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
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
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 0 1 1 1 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 0 1 1 k1 y k 0 1 k1 63
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 1 1 1 1 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
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 1 1 1 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
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 ) 0 1 1 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
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 0 1 1 α E 1 1 Bcaus E = E E 1 1 = 1 1 0 = 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
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 0 1 1 or E(SSB) = β E 1 1 1 = β E E 1 1 = 1 1 = 1 E(SSB) = 1 SSB E 1 1 1 1 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 1 1 1 1 1 1 1 1 1 or E (MSSAB) 1 1 1 1 68 Undr H 0AB, th an su of suars du to ntracton btwn Factor A and B s an unbasd stat of
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 6 4 4 7 4 6 8 B 10 8 7 7 9 7 9 1 8 8 C 7 6 9 9 7 4 6 D 8 4 6 7 9 7 10 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
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 = 6+ 4 + + + 4 + + 7 + 4 + 6 + 8 + 10 + 8 + 7 + 7 + 9 + 7 + 9 + 1 + 8 + 8 + 7 + + 6 + + 9 + 9 + 7 + + 4 + 6 + 8 + 4 + 6 + + + 7 + 9 + 7 + 10 = 68 N = 40 G 68 68 CF = 179 6 N 40 Raw Su of Suars (RSS) = 6 + 4 ++ 10 = 1946 Total Su of Suars (TSS) = RSS - CF = 1946-1796 =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 4 30 4 B 41 44 8 C 3 31 63 D 8 38 66 Total 1 143 68 4 10 8 10 63 10 66 10 Su of Suars du to Machn SSM CF 1 0 = 18466 1796 = 10 143 0 Su of Suars du to Shfts SSS CF = 18037-1796 = 81 70 Su of Suars du to Intracton (SSMS) 4 41 3 8 30 44 31 38 CF SSM SSS = 1861 1796 10 81 = 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 = 104-10 -81-6 = 848
For SSM MSSM df SSS MSSS df SSE MSSE df 81 1 SSMS MSSMS df 10 3 848 31 17 81 6 6 3 167 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 = 6 4 6 : Man ffct of Shft A = Shft B, s F Slarly, for tstng 81 6 306 H 0AB 167 F 0817 6 : 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) 3 10 17 Du to Shft 1 81 81 Du to Intracton 3 6 167 Du to Error 3 848 6 Total 39 104 F-tst or Varanc Rato 17 64 6 81 6 167 6 306 0817 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
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, 9 1066, 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 = 11764483 RSS = 18806 TSS = 11174 SSA = 1107070 SSB = 97 SSAB = 116 SSE = 6389
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 1 1107070 1107070 1107070/34 = 0793 F(1,1) = 439 Cak Mx 97 1478 1478/34 = 8 Intracton 116 63 63/34 = 106 F(,1) = 1941 F(,1) = 1941 Error 1 6389 34 Total 17 11174 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
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) 1 3 4 6 7 8 9 10 11 1 14 16 0 4 30 1 161 00 16 30 34 37 39 41 4 43 44 4 46 48 49 0 4 4,0 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 6,366 181 1900 1916 19 1930 1933 1936 1937 1938 1939 1940 1941 194 1943 1944 194 1946 190 9849 9900 9917 99 9930 9933 9934 9936 9938 9940 9941 994 9943 9944 994 9946 9947 990 3 1013 9 98 91 901 894 888 884 881 878 876 874 871 869 866 864 86 83 341 308 946 871 84 791 767 749 734 73 713 70 69 683 669 660 60 61 4 771 694 69 639 66 616 609 604 600 96 93 91 87 84 80 77 74 63 0 1800 1669 198 1 11 1498 1480 1466 144 144 1437 144 141 140 1393 1383 1346 661 79 41 19 0 49 488 48 478 474 470 468 464 460 46 43 40 436 166 137 106 1139 1097 1067 104 107 101 100 996 989 977 968 9 947 938 90 6 99 14 476 43 439 48 41 41 410 406 403 400 396 39 387 384 381 367 1374 109 978 91 87 847 86 810 798 787 779 77 760 7 739 731 73 688 7 9 447 43 41 397 387 379 373 368 363 360 37 3 349 344 341 338 33 1 9 84 78 746 719 700 684 671 66 64 647 63 67 61 607 98 6 8 3 446 407 384 369 38 30 344 339 334 331 38 33 30 31 31 308 93 116 86 79 701 663 637 619 603 91 8 74 67 6 48 36 8 0 486 9 1 46 386 363 348 337 39 33 318 313 310 307 30 98 93 90 86 71 106 80 699 64 606 80 6 47 3 6 18 11 00 49 480 473 464 431 10 496 410 371 348 333 3 314 307 30 97 94 91 86 8 77 74 70 4 1004 76 6 99 64 39 1 06 49 48 478 471 460 4 441 433 4 391 11 484 398 39 336 30 309 301 9 90 86 8 79 74 70 6 61 7 40 96 70 6 67 3 07 488 474 463 44 446 440 49 41 410 40 394 360 1 47 388 349 36 311 300 9 8 80 76 7 69 64 60 4 0 46 30 933 693 9 41 06 48 46 40 439 430 4 416 40 398 386 378 370 336 13 467 380 341 318 30 9 84 77 7 67 63 60 1 46 4 38 1 907 670 74 0 486 46 444 430 419 410 40 396 38 378 367 39 31 316 14 460 374 334 311 96 8 77 70 6 60 6 3 48 44 39 3 31 13 886 61 6 03 469 446 48 414 403 394 386 380 370 36 31 343 334 300 1 44 368 39 306 90 79 70 64 9 1 48 43 39 33 9 07 868 636 4 489 46 43 414 400 389 380 373 367 36 348 336 39 30 87 16 449 363 34 301 8 74 66 9 4 49 4 4 37 33 8 4 0 01 83 63 9 477 444 40 403 389 378 369 361 3 34 337 3 318 310 7 17 44 39 30 96 81 70 6 0 4 41 38 33 9 3 19 1 196 840 611 18 467 434 410 393 379 368 39 3 34 33 37 316 308 300 6 18 441 3 316 93 77 66 8 1 46 41 37 34 9 19 1 11 19 88 601 09 48 4 401 38 371 360 31 344 337 37 319 307 300 91 7 19 438 3 313 90 74 63 48 43 38 34 31 6 1 1 11 07 188 818 93 01 40 417 394 377 363 3 343 336 330 319 31 300 9 84 49 0 43 349 310 87 71 60 4 40 3 31 8 3 18 1 08 04 184 810 8 494 443 410 387 371 36 34 337 330 33 313 30 94 86 77 4 1 43 347 307 84 68 7 49 4 37 3 8 0 1 09 0 00 181 74 80 78 487 437 404 381 36 31 340 331 34 317 307 99 88 80 7 36
TABLE (Contnud) Dgrs of Two-Way Anova wth Obsrvatons Pr Cll Frdo: Dgrs of Frdo: Nurator Dnonator 1 3 4 6 7 8 9 10 11 1 14 16 0 4 30 430 344 30 8 66 47 40 3 30 3 3 18 13 07 03 198 178 794 7 48 431 399 376 39 34 33 36 318 31 30 94 83 7 67 31 3 48 34 303 80 64 3 4 38 33 8 4 0 14 10 04 00 196 176 788 66 476 46 394 371 34 341 330 31 314 307 97 89 78 70 6 6 4 46 340 301 78 6 1 43 36 30 6 18 13 09 0 198 194 173 78 61 47 4 390 367 30 336 3 317 309 303 93 8 74 66 8 1 44 338 99 76 60 49 41 34 8 4 0 16 11 06 00 196 19 171 777 7 468 418 386 363 346 33 31 313 30 99 89 81 70 6 4 17 6 4 337 98 74 9 47 39 3 7 18 1 10 0 199 19 190 169 77 3 464 414 38 39 34 39 317 309 30 96 86 77 66 8 0 13 7 41 33 96 73 7 46 37 30 0 16 13 08 03 197 193 188 167 768 49 460 411 379 36 339 36 314 306 98 93 83 74 63 47 10 8 40 334 9 71 6 44 36 9 4 19 1 1 06 0 196 191 187 16 764 4 47 407 376 33 336 33 311 303 9 90 80 71 60 44 06 9 418 333 93 70 4 43 3 8 18 14 10 0 00 194 190 18 164 760 4 44 404 373 30 333 30 308 300 9 87 77 68 7 49 41 03 30 417 33 9 69 3 4 34 7 1 16 1 09 04 199 193 189 184 16 76 39 41 40 370 347 330 317 306 98 90 84 74 66 47 38 01 384 99 60 37 1 09 01 194 188 183 179 17 169 164 17 1 146 100 664 460 378 33 30 80 64 1 41 3 4 18 07 199 187 179 169 100 7