,!it!also!controls!the!power!of!the!test,!that!is,!the!
complement! of! the! type" 2! error! probability! (1! "! $)! (i.e.,! the!
probability!of!correctly!rejecting! H 0 !when!it!is!in!fact!false).! An! a! priori! analysis! is! used! to! determine! the! necessary! sample! size! N! of! a! test! given! a! desired! #! level,! a! desired! power! level! (1! "! # ),! and!the!size!of!the!effect!to!be!detected! (i.e.,!a!measure!of!the!difference!between!the! H 0 !and!the! H 1).! In! contrast,! a! post"hoc! analysis! is! typically! performed! after !a!study!has!been!conducted!so!that!the!sample!size! N !is! already! a! matter! of! fact.! Given! N ,! !,! and! a! specified! effect! size,!this!type!of!analysis!returns!the!power!(1!–!! ),!or!the!!! error! probability! of! the! test.! Obviously,! post"hoc! analyses!
are! less! ideal! than! a"priori! analyses! because! only! #! is!
controlled,! not! $.! Both! $! and! its! complement! (1! "! $)! are!
assessed! but! not! controlled! in! post"hoc! analyses.! Thus,! post" hoc! power! analyses! can! be! characterized! as! instruments! providing!for!a!critical!evaluation!of!the!(often!surprisingly! large)!error!probability!! !associated!with!a!false!decision!in! favor!of!the! H 0.!! The! third! type! of! power! analysis! provided! by! GPower ,! compromise! power! analysis,! provides! a! pragmatic! solution! to!the!frequently!encountered!problem!that!the!ideal!sample! size! N !calculated!by!an!a"priori!power!analysis!exceeds!the! available! resources! (Erdfelder,! 1984).! For! example,! clinical! investigators! are! sometimes! interested! in! diseases! or! disorders!of!a!very!low!prevalence!for!which!the!number!of! available! participants! is! small.! In! spite! of! these! suboptimal! circumstances,!a!fair!decision!between! H 0 !and! H 1 !is!possible.! For! this! situation,! a! reasonable! compromise! between! a! preferably!small!#!and!a!preferably!large!power!(1!–! # )!has! to!be!found.!To!this!end,!a!decision!has!to!be!made!of!how! important! # !should!be!in!comparison!to!#.!This!weighting!is! expressed!by!the!factor! q !( q !=! #! /!#).!Based!on! N ,! q ,!and!the! specified! effect! size,! the! compromise! power! analysis! then! determines! #! and! # ,! and! the!associated!critical!value!of!the! relevant! test! statistic.! In! other! words,! compromise! power!
analyses! control! the! error! probability! ratio! q! =! $/#.! Both! #!
and! $! are! assessed! given! a! fixed! error! probability! ratio! q .!
Note! that! compromise! analyses! can! also! be! very! useful! when! the! available! N! is! “too! large”.! For! example,! in! goodness"of"fit! tests,! very! large! sample! sizes! are! not! unusual.!Under!these!conditions,!even!negligible!deviations! of! the! empirical! data! structure! from! the! data! structure! implied! by! the! model! ( H 0 )! may! lead! to! model! rejections! if!
conventional!significance!levels!like! #!=!.05!are!used.!In!such!
situations,!compromise!power!analyses!provide!users!with!a! method!to!find!more!reasonable,!strict!decision!criteria!such! that! effect! sizes! of! interest! are! detected! with! balanced!
probabilities! #!and! $!consistent!with!the!user"defined!error!
probability!ratio! q !=! $/#.!
Examples!of!statistical!power!analyses!with!GPower! We! will! present! examples! of! statistical! power! analysis! for! the! three! most! often! applied! statistical! tests! in!
psychological! research,! that! is! t ",! F ",! and!!^2 "tests.! We! will!
describe!how!to!obtain!calculations!of!sample!size!(in!case!of! a! priori! analyses),! statistical! power! (in! case! of! post"hoc! analyses),! and!!! and_! !!_ values! (in! case! of! compromise! analyses)! using! the! GPower! program.! GPower! exists! in! two! versions! that! are! equivalent! in! their! numerical! implementation.! One! version! is! MS"DOS! compatible! and! may! be! run! under! Windows;! the! other! version! has! been! designed! for! Mac! OS! 7! to! 9! and! may! be! run! in! the! classic! mode!of!Mac!OS!X.!All!explanations!and!figures!refer!to!the! Macintosh!version;!however,!given!that!the!user!interface!of! the! two! versions! is! very! similar,! no! difficulties! should! emerge! in! following! the! descriptions! for! users! of! the! MS" DOS!version^2 "^ .!
Power!analyses!for!t"tests!
Independent!samples!t"test! A!frequently!cited!study!by!Warrington!and!Weiskrantz! (1970,!Experiment!2)!compared!the!memory!performance!of! amnesic!patients!with!that!of!control!subjects.!In!addition!to! commonly! used! direct! memory! tests,! such! as! a! recall! test,! indirect!memory!measures,!such!as!a!word"stem!completion! test,! were! used.! Indirect! tests! are! thought! to! measure!after" effects!of!experiences!without!giving!the!explicit!instruction! to! remember.! Whereas! the! amnesic! patients! performed! worse! than! controls! in! the! recall! test! (means! of! 8! vs.! 13),! there! was! no! significant! difference! between! the! groups! in! the!word"stem!completion!test!(means!of!14.5!vs.!16).!! Do! these! results! prove! that! amnesics! are! as! good! as! controls!in!indirect!test!performance,!at!least!with!respect!to! word"stem! completion?! Looking! at! the! sample! means,! we! note! a! difference! between! amnesics! and! controls! in! the! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!! (^2)! Program!users!can!select!between!an!accuracy!mode!and!a!speed! mode! (see! the! «I! prefer…»! option! in! Figure! 1).! Whereas! accuracy! mode!calculations!are!based!on!the!actual!noncentral!distribution!of! the!relevant!test!statistic,!speed!mode!calculations!approximate!this! distribution! by! other! types! of! distributions.! The! computational! capability! of! present"day! computers! allows! using! the! accuracy! mode!without!hesitation.!
=!(! 1 !"!! 2 ) /! ("!into!(! 1 !"!! 2 ) = d !× " !and!by!inserting!the!values!
of!the!example,!2.1694!×! 3 !=!6.5082).!This!result!implies!that!a! population! mean! difference! not! less! than! 6.5082! words! in! favor! of! the! control! group! would! have! been! necessary! to! achieve!a!power!of!.95.!! Alternatively,!if!we!want!to!detect!an!effect!of!size! d !=!0.5!
with! n 1! =! 4,! n 2! =! 8,! and! equally! large! #! and!!! error!
probabilities! ( q! =! 1),! the! «Compromise»! option! has! to! be! chosen! as! type! of! power! analysis! (see! Figure! 2).! Here,! we! specify! the! «Beta/alpha! ratio»! as! “1”! if! we! consider! both! types!of!error!as!equally!serious.^4 "^ !Then! GPower !returns!!!=!!! =! .3422! (associated! with! a! critical! value! of! t (10)! =! 0.4186).! Under! the! prevailing! circumstances,! choosing! this! significance!level!is!the!best!possible!decision.!However,!this! statistical! test! is! hardly! any! better! than! tossing! a! coin! to! decide!whether!to!accept!or!reject! H 0 .!!
Paired!samples!t"test!
In! succession! of! Gesell! and! Thompson’s! (1929)! work,! a! number! of! experiments! with! monozygotic! pairs! of! twins! have! been! conducted! giving! one! randomly! chosen! twin! training!of!specific!motor!skills!while!the!other!one!did!not! obtain! any! training! program.! This! allowed! for! a! controlled! investigation! of! whether! certain! abilities! (e.g.! learning! to! walk,! control! of! the! bladder)! develop! in! a! process! of! maturation! or! whether! environmental! influences! can! promote!or!impair!this!development.! Imagine! that! we! want! to! replicate! such! a! twin! study! which!shall!be!analyzed!with!a!paired!samples! t" test.!Let!us! further!assume!that!there!is!only!a!pool!of! 20 !pairs!of!twins! available.!What!are!reasonable!error!probabilities!we!have!to! accept!for!our!statistical!test?! X! and! Y! denote! the! age! at! which! the! trained! and! the! untrained! twin,! respectively,! will! master! a! special! motor! ability.! H 0! of! the! one"tailed! paired! samples! t "test! is! characterized! by! " x"y! =! " x "! " y! %! 0,! with! " x"y! denoting! the! population! mean! of! the! age! differences! of! each! twin! pair.! The!effect!size! d z!is!defined!as:!
! (^2 2 2 ) 2cov 2
x y x y x y z x y (^) x y xy x y x y
d
with! # x"y! being! the! standard! deviation! of! the! ( X! –! Y )! differences,! covxy! being! the! covariance,! and! "! being! the! (positive)! correlation! between! the! X! and! Y! values! in! the! population! given! H 1! is! true.! Other! things! being! equal,! the! larger!the!correlation! $ ,!the!smaller!the!denominator!will!be,! and! the! larger! will! be! the! effect! size! index! d z.! If! H 1! is! true,!
! " (^)! 4! Alternatively,! values! of! q! >! 1! could! be! used! if! a! type" 2! error! is!
considered!less!serious!than!a!type" 1 !error.!
the! distribution! of! our! test! statistic! is! the! noncentral! t " distribution! with! N! –! 1! degress! of! freedom! ( N! denotes! the! number! of! twin! pairs,! i.e.! the! measurement! pairs)! and! a! noncentrality!parameter!
! x^ y z x y
N d N
Let!us!assume!that!on!average!the!developmental!difference! in!a!specific!motor!skill!amounts!to! 2 !months.!For!a!specific! motor!skill,!the!standard!deviation!of!the!age!difference!may! amount! to! 4! months.! Hence,! following! Equation! (2),! the! effect!size!to!be!detected!with!this!replication!study!equals! dz! =!2/4!=!0.5.!Because!we!want!to!decide!upon!the!size!of!the!!! and!! !error!probabilities!given! N !and! dz !are!fixed,!we!need! the! «Compromise»! analysis! in! GPower .! The! option! « t "Test! (means)»!we!have!used!in!the!previous!example!is!based!on! independent! samples! and! calculates! the! degrees! of! freedom! as! N !–!2.!This!is!no!longer!adequate!for!the!current!situation! because!the!twin!data!are! dependent .!For!a!paired!samples! t " test!there!are! N !–! 1 !degrees!of!freedom.!Therefore,!we!have! to! choose! the! option! «Other! t"Tests»! for! which! the! degrees! of! freedom! can! be! determined! independently! of! N .! The! hypothesis! is! directional! again—we! want! to! know! whether! trained! twins! are! beyond! their! untrained! siblings! in! their! motor! skill! development—so! that! we! choose! the! «one" tailed»! option.! In! «Other! t"Tests»,! the! to"be"specified! effect! size!is!labelled! f !instead!of! d. 5 "^ !The!noncentrality!parameter! is!calculated!as!follows:!
! % ' f & N .! (4)!
Comparing!Equations!(3)!and!(4)!we!see!that!the!calculated! dz !value!(i.e.!0.5)!can!be!inserted!for!the!effect!size !f !to!obtain! the!correct!noncentrality!parameter!for!matched"pairs! t "tests! using!the!«Other!t"Tests»!option.!« N »!has!to!be!set!to! 20 !(20! pairs! of! twins! were! available).! If!!! and!!! shall! be! of! same! size,!the!»Beta/alpha!ratio»!option!again!has!to!be!set!to!“1”.! The!test!has! N !–! 1 !=! 19 !degrees!of!freedom!(«DF!for!t"Test»).!
GPower !returns!a!noncentrality! parameter!of! %!=!2.2361!and!
recommends! to! choose!!! =!!! =! .1357.! For! this! situation! the! power!is! 1 !"!! !=!.8643.!In!order!to!reject! H 0 !(i.e.,!in!order!to! reject! the! hypothesis! of! no! differences! between! the! twins,! which! implies! rejecting! the! maturation! hypothesis)! and! to! accept!the! H 1 !(i.e.,!to!accept!the!“environmental!influences”! hypothesis),! the! empirical! t "value! has! to! exceed! the! critical! value! t (19)! =! 1.1328.! Even! though! this! result! is! less! devastating! than! that! of! the! previous! example,! there! is! nevertheless! a! large! error! probability! associated! with! each! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "^5 !The!reason!for!using!the!symbol! f !rather!than! d !is!that!the!«Other!t" Tests»! option! of! GPower! has! been! designed! to! provide! power! analyses!for!any!type!of! t "test,!not!just !t" tests!for!means.!
decision! possible.! In! order! to! reduce! this! error! probability,! the!sample!size!would!have!to!be!increased.!To!what!extend! we! would! have! to! increase! the! sample! size! can! be! incrementally! determined! with! the! «Post"hoc»! analysis! option.!A!«Post"hoc»!analysis!again!with!the!options!«Other! t"Tests»!and!«one"tailed»!as!well!as!the!specifications!! !=!.05,! f! =! 0.5,! N! =! 45,!and! df !=! 44 !returns!a!power!value!of! 1 !"!! !=! .9512.! A! sample! of! this! size! is! necessary! to! obtain! a! power! level!above!.95.!
Figure 3: GPower display of an a priori power analysis for a t -test (correlations) situation. For details see text.
In! the! above! example,! pairs! of! twins! provided! the! dependent! data.! As! a! matter! of! course,! we! would! have! to! proceed!analogously!for!other!kinds!of!dependent!data,!for! example,!for!repeated!measurements!of!the!same!subjects.!
t"test!for!correlations^6^ "!
Berry! and! Broadbent! (1984)! investigated! the! relation! between! task! performance! in! controlling! a! computer! simulation!and!verbalizable!knowledge!about!the!simulated! system.! Experiment! 1! found! a! negative! correlation! (in! the! range!of!".25!and!".30)!between!both!variables.!The!better!the! participants! controlled! the! simulation,! the! worse! they!were! able!to!provide!information!about!the!simulated!system.!! However,! this! negative! correlation! was! not! statistically! significant.!The!authors!attributed!the!lack!of!significance!to! the! small! sample! size! ( N! =! 12).! But! how! large! was! the! probability! to! find! a! correlation! of! a! specified! size! in! this! study?!! Have! a! look! at! the! definition! of! the! noncentrality! parameter! % for_! t_ "tests! for! correlations! between! two! variables:!
!
2 1 2 N
with! $ !as!the!population!correlation!associated!with! H 1,!and! N! as! the! sample! size—that! is! the! number! of! measurement!
pairs.! According! to! Cohen! (1988)! correlations! of! ,!=!.30!are!
defined!as!medium!sized!effects.!What!are!the!odds!to!find! an!effect!of!this!size!in!an!experiment!like!the!one!described! by! Berry! and! Broadbent! (1984)?! We! have! to! choose! «Post"
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! " (^)! 6 !We!would!like!to!thank!Dave!Kenny!for!making!us!aware!of!the!
fact! that! the_! t"_ test! (correlations)! power! analyses! of! GPower! are! correct! only! in! the! point"biserial! case! (i.e.,! correlations! between! a! binary! variable! and! continuous! variable,! the! latter! being! normally! distributed! for! each! value! of! the! binary! variable).! For! correlations! between! two! continuous! variables! following! a! bivariate! normal! distribution,! the_! t"_ test! (correlations)! procedure! slightly! overestimates! power.! The! procedures! for! correlations! available! in! GPower! 3! provide! an! exact! solution! to! this! problem! (Faul! et! al.,! 2007).! !
hoc»!as!type!of!power!analysis!and!«t"Test!(correlations)»!as! type!of!test.!The!test!is!«one"tailed»!because!we!want!to!test!
H 0:! ,! -! 0 !versus! H 1 :! ,!<!0.!Enter! r !=!.30!as!effect!size!measure,!
N! =! 12! as! sample! size,! and!!! =! .05.! The! noncentrality!
parameter! turns! out! to! be! %! =! 1.0894,! a! t "value! of! t (10)! =! "
1.8125! or! below! denotes! a! significant! result.! However,! statistical!power!is!only! 1 !"!! !=!.2648.!Berry!and!Broadbent’s! (1984)! explanation! for! not! finding! a! significant! correlation! seems! to! be! very! plausible:! their! study! lacked! statistical! power.!But!how!large!should!the!sample!be!in!order!to!find! medium!effects!with!a!power!of!.95?!Change!the!settings!to! «A!priori»!type!of!analysis,!enter!.05!as! !, .95!as!power,!and! .30! as! effect! size! r! (see! Figure! 3).! The! required! sample! size! amounts! to! N! =! 111.! The! critical! t" value! equals! t (109)! =! "
1.6590,!the!noncentrality!parameter!is! %!=!3.3133.!!
Power!analyses!for!F"tests! We! will! restrict! our! descriptions! to! power! analyses! for! analyses!of!variance!for!fixed!effects.!These!analyses!can!be! conducted! with! the! «F"Test! (ANOVA)»! option.! We! will! describe! neither! the! «F"Test! (MCR)»! option! (for! F "tests! in! multiple! regression/correlation! analyses)! nor! the! «Other! F" Tests»! option! (for! approximate! F "tests! for! fixed! factors! in! mixed! models! and! approximate! multivariate! analysis! of! variance!(MANOVA)! F "tests).! The!effect!size!index!of!relevance!is!the!index! f !or! f^2 .!The! relation!between! f^^2 !and!the!noncentrality!parameter!^ &^ of!the! noncentral! F "distribution!is!given!by:! !. ' f^2 & n k & ' f^2 & N ,! (6)! with! n! denoting! the! number! of! subjects! in! each! of! the! k! groups.!The!effect!size!index! f !is!defined!as:!
!
2 f 1 2
with! '^2! as! the! amount! of! the! total! population! variance! explained!by!the!group!differences!specified!in! H 1.!In!case!of! unequal! group! sample! sizes! n j,!! the! effect! size! index! f! is!
Figure 5: GPower effect size computation of ( based on the therapy success rates under H (^) 0 and H (^) 1. For details see text.
Table!1.!Cell!probabilities!in!the! 2 !×! 2 !contingency!table!under! the!H 1 !with!type!of!therapy!and!therapy!success!as!row!and! column!variables,!respectively.!
!! Therapy!success!!
!! Success! Failure! (^3)!
Type!of!!^ X!^ .88!×!.5!=!!!.440!^ .12!×!.5!=!!!.060!^ .500! therapy! Y! .79!×!.5!=!!!.395!^ .21!×!.5!=!!!.105!^ .500!
! (^) 3" .835!^ .165!^ 1.000!
following! Cohen’s! (1988)! suggestions! to! calculate! statistical! power! for! complex! designs.! Evidence! has! been! provided! that! Cohen’s! power! assessment! procedure! for! interaction! tests!is!flawed!(Bradley,!Russell,!&!Reeve,!1996;!Erdfelder!et! al.,!1996;!Koele,!1982,!footnote!1).!
Power!analyses!for!!^2 "tests!
Two! types! of_!!^2 "tests! are! commonly! applied! in! psychological! research! (cf.! Cohen,! 1988):! (a)! contingency! tests! (also! called! independence! tests)! assessing! deviations! ( H 1)! from! stochastic! independence! ( H 0 )! of! two! or! more! categorical! variables! and! (b)! goodness"of"fit! tests! of! a! theoretical! distribution! to! frequency! data.! Statistical! power! computations!are!based!on!the!noncentral!_!^2 "distribution!for! both! cases! (Johnson! &! Kotz,! 1970,! Chap.! 28).! Its! noncentrality!parameter! !. ' 42 & N! (9)! is! the! product! of! the! sample! size! N! and! the! squared! effect! size!index!
! 1 0 2 1 0
m ( (^) i i )
i i
p p p
4 '
0 .!^ (10)!
In!Equation!(10),! m !denotes!the!number!of!categories,! p 0i !the! probability!of!category! i !under! H 0 ,!and! p 1i !the!probability!of! category! i !under! H 1.!! Let!us!have!a!look!at!a!contingency!test!of!the!following! type:! The! success! rate! of! therapy! X! is! quite! large! with! p x! =! .88.! Unfortunately,! therapy! X! is! also! very! expensive.! Assume! that! a! new! therapy! Y! would! be! much! cheaper.! Of! course,! this! new! therapy! should! only! be! applied! if! its! success!rate!is!not!(significantly)!smaller!than!that!of!therapy! X.! This! situation! corresponds!to!a!test!of! H 0:! p y!-! p x!against! H 1:! p y! <! p x! which! can! be! tested! with! a! one"tailed!!^2 " contingency! test! for! 2! ×! 2! contingency! tables.! Type! of! therapy!(X!vs.!Y)!functions!as!the!row!variable!and!therapy! success! (success! vs.! failure)! as! the! column! variable.! Half! of! the!sample!is!assigned!to!therapy!X!and!Y,!respectively.!We! want! to! detect! a! disadvantage! of! therapy! Y! given! there! is!
one! with! a! high! degree! of! certainty.! In! other! words,! if! the! statistical! test! does! not! reveal! a! difference! between! the! two! therapies,! we! want! to! be! very! sure! that! there! really! is! no! difference.!Therefore,!statistical!power!is!set!to! 1 !"!! !=!.95.!By! contrast,! we! accept! a! risk! of!!! =! .20! to! incorrectly! rejecting! therapy!Y!as!less!efficient!than!X.!By!definition,!we!will!call! therapy! Y! less! efficient! than! X! only! if! its! success! rate! undershoots! the! success! rate! of! X! by! at! least! .09.! With! a! success! rate! for! therapy! X! of! p x! =! .88,! this!implies!a!success! rate! for! therapy! Y! of! p y! =! .88! "! .09! =! .79.! The! H 1! cell! probabilities!of!the! 2 !×! 2 !contingency!table!implied!by!these! specifications!are!displayed!in!Table!1.!! What!sample!size! N !is!needed!for!this!test!situation?!To! answer! this! question! we! choose! «A! priori»! as! type! of! analysis! and! «Chi"square! Test»! as! type! of! test.! GPower! calculations! are! based! on! a! nondirectional!!^2 "test! situation.! For! the! directional! test! problem! we! face,!!! is! set! to! .40! instead!of!.20.!Power!is!set!to!.95.!The!effect!size!measure! 5! which!corresponds!to!the!alternative!hypothesis!specified!by! p x! =! .88! and! p y! =! .79! can! be! calculated! in! GPower! with! the! «Calc! “ 5 ”»! option! button.! A! submenu! appears! (see! Figure! 5).!Because!50%!of!the!sample!is!treated!under!X!and!Y,!the! cell! probabilities! of! the! contingency! table! under! H 1! yield!! .880! ×! .5! =! .440! and! .120! ×! .5! =! .060,! respectively,! as! the! success! and! failure! rates! with! therapy! X! (see! Table! 1).! Analogously,!we!obtain!.790!×!.5!=!.395!and!.210!×!.5!=!.105,! respectively,!as!the!success!and!failure!rates!with!therapy!Y.! H 0 !predicts!statistical!independence!of!therapy!and!outcome! given! the! same! marginal! means.! This! implies! identical! success! probabilities! (.835! ×! .5! =! .4175)! and! failure! probabilities! (.165! ×! .5! =! .0825)! for! both! therapies.! After! inserting! the! H 0! and! H 1! probabilities! in! the! «Calc! “ 5 ”»!
submenu!the!effect!size!5"is!calculated!as!given!in!Equation! (10).! We! thus! obtain! 5! =! 0.1212."! Finally,! df! =! 1! has! to! be! specified!in!the!main!window.!The!noncentrality!parameter! equals! & !=!6.1696.!The!a!priori!analysis!returns!a!necessary! N! of! 420 !and!a!critical!!^2 "value!of!!^2 (1) !=!0.7083.!If!the!!^2 "statistic! exceeded! this! critical! value! and! the! sample! success! rate! of! therapy! Y! were! smaller! than! that! of! therapy! X,! we! would! accept! H 1 .!The!new!therapy!Y!would!have!to!be!rejected.!If! the!!^2 "statistic!did!not!exceed!this!critical!value,! H 0 !would!be! maintained.! The! less! expensive! therapy! Y! could! be! used.! Note! that! all! computations! are! approximations! because! the! exact! distribution! of! the!!^2 "statistic! is! only! a!!^2 "distribution! for!the!asymptotic!case,!that!is,!for! N! $ %.!However,!with! N! =! 420,! the! deviation! from! the! asymptotic! distribution! is! negligibly!small^8 "^ .!
Conclusion!
Statistical!power!considerations!are!indispensable!for!the! evaluation! of! statistical! decisions! as! well! as! for! designing! studies.!With! GPower !we!introduced!an!easy!to!use!software! tool! that! facilitates! the! implementation! of! various! kinds! of! power!analyses.! This!paper!was!restricted!to!power!analyses!of!the!most! frequently!used!statistical!tests.!We!recommend!the!work!of! O’Brian!and!Muller!(1993)!as!well!as!of!Chartier!and!Allaire! (in! press,! this! issue)! to! those! interested! in! power! analyses! for!repeated!measures!designs!and!for!multivariate!analysis! of! variance! (MANOVA)! F "tests.! Power! analyses! for! these! situations! can! be! accomplished! in! GPower! using! the! «Other! F"Tests»!option,!but!care!has!to!be!taken!to!correctly!specify! the! noncentrality! parameter.! Readers! seeking! information! about! power! analyses! for! random"effect! ANOVAs! are! referred!to!Koele!(1982).!!
References!
Berry,!D.!C.,!&!Broadbent,!D.!E.!(1984).!On!the!relationship! between! task! performance! and! associated! verbalizable! knowledge.! The! Quarterly! Journal! of! Experimental! Psychology! A:! Human! Experimental! Psychology ,! 36A ,! 209 " 231.! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!! " (^)! 8! A! Monte! Carlo! study! with! 10.000! random! samples—each! of!
sample!size! N !=!420—drawn!from!the!population!under! H 1 !resulted! in! an! empirical! estimation! of! statistical! power! of! .9514.! This! estimation! arose! when! each! occurrence! in! which! the! test! statistic! exceeded!the!critical!value!of!0.7083!was!counted!as!a!rejection!of! H 0 .! If! only! the! subset! of! test! statistics! were! counted! for! which! the! sample!success!rate!of!therapy!X!was!larger!than!that!of!therapy!Y,! power!estimation!equaled!.9509.!The!results!show!that!the!statistical! power! considerations! stay! valid! for! the! “one"tailed”! decision! rule! applied!in!the!above!example.!
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