Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap !-Analysis of Variance - ANOVAChapter 13-14Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!Chapter 'oalsAfter completing this chapter, you should be able to: (ecogni)e sit*ations in +hich to *se analysis of ,arianceUn-erstan- -ifferent analysis of ,ariance -esigns#erfor. a single-factor hypothesis test an- interpret res*ltsCon-*ct an- interpret post-hoc .*ltiple co.parisons proce-*resAnaly)e t+o-factor analysis of ,ariance testsStatistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-3Chapter O,er,ie+Analysis of Variance /ANOVA01-test2*3ey-4ra.er testOne-5ay ANOVA2+o-5ay ANOVA %nteractionEffectsStatistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-413&1 'eneral ANOVA Setting%n,estigator controls one or .ore in-epen-ent ,aria6lesCalle- factors /or treat.ent ,aria6les0Each factor contains t+o or .ore le,els /or gro*ps or categories7classifications0O6ser,e effects on the -epen-ent ,aria6le(esponse to le,els of in-epen-ent ,aria6leExperi.ental -esign8 the plan *se- to collect the -ataStatistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-913&3 Co.pletely (an-o.i)e- :esignExperi.ental *nits /s*6;ects0 are assigne- ran-o.ly to treat.entsS*6;ects are ass*.e- ho.ogeneo*sOnly one factor or in-epen-ent ,aria6le5ith t+o or .ore treat.ent le,elsAnaly)e- 6y one-factor analysis of ,ariance /one-+ay ANOVA0Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-$ypotheses of One-5ay ANOVA All pop*lation .eans are e=*al i&e&, no treat.ent effect /no ,ariation in .eans a.ong gro*ps0 At least one pop*lation .ean is -ifferent i&e&, there is a treat.ent effect :oes not .ean that all pop*lation .eans are -ifferent /so.e pairs .ay 6e the sa.e0 c 3 ! 1 "? ? ? ? 8 $ = = = = sa.e the are .eans pop*lation the of all Not 8 $1Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-@One-1actor ANOVA All Means are the sa.e82he N*ll $ypothesis is 2r*e /No 2reat.ent Effect0c 3 ! 1 "? ? ? ? 8 $ = = = = sa.e the are ? all Not 8 $i 13 ! 1? ? ? = =Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-AOne-1actor ANOVA At least one .ean is -ifferent82he N*ll $ypothesis is NO2 tr*e /2reat.ent Effect is present0c 3 ! 1 "? ? ? ? 8 $ = = = = sa.e the are ? all Not 8 $i 13 ! 1? ? ? =3 ! 1? ? ? or(continued)Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-1"#artitioning the Variation2otal ,ariation can 6e split into t+o parts8SS2 B 2otal S*. of S=*ares /2otal ,ariation0SSA B S*. of S=*ares A.ong 'ro*ps /A.ong-gro*p ,ariation0SS5 B S*. of S=*ares 5ithin 'ro*ps /5ithin-gro*p ,ariation0SS2 B SSA C SS5Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-11#artitioning the Variation2otal Variation B the aggregate -ispersion of the in-i,i-*al -ata ,al*es across the ,ario*s factor le,els /SS205ithin-'ro*p Variation B -ispersion that exists a.ong the -ata ,al*es +ithin a partic*lar factor le,el /SS50A.ong-'ro*p Variation B -ispersion 6et+een the factor sa.ple .eans /SSA0SS2 B SSA C SS5(continued)Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-1!#artition of 2otal VariationVariation Due to Factor (SSA)Variation Due to Random Sampling (SSW)otal Variation (SS)Co..only referre- to as8S*. of S=*ares 5ithinS*. of S=*ares ErrorS*. of S=*ares Unexplaine-5ithin 'ro*ps VariationCo..only referre- to as8S*. of S=*ares Det+een S*. of S=*ares A.ongS*. of S=*ares Explaine-A.ong 'ro*ps VariationBCStatistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-132otal S*. of S=*ares= = =c1 ;n1 i!i;;0 E E / SS25here8SS2 B 2otal s*. of s=*aresc B n*.6er of gro*ps /le,els or treat.ents0n; B n*.6er of o6ser,ations in gro*p ;Ei; B ith o6ser,ation fro. gro*p ; E B gran- .ean /.ean of all -ata ,al*es0SS2 B SSA C SS5Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-142otal Variation!roup " !roup # !roup $Response, %X!cn!1!!110 E E / &&& 0 E E / 0 E E / SS2c + + + =(continued)Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-19A.ong-'ro*p Variation5here8SSA B S*. of s=*ares a.ong gro*psc B n*.6er of gro*ps or pop*lationsn; B sa.ple si)e fro. gro*p ; E; B sa.ple .ean fro. gro*p ; E B gran- .ean /.ean of all -ata ,al*es0!;c1 ;;0 E E / n SSA = =SS2 B SSA C SS5Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-1A.ong-'ro*p Variation!roup " !roup # !roup $Response, %X1X2X3X2 22 221 10 x x / n &&& 0 x x / n 0 x x / n SSAc c + + + =(continued)Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-1@5ithin-'ro*p Variation5here8SS5 B S*. of s=*ares +ithin gro*psc B n*.6er of gro*psn; B sa.ple si)e fro. gro*p ; E; B sa.ple .ean fro. gro*p ;Ei; B ith o6ser,ation in gro*p ;!;i;n1 ic1 ;0 E E / SS5; = = =SS2 B SSA C SS5Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-1A5ithin-'ro*p VariationS*..ing the ,ariation +ithin each gro*p an- then a--ing o,er all gro*psic nSS5MS5=Mean S=*are 5ithin B SS57-egrees of free-o.!;i;n1 ic1 ;0 E E / SS5; = = =(continued)Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!"5ithin-'ro*p Variation!roup " !roup # !roup $Response, %1X2X3X!ccn!!1!!1110 E E / &&& 0 E E / 0 E x / SS5c + + + =(continued)Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!1O6taining the Mean S=*aresc nSS5MS5=1 cSSAMSA=1 nSS2MS2=Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!!One-5ay ANOVA 2a6leSource of Variationdf SS&S(Variance)Among !roupsSSA &SA 'Within !roupsn ( c SSW &SW 'otal n ( "SS 'SSA)SSWc ( "&SA&SWF ratioc B n*.6er of gro*psn B s*. of the sa.ple si)es fro. all gro*ps-f B -egrees of free-o.SSAc ( "SSWn ( cF 'Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!313&9 One-1actor ANOVA1 2est Statistic2est statistic MSA is .ean s=*ares a.ong ,ariancesMSW is .ean s=*ares +ithin ,ariances:egrees of free-o.-f1 B c F 1/c B n*.6er of gro*ps0-f! B n F c/n B s*. of sa.ple si)es fro. all pop*lations0MS5MSA1 =$"8 ?1B ?! B G B ?c$18 At least t+o pop*lation .eans are -ifferentStatistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!4%nterpreting One-1actor ANOVA 1 Statistic2he 1 statistic is the ratio of the a.ong esti.ate of ,ariance an- the +ithin esti.ate of ,ariance2he ratio .*st al+ays 6e positi,e df1 B c -1 +ill typically 6e s.all df! B n - c+ill typically 6e large:ecision (*le8(e;ect $" if 1 H 1U, other+ise -o not re;ect $"" B &"9(e;ect $":o not re;ect $"F* Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!9One-1actor ANOVA 1 2est Exa.pleIo* +ant to see if three -ifferent golf cl*6s yiel- -ifferent -istances& Io* ran-o.ly select fi,e .eas*re.ents fro. trials on an a*to.ate- -ri,ing .achine for each cl*6& At the &"9 significance le,el, is there a -ifference in .ean -istanceJ+lub "+lub #+lub $!94!34!""!!3>!!>!"!!>!"0! C 9 /!"9&@ F !!>0!B 4>19A3&3!39@&!MS5MSA1 = = =+ritical Value:F* ' $-/0Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-!AS*&&AR1Groups Count Sum Average VarianceCl*6 1 9 1!4< !4A&! 1"@&!Cl*6 ! 9 113" !!< >>&9Cl*6 3 9 1"!A !"9&@ A4&!A23VASource of VariationSS df MS F P-value F critDet+een 'ro*ps4."5-4 # #$,/-# #,-#., 4&AAE-"9 $-/05ithin 'ro*ps"""0-5 "# 0$-$2otal 9@3ual:;: the interaction of A and 7 is e>ual to ?ero:": interaction of A and 7 is not ?eroF est for Factor A 8ffect:;: uaresDegrees of Freedom&ean S>uaresFStatisticFactor A SSA r B "&SA B SSA 7/r F 10&SA&S8Factor 7 SS7 c B "&S7B SSD 7/c F 10&S7&S8A7(9nteraction)SSA7 (r B ")(c B ")&SA7B SSAD 7 /r F 10/c F 10&SA7&S88rror SS8 rc(n B ")&S8 ' SSE7rc/nQ F 10otal SS n B "Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-4Exa.ples8%nteraction ,s& No %nteractionNo interaction81actor D Ke,el 11actor D Ke,el 31actor D Ke,el !1actor A Ke,els1actor D Ke,el 11actor D Ke,el 31actor D Ke,el !1actor A Ke,elsMean (esponseMean (esponse%nteraction is present8Statistics for Managers Using Microsoft Excel, 4e!""4 #rentice-$all, %nc&Chap 13-4@Chapter S*..ary:escri6e- one-+ay analysis of ,ariance2he logic of ANOVAANOVA ass*.ptions1test for -ifference inc.eans2he 2*3ey-4ra.er proce-*re for .*ltiple co.parisons:escri6e- t+o-+ay analysis of ,arianceExa.ine- effects of .*ltiple factors Exa.ine- interaction 6et+een factors