Topic 3. Single factor ANOVA [ST&D Ch. 7]

Transcription

1 Topic 3. Single facor ANOVA [ST&D Ch. 7] "The analyi of variance i more han a echnique for aiical analyi. Once i i underood, ANOVA i a ool ha can provide an inigh ino he naure of variaion of naural even" Sokal & Rohlf (1995, BIOMETRY The F diribuion [ST&D p. 99] From a normally diribued populaion (or from populaion wih equal variance σ : 1. Sample n 1 iem and calculae heir variance. Sample n iem and calculae heir variance 3. Conruc a raio of hee wo ample variance Thi raio will be cloe o 1 and i expeced diribuion i called he F- diribuion, characerized by wo value for df ( 1 = n 1 1, = n / F(1,40 F(8,6 F(6,8 Fig 1. Three repreenaive F- diribuion. 1 F 3 4 F value: proporion of he F-diribuion o he righ of he given F-value wih df 1 = 1 for he numeraor and df = for he denominaor. Thi F e can be ued o e if wo populaion have he ame variance: uppoe we pull 10 obervaion a random from each of wo populaion and wan o e H 0 : 1 v. H 1 : 1 (a wo-ailed e. F 0.05,[ df19, df9] 4.03 Inerpreaion: The raio 1 / from ample of en individual from normally diribued populaion wih equal variance, i expeced o be larger han 4.03 (F /=0.05 or lower han 0.4 (F 1-/=0.975 no in Table by chance in only 5% of he experimen. In pracice, hi e i rarely ued o e difference beween variance becaue i i very eniive o deparure from normaliy. I i beer o ue Levene e. 1

2 3. 3. Teing he hypohei of equaliy of wo mean [ST&D p ] The raio beween wo eimae of can be ued o e difference beween mean: H o : 1 - = 0 veru H A : 1-0 How can we ue variance o e difference beween mean? By being creaive in how we obain eimae of. F eimae. of.. from. mean eimae. of.. from. individual The denominaor i an eimae of from he individual wihin each ample. If here are muliple ample, i i a weighed average of hee ample variance. The numeraor i an eimae of from he variaion of mean among ample. Recall: Y, o n n Y Thi implie ha mean may be ued o eimae by muliplying he variance of ample mean by n. So F can be expreed a: F among wihin n Y When he wo populaion have differen mean (bu ame variance, he eimae of baed on ample mean will include a conribuion aribuable o he difference beween populaion mean and F will be higher han expeced by chance. Noe: an F e wih 1 df in he numeraor ( reamen i equivalen o a e: F (1, df, 1 - = df, 1- /

3 Nomenclaure: o be conien wih ST&D ANOVA we will ue r for he number of replicaion and n for he oal number of experimenal uni in he experimen Example : Table 1. Yield (100 lb./acre of whea varieie 1 and. Varieie Replicaion Y i. Y i. i Y 1. = Y. = Y.. = 185 Y.. = 18.5 = reamen and r = 5 replicaion, We will aume ha he wo populaion have he ame (unknown variance. 1 Eimae he average variance wihin ample or experimenal error ( r1 1 1 ( r 1 pooled = 4* * 4.0 / (4 + 4= 5.5 ( r 1 ( r 1 1 Eimae he beween ample variabiliy. Under H o, Y 1. andy. eimae he ame populaion mean. To eimae he variance of mean: ( Yi. Y.. i1 = [( ( ] / (-1= 4.5 Y 1 Therefore he beween ample eimae i b = r Y = 5 * 4.5=.5 Thee wo variance are ued in he F e a follow: F = b / w =.5/5.5 = 4.9 under our aumpion (normaliy, equal variance, ec., hi raio i diribued according o an F (-1, (r-1 diribuion. Thi reul indicae ha he variabiliy beween he ample i 4.9 ime larger han he variabiliy wihin he ample. b =1 ( reamen 1 w = (r-1 = 4 + 4= 8. From Table A.6 ( p. 614 ST&D criical F 0.05, 1, 8 = 5.3. Since 4.9 < 5.3 we fail o rejec H 0 a =0.05 ignificance level. An F value of 4.9 or larger happen ju by chance abou 7% of he ime for hee degree of freedom. 3

4 The linear addiive model Model I ANOVA or fixed model: Treamen effec are addiive and fixed by he reearcher Y ij = + i + ij Y ij i compoed of he grand mean of he populaion plu an effec for he reamen i plu a random deviaion ij. = average of reamen ( 1,, herefore i = 0 Hypohei: H 0 : i 1 =... = i = 0 v. H A : ome i 0 The daa repreen he model a: Yij =Y.. + (Y i. -Y.. + (Yij - Y i. 1. Experimenal error are random, independen and normally diribued abou a zero mean, and wih a common variance.. When H o i fale hree will be an addiional componen in he variance beween reamen = r i /(-1 3. The reamen effec i muliplied by r becaue i a variaion among mean baed on r replicaion. The Compleely Random Deign CRD CRD i he baic ANOVA deign. A ingle facor i varied o form he differen reamen. Thee reamen are applied o independen random ample of ize n. The oal ample ize i r*= n. H 0: 1 = = 3 =... = again H 1 : no all he i ' are equal. 4

5 The reul are uually ummarized in an ANOVA able: Source df Definiion SS MS F Treamen - 1 SST SST/(-1 MST/ r( Y i. Y.. MSE Error Toal n - 1 (r-1=n- i i1 i, j n j1 ( Y Y i. TSS ij ( Y ij Y.. SST TSS SSE/(n- SST: muliplicaion by r reul in MST eimaing raher han /r TSS = SST + SSE We can decompoe he oal TSS ino a porion due o variaion among group and anoher porion due o variaion wihin group. The degree of freedom ( are alo addiive: (n-+(-1= n-1 MST= SST/(-1, he mean quare for reamen I i an independen eimae of when H 0 i rue. MSE= SSE/(n- i he mean quare for error I give he average diperion of he iem around he group mean. I i an eimae of a common I he wihin variaion or variaion among obervaion reaed alike. MSE eimae he common if he reamen have equal variance. F=MST/MSE. We expec F approximaely equal o 1 if no reamen effec. MST r i /( 1 Expeced MSE 5

6 Some advanage of CRD (ST&D 140 Simple deign Can accommodae well unequal number of replicaion per reamen Lo of informaion due o miing daa i maller han in oher deign The number of d.f. for eimaing experimenal error i maximum Can accommodae unequal variance, uing a Welch' variance-weighed ANOVA (Biomerika 1951 v38, 330. The diadvanage of CRD (ST&D 141 The experimenal error include he enire variaion among e.u. excep ha due o reamen. Aumpion aociaed wih ANOVA Independence of error: Guaraneed by he random allocaion of e.u. Normal diribuion: Shapiro and Wilk e aiic W (ST&D p.567, and SAS PROC UNIVARIATE NORMAL. Normaliy i rejeced if W i ufficienly maller han 1. Homogeneiy of variance: o deermine if he variance i he ame wihin each of he group defined by he independen variable. Levene' e: i more robu o deviaion from normaliy. Levene e i an ANOVA of he quare of he reidual of each obervaion from he reamen mean. Original daa Reidual T1 T T3 T1 T T Treamen mean 6

7 Power The power of a e i he probabiliy of deecing a real reamen effec. To calculae he ANOVA power, we 1 calculae he criical value (a andardized meaure of he expeced difference among mean in uni. depend on: The number of reamen (k The number of replicaion (r -> When r Power The reamen difference we wan o deec (d -> When d Power An eimae of he populaion variance ( = MSE The probabiliy of rejecing a rue null hypohei (. r MSE wih i = µ i - µ In a CRD we can implify hi general formula if we aume all i =0 excep he exreme reamen effec µ (k µ (1. If d= µ (k - µ (1 i = d/ k i i k ( d / ( d / k d / 4 d k / 4 d / k d k And he approximae formula implifie o d k * * r MS error Enering he char for 1 = df = k-1 and (0.05 or 0.01 he inercepion of and = df = k(n-1 give he power of he e. Example: Suppoe ha one experimen ha k=6 reamen wih r= replicaion each. The difference beween he exreme mean wa 10 uni, MSE= 5.46, and he required = 5%. To calculae he power: d * r * MSE 10 * (6* Ue Char v 1 = k-1= 5 and ue he e of curve o he lef ( = 5%. Selec curve v = k(n-1= 6. The heigh of hi curve correponding o he abcia of =1.75 i he power of he e. In hi cae he power i lighly greaer han

8 Sample ize To calculae he number of replicaion n for a given and power: a Specify he conan, b Sar wih an arbirary number of n o compue, c Ue Pearon and Harley char o find he power, d Ierae he proce unil a minimum r value if found which aifie a required power for a given level. Example: Suppoe ha 6 reamen will be involved in a udy and he anicipaed difference beween he exreme mean i 15 uni. Wha i he required ample ize o ha hi difference will be deeced a = 1% and power = 90%, knowing ha = 1? (noe, k = 6, = 1%, = 10%, d = 15 and MSE = 1. d k * * r MS error n df (1- for =1% 6(-1= (3-1= (4-1= Thu 4 replicaion are required for each reamen o aify he required condiion. 8

9 Subampling: he need deign If meaure of he ame experimenal uni vary oo much, he experimener can make everal obervaion wihin each experimenal uni. Such obervaion are made on ubample or ampling uni. Example Sampling individual plan wihin po where he po (e.u.. Sampling individual ree wihin an orchard plo (e.u.. Hierarchical way or need analyi of variance. Can have muliple level Applicaion of need ANOVA: Tr. Po Plan a Acerain he magniude of error a variou age of an experimen or an indurial proce. b Eimae he magniude of he variance aribuable o variou level of variaion in a udy (e.g. quaniaive geneic. c Dicover ource of variaion in naural populaion or yemaic udie Linear model for ubampling: Y ijk = + i + j (i + k (ij Two random elemen are obained wih each obervaion: and. The k (ij are he error aociaed wih each ubample. The k (ij are aumed normal wih mean 0 and variance. The ubcrip j(i indicae ha he j h level of facor B (po i need under he i h level of facor A (reamen. Thi mean ha oher meaure below ha level are no real replicaion. Thi i repreened in he daa a Y ijk =Y... + (Y i.. -Y... + (Y ij. - Y i.. + (Y ijk - Y ij.. The do noaion: he do replace a ubcrip and indicae ha all value covered by ha ubcrip have been added 9

10 Need ANOVA wih equal ubample number: compuaion Example: ST&D p. 159: 6 reamen and 3 po need under each level of reamen (replicaion, and 4 ubample. Variable: em growh Noe ha he Po number i ju an ID (no a reamen Treamen 1 Treamen T 1 : low T/ 8 h T :low T/ 1 h T 3 :low T/ 16 h T 4 :high T/ 8 h T 5 :high T/ 1 h T 6 :high T/ 16 h Po number Po number Po number Po number Po number Po number Plan N o Po oal = Y ij Treamen oal = Y i Treamen mean= Y i In hi example = 6, r = 3, and = 4, and n = r = 7 CRD Po 1 Po Po 3 Po 1 Po Po 3 r r ( Yij Y r( Yi Y ( Yij Yi i1 j1 i1 i1 j or TSS = SST + SSE. Degree of freedom: TSS = n-1, SST =-1, and + SSE =n-, repecively. In he need deign he SST i unchanged bu he SSE i furher pariioned ino wo componen. The reuling equaion can be wrien Need CRD r i1 j1 k 1 ( Y ijk Y... r i1 ( Y i.. Y... i1 r j1 ( Y ij. Y i.. i1 r j1 k 1 ( Y ijk Y ij. or SS = SST + SSEE + SSSE SSE= SSEE + SSSE 10

11 SSEE: Sum of quare due o experimenal error: variaion among plo (po wihin reamen. SSSE: Sum of quare due o ampling error: variaion among ubample (plan wihin plo (po. ANOVA able: Source of variaion df SS MS F Expeced MS Treamen ( i - 1 = 5 SST SST/ 5 MST/MSEE /5 Exp. Error (e j(i (r - 1 = 1 SSEE SSEE/ 1 MSEE/MSSE +4 Samp. Error (d k (ij r ( - 1 = 54 SSSE SSSE/ 54 Toal r - 1 = 71 SS In eing a hypohei abou populaion reamen mean, he appropriae divior for F i he experimenal error MS ince i include variaion from all ource ha conribue o he variabiliy of reamen mean excep reamen. To eimae he differen componen of variance in he po experimen: Variance Sum of Mean Variance Percen of Source df Square Square componen oal Toal % rm % po % error % MSSE= 0.93 MSEE= +4 = (MSEE - /4= ( /4= 0.30 MST= +4 +Tr Tr= (MST MSEE/1= ( /1=.81 SAS CODE proc GLM; cla rm po; model growh= rmn po (rm; random po (rm; e h=rm e=po(rm; proc VARCOMP; cla rm po; model growh= rm po (rm; Po(rm: Po only ha meaning wihin reamen (i an ID variable. Po 1 in reamen 1 i no relaed o Po 1 in reamen! 11

12 The opimal allocaion of reource ST&D 173 or Biomery Sokal & Rohlf pg. 309 One of he main reaon o do a need deign i o inveigae how he variaion i diribued beween experimenal uni and ubample. The relaive efficiency i no meaningful, unle he relaive co of obaining he wo deign are aken ino conideraion. If one deign i wice a efficien a anoher bu a he ame ime i en ime a expenive we migh no chooe i. Co funcion. C N ub * Neu ( CSUB Neu ( Ceu To calculae he number of ubample (N per experimenal uni ha will reul in imulaneou minimal co and minimal variance he following formula may be ued: N ub The opimum number of ubample will increae when he relaive co of he ubample i low and he variance wihin he experimenal uni i high (S SUB. C C e. u. ub * * ub e. u. 3* * 0.30 Example Po: $3, plan $1 hen: N ub 3 opimum 3 plan per po If C eu = C SUB SUB N ubample are valuable only if e.u. beween ubample > beween e.u. In erm of efficiency, ubampling i only ueful when he variaion among ubample i larger han he variaion among experimenal uni and/or he co of he ubample i maller han he co of he experimenal uni. 1