UNIT 8 TWO-WAY ANOVA WITH m OBSERVATIONS PER CELL

Transcription

1 UNIT 8 TWO-WAY ANOVA WITH OBSERVATIONS PER CELL Two-Way Anova wth Obsrvatons Pr Cll Structur 81 Introducton Obctvs 8 ANOVA Modl for Two-way Classfd Data wth Obsrvatons r Cll 83 Basc Assutons 84 Estaton of Paratrs 8 Tst of Hyothss 86 Dgrs of Frdo of Varous Su of Suars 87 Exctatons of Varous Su of Suars 88 Suary 89 Solutons/Answrs 81 INTRODUCTION In th analyss of varanc tchnu, f xlanatory varabl s only on and dffrnt lvls of ndndnt varabl s undr consdraton thn t s calld on-way analyss of varanc and a tst of hyothss s dvlod for th ualty of svral an of dffrnt lvls of a factor/ndndnt varabl/ xlanatory varabl But f w ar ntrstd to consdr two ndndnt varabls for analyss n lac of on, and abl to rfor th two hyothss for th lvls of ths factors ndndntly (thr s no ntracton btwn ths two factors) Th abov analyss has bn gvn n th Unts 6 and 7 rsctvly But f w ar ntrstd to tst th ntracton btwn two factors and w hav ratd obsrvatons thn th two-way analyss of varanc wth obsrvaton r cll s consdrd If thr ar xactly sa nubrs of obsrvatons n th cll thn t s calld balanc In ths unt, a athatcal odl for two-way classfd data wth - obsrvatons r cll s gvn n Scton 8 Th basc assutons ar gvn n Scton 83 whras th staton of aratrs s gvn n Scton 84 Tst of hyothss for two-way ANOVA s xland n Scton 8 and dgrs of frdo of varous su of suars ar dscrbd n Scton 86 Th xctd valus of su of suars for two factors and thr ntractons ar drvd n Scton 87 Obctvs Aftr studyng ths unt, you would b abl to dscrb th ANOVA odl for two-way classfd data wth obsrvatons r cll; dscrb th basc assutons for th gvn odl; obtan th stats of th aratrs of th gvn odl; 61

2 Analyss of Varanc dscrb th tst of hyothss for two-way classfd data wth obsrvatons r cll; drv th xctatons of th varous su of suars; and rfor to tst th hyothss for two-way classfd data wth obsrvatons r cll 8 ANOVA MODEL FOR TWO-WAY CLASSIFIED DATA WITH OBSERVATIONS PER CELL 6 In Unt 7, t was sn that w cannot obtan an stat of, or ak a tst for th ntracton ffct n th cas of two-way classfd data wth on obsrvaton r cll Ths s ossbl, howvr, f so or all of th clls contan or than on obsrvatons W shall assu that thr s an ual nubr of () obsrvatons n ach cll Th obsrvatons n th (, ) th cll wll b dnotd y 1, y,, y Thus, y k s th k th obsrvaton for th lvl of factor A and th lvl of factor B, = 1,,, ; = 1,,, & k = 1,,, Th athatcal odl y k = µ + k whr µ s th tru valu for th (, ) th cll and k s th rror k ar assud to b ndndntly dntcal norally dstrbutd, ach wth an zro and varanc σ Th tabl of obsrvatons can b dslayd as follows: A/B B 1 B B B Total Total A 1 y 111 y 11 y 11 y 1 y 11 y 1 y 11 y 1 y 11 y 1 y 1 y 11 A y 11 y 1 A A y 1 y 11 y 1 y 1 y 11 y 1 y 1 y 1 y 1 y y y 1 y y y 1 y y y 1 y 1 y y y 1 y y y 1 y y y 1 y 1 y Y y 1 y y y 1 y y y 1 y 1 y y y 1 y y y 1 y y Total y 1 y y y y y y y

3 Th odl can b wrttn as y k = µ + (µ -µ) + (µ -µ) + (µ -µ -µ +µ) + k Two-Way Anova wth Obsrvatons Pr Cll = µ + α + β + (αβ) + k whr, µ s gnral an ffct, α s th ffct of th lvl of th factor A, s th ffct of th lvl of factor B, (αβ) s th ntracton ffct btwn th lvl of A factor and th lvl of B factor whr, 1 α 0, β 0, αβ 0, αβ y = Su of all th obsrvatons y = Total of all obsrvatons n th th lvl of factor A y = Total of all obsrvatons n th th lvl of factor B 83 BASIC ASSUMPTIONS Followng assutons should b followd for vald and rlabl tst rocdur for tstng of hyothss as wll as for staton of aratrs 1 All th obsrvatons yk ar ndndnt Dffrnt ffcts ar addtv n natur 3 k ar ndndnt and dntcaly dstrbutd as noral wth an zro and constant varanc 84 ESTIMATION OF PARAMETERS Th last suar stats for varous ffcts, obtand by nzng th rsdual su of suars E y k 1 1 k1 by artally dffrntatng E wth rsct to μ, α (=1,,, ), β (=1,,, ) and (αβ) for all = 1,,, ; =1,,, and uatng ths uatons ual to zro Ths uatons ar calld noral uatons Soluton of ths noral uatons rovd th stats of ths aratrs [μ, α, β, (αβ) ] E E y k k1 y k 0 1 k1 63

4 Analyss of Varanc E y k 0 1 k1 E y k 0 k1 1 1 k1 Ths uatons gv, y 1 k 1 y y k k y y y k 1 k 1 Slarly, y y ( ) y y y y Substtutng th valus of,, and ( α βˆ ), n th odl and thn slct th valu of k such that both th sds ar ual, so y k y y y y y y y y y y y k or y k y y y y y y y y y y y k Suarng and sung both th sds ovr, & k, thn w gt y k y y y y y 1 1 k1 1 y y y y y k y 1 as usual roduct trs vansh k1 Total Su of Suars = Su of Suars du to Factor A+ Su of Suars du to Factor B + Su of Suars du to Intracton A and B + Su of Suars du to Error or TSS = SSA + SSB + SSAB + SSE 8 TEST OF HYPOTHESIS 64 Thr ar thr hyothss whch ar to b tstd ar as follows: H 0A : α 1 = α = = α = 0 H 1A : α 1 α α 0 H 0B : β 1 = β = = β = 0 H 1B : β 1 β β 0

5 H 0AB : (αβ) = 0 for all and or A and B ar ndndnt to ach othr H 1AB : (αβ) 0 Th arorat tst statstcs for tstng th abov hyothss s: SSA F SSE 1 MSSA 1 MSSE If ths valu of F s gratr than th tabulatd valu of F wth [(-1), (- 1)] df at α lvl of sgnfcanc so w rct th null hyothss, othrws w ay acct th null hyothss Slarly, tst statstcs for scond and thrd hyothss ar SSB F SSE SS AB F SSE 1 MSSB 1 MSSE MSSE MSSAB Two-Way Anova wth Obsrvatons Pr Cll For ractcal ont of vw, frst w should dcd whthr or not H 0AB can b rctd at an arorat lvl of sgnfcanc by usng abov F If ntracton ffcts ar not sgnfcant th factor A and factor B ar ndndnt thn w can fnd th bst lvl of A and bst lvl of B by ultl coarson thod usng t-tst On th othr hand, f thy ar found to b sgnfcant, thr ay not b a sngl lvl of factor A and sngl lvl of factor B that wll b th bst n all stuatons In ths cas, on wll hav to coar for ach lvl of B at th dffrnt lvls of A and for ach lvl of A at th dffrnt lvls of B Th abov analyss can b shown n th followng ANOVA tabl: Sourss of Varaton ANOVA Tabl for Two-way Classfd Data wth Obsrvatons r Cll DF SS MSS F Btwn th lvls of A -1 SSA y y 1 MSSA = SSA / (-1) F = MSSA / MSSE Btwn th lvls of B -1 SSB y y 1 MSSB = SSB / (-1) F = MSSB / MSSE Intracton AB (-1) (-1) SSAB MSSAB = SSAB / (-1)(-1) y y y y 1 1 F = MSS(AB) / MSSE Error (-1) TSS y k y 1 1 k1 MSSE = SSE / (-1) Total -1 6

6 Analyss of Varanc Sts for Calculatng Varous Sus of Suars 1 Calculat G = Grand Total = Total of all obsrvatons = y 1 1 k1 k Dtrn N = Nubr of obsrvatons 3 Fnd Corrcton Factor (CF) = G /N 4 Raw Su of Suars ( RSS) = y 1 1 k1 Total Su of Suars (TSS) = RSS - CF 6 Su of Suars du to Factor A (SSA) = {y 1 / + y / + + y / + + y /} CF 7 Su of Suars du to Factor B (SSB) = {y 1 /+y / + +y / + + y /} CF 8 Su of Suars du to Mans (SSM) = {y 1 / + y / + + y k / + + y /} CF 9 Su of Suars du to Intraton AB(SSAB) = SSM SSA - SSB 10 Su of Suars du to Error (SSE) = TSS -SSA-SSB-SSAB 11 Calculat MSSA = SSA/df 1 Calculat MSSB = SSB/df 13 Calculat MSSAB = SS(AB)/df 14 Calculat MSSE = SSE/df k 1 Calculat F A = MSSA/MSSE ~ F (-1), (-1) 16 Calculat F B = MSSB/MSSE ~ F (-1), (-1) 17 Calculat F AB = MSS(AB)/MSSE ~ F ((-1)(-1), (-1) 86 DEGREES OF FREEDOM OF VARIOUS SUM OF SQUARES Total su of suars (TSS) consdrs th obsrvatons so th dgrs of frdo for TSS ar (-1) On dgr of frdo s lost du to th rstrcton that (y y ) k1 k Th dgrs of frdo for su of suars du to factor A s (-1) bcaus t has lvls Slarly, th dgrs of frdo for su of suars du to factor B s (-1) bcaus t has lvls, undr consdraton Su of suars du to ntracton of factors A and B s (-1) (-1) and th dgrs of frdo for su of suars du to rrors s (-1) Thus arttonng of dgrs of frdo s as follows: 66 (-1) = (-1) + (-1) + (-1)(-1) + (-1) whch ls that th df ar addtv

7 87 EXPECTATIONS OF VARIOUS SUM OF SQUARES Two-Way Anova wth Obsrvatons Pr Cll 871 Exctd Valu of Su of Suars du to Factor A E (SSA) = E y y Substtutng th valu of y and 1 y fro th odl, w gt E 1 E 1 or E (SSA) = = E α α 1 E (SSA) = α E α E 1 1 Bcaus E = E E 1 1 = = 1 E SSA 1 ( 1) SSA E 1 ( 1) 1 ( 1) or E MSSA 1 Undr H0A th MSSA s an unbasd stat of 87 Exctd Valu of Su of Suars du to Factor B Procdng slarly, or by sytry, w hav E(SSB) = E y y 1 67

8 Analyss of Varanc Substtutng th valu of y and y fro th odl w gt E(SSB) = E 1 or E(SSB) = β β or E(SSB) = β E or E(SSB) = β E = β E E 1 1 = 1 1 = 1 E(SSB) = 1 SSB E or E(MSSB) = 1 1 Undr H 0B th MSSB s an unbasd stat of Slarly you can obtan th xctd valu of SSAB, whch wll b E E SSAB 1 SSAB or E (MSSAB) Undr H 0AB, th an su of suars du to ntracton btwn Factor A and B s an unbasd stat of

9 873 Exctd Valu of Su of Suars du to Error Procdng slarly, or by sytry, w hav E (SSE) = E y k y 1 1 k1 Substtutng th valu of yk and y fro th odl, w hav Two-Way Anova wth Obsrvatons Pr Cll or E SSE ( E(SSE) = E k 1) or E (MSSE) = 1 1 k1 = E k 1 1 k 1 = E k 1 1 k1 = E k 1 1 k1 1 1 = E k E 1 1 k1 1 1 = / = 1 Hnc, an su of suars du to rror s an unbasd stat of Exal 1: A anufacturr wshs to dtrn th ffctvnss of four tys of achns (A, B, C and D) n th roducton of bolts To accuulat ths, th nubrs of dfctv bolts roducd for ach of two shfts n th rsults ar shown n th followng tabl: Machn Frst shft Scond Shft M T W Th F M T W Th F A B C D Prfor an analyss of varanc to dtrn at % lvl of sgnfcanc, whthr thr s a dffrnc (a) Btwn th achns and (b) Btwn th shfts 69

10 Analyss of Varanc Soluton: Thr ar two factors th achn and shft Th lvls of achn ar four and lvls of shft ar two Th Coutaton rsults ar as follows: G = = 68 N = 40 G CF = N 40 Raw Su of Suars (RSS) = = 1946 Total Su of Suars (TSS) = RSS - CF = =104 Su of suar du to achns and du to shfts can b calculatd by consdrng th followng two-way tabl: Machn Shft Total I Shft II Shft A B C D Total Su of Suars du to Machn SSM CF 1 0 = = Su of Suars du to Shfts SSS CF = = Su of Suars du to Intracton (SSMS) CF SSM SSS = = 6 Fnaly, th Su of Suars du to rror s foundd by subtractng th SSM, SSS and SSSM fro TSS SSE = TSS-SSM SSS SSMS = = 848

11 For SSM MSSM df SSS MSSS df SSE MSSE df 81 1 SSMS MSSMS df tstng H0 A : Man ffct of Machn A= Machn B = Machn C = Machn D, s Two-Way Anova wth Obsrvatons Pr Cll For tstng H 0B 17 F = : Man ffct of Shft A = Shft B, s F Slarly, for tstng H 0AB 167 F : Intracton ffct of Machn and Shft, s ANOVA Tabl for Two-way Classfd Data - Obsrvaton r Cll Sourcs of Varaton Du to Machnry Dgrs of Frdo (DF) Su of Suars (SS) Man Su of Suars (MSS) Du to Shft Du to Intracton Du to Error Total F-tst or Varanc Rato Th tabulatd valu of F at 3 and 3 dgrs of frdo at % lvl of sgnfcanc s 90 Th coutd valu of F for ntracton s 0817 so th avrag rforancs n dffrnt shfts ar not sgnfcant Thr s a sgnfcant dffrnc aong achns, snc th calculatd valu of F for achns s 64 and th crtcal valu (tabulatd valu) of F s 90 Th tabulatd valu for shfts s 41 Th calculatd valu of F for shfts s 306 Hnc, thr s no dffrnc du to shfts 71

12 Analyss of Varanc E1) An xrnt s rford to dtrn th ffct of two advrtsng caagns on thr knds of cak xs Sals of ach x wr rcordd aftr th frst advrtsng caagns and thn aftr th scond advrtsng caagn Ths xrnt was ratd thr ts for ach advrtsng caagn and got th followng rsults: Caagn I Caagn II Mx1 74, 64, 0 109, 1086, 106 Mx 4, 73, 1 108, 1073, 998 Mx3 76, 40, , 104, 10 Prfor an analyss of varanc to dtrn at % lvl of sgnfcanc, whthr thr s a dffrnc (a) Btwn th cak xs and (b) Btwn th caagns 88 SUMMARY In ths unt, w hav dscussd: 1 Th ANOVA odl for two-way classfd data wth obsrvatons r cll; Th basc assutons for th gvn odl; 3 How to obtan th stats of th aratrs of th gvn odl; 4 How to tst th hyothss for two-way classfd data wth obsrvatons r cll; How to drv th xctatons of th varous su of suars; and 6 Nurcal robls to tst th hyothss for two-way classfd data wth obsrvatons r cll 89 SOLUTIONS /ANSWERS 7 E1) For st u an ANOVA Tabl for ths robl, th coutaton rsults ar as follows: Grand Total G = 14 N = 18 Corrcton Factor (CF) = (14 14) /18 = RSS = TSS = SSA = SSB = 97 SSAB = 116 SSE = 6389

13 ANOVA Tabl Two-Way Anova wth Obsrvatons Pr Cll Sourcs of Varaton DF SS MSS F-Calculatd F-Tabulatd at % lvl of sgnfcanc Advrtsng caagn /34 = 0793 F(1,1) = 439 Cak Mx /34 = 8 Intracton /34 = 106 F(,1) = 1941 F(,1) = 1941 Error Total Snc coutd valu of F for cak x and ntracton ar 8 and 106 rsctvly whch ar lss than corrsondng tabulatd valu so thy ar not sgnfcant Whras th calculatd valu of F for advrtsng caagn s gratr than corrsondng tabulatd valu so thr s a sgnfcant dffrnc aong advrtsng caagn 73

14 Analyss of Varanc TABLE: Th F Tabl Valu of F Corrsondng to % (Noral Ty) and 1% (Bold Ty) of th Ara n th Ur Tal Dgrs of Frdo: Dgrs of Frdo (Nurator) (Dnonator) ,0 4,999,403,6,764,89,98,981 6,0 6,06 6,08 6,106 6,14 6,169 6,08 6,34 6,8 6,

15 TABLE (Contnud) Dgrs of Two-Way Anova wth Obsrvatons Pr Cll Frdo: Dgrs of Frdo: Nurator Dnonator