Analysis of Variance and Desiggpn of Experimenteriment--II MODULE ––VV LECTURE - 29 FACTORIAL EXPERIMENTS Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 2 Sums of squares Suppose 2n factorial experiment is carried out in a randomized block design with r replicates. Denote the total yield (output) from r plotes (experimental units) receiving a particular treatment combination by the same symblbol w ithin a square bktFbracket. For examp le, [ab] deno tes the to ta l y ie ld from the p lo ts rece iv ing the tttreatmen t combination (ab). I2In a 22 ftilfactorial exper imen tthftilfftttlt, the factorial effect totals are []Aabba=−+− [ ][][][1] [ab] = treatment total, i.e. sum of r observations in which both the factors A and B are at second level. [a] = treatment total, i.e., sum of r observations in which factor A is at second level and factor B is at first level. [b] = treatment total, i.e., sum of r observations in which factor A is at first level and factor B is at second level. [1] = treatment total, i.e., sum of r observations in which both the factors A and B are at first level. Thus r ⎡⎤ []Ayyyy=−+−∑ ⎣⎦iab() ib () ia () i (1) i=1 ' = A AAy (say). where A A is a vector of +1 and -1 and y A is a vector denoting the responses from ab, b, a and 1. Similarly, other effects can also be found. 3 The sum of squares due to a particular effect is obtained as 2 []Total yield . Total number of observations In a 22 factorial experiment in an RBD, the sum of squares due to A is ()A'2y SSA = AA . r22 Ina2n factorial experiment in an RBD, the divisor will be r. 2n . If latin square design is used based on 2n x2n Latin square , then r is replaced by 2n. 4 Yates method of computation of sum of squares Yates method gives a systematic approach to find the sum of squares. We are not presenting here the complete method. Only the part which is used for computing only the sum of squares is presented and the method to verify them is not presentdted. It has following steps: 1. First write the treatment combinations in the standard order in the column at the beginning of table, called as treatment column. 2. Find the total yield for each treatment. Write this as second column of the table, called as yield column. 3. Obtain columns (1), (2),…,(n) successively (i) obtain column (1) from yield column a) upper ha lf is o bta ine d by a dding y ie lds in pairs. b) second half is obtained by taking differences in pairs, the difference obtained by subtracting the first term of pairs from the second term. (ii) The columns (2), (3),…, (n) are obtained from preceding ones in the same manner as used for getting (1) from the yield columns. 4. This process of finding columns is repeated n times in 2n factorial experiment. []column()n 2 5. Sum of squares due to interaction = . Total number of observations 5 EampleExample: Yates procedu re for a 22 factorial ex periment Treatment Yield (1) (2) combinations (total from all r replicates) (1) (1) (1) + (a) (1) + (a) + (b)+ (ab) = [M ] a (a) (b) + (ab) -(1) + (a) - (b) + (ab) = [A] b (b) (a) - (1) -(1) - (a) + (b) + (ab) = [B] ab (ab) (ab) - (b) (1) - (a) - (b) +(+ (ab)=[) = [AB] Note: The columns are repeatedly obtain 2 times due to 22 factorial experiment. 2 []A Now SSA = 4r 2 [B] SSB = 4r 2 []AB SSAB = . 4r 6 Example: Yates procedure for a 23 factorial experiment Treatment Yield (total from all r replicates) (()1) (()2) (()3) (()4) (()5) (()6) 1 (1) ua1 = (1)+ ( ) vuu112= + wvv112= + []M a ()a []A ubab2 = ()+ ( ) vuu234= + wvv234= + b (b) ucac3 = ()+ ( ) vuu356= + wvv356= + []B ab ()ab u4 = ()( bc+ abc ) vuu478= + wvv478= + []AB c ()ac ua5 = ()− (1) vuu527= − wvv521= − []C ac (ac ) uabb6 = ()()− vuu643= − wvv643= − []AC bc ()bc uacc7 = ()()− vvu765= − wvv765= − []BC abc ()abc u8 = ()() abc− bc vuu887= − wvv887= − []ABC The sum of squares are obtained as follows when the design is RBD: 2 []Effect SS() Effect = r.23 FthFor the analilysis o f2f 2n fac tor ia l exper imen t, the ana lys is o f var iance invo lves the par titioni ng of t reat ment sum of squares so as to obtain the sum of squares due to main and interaction effects of factors. These sum of squares are mutually orthogonal, so Total SS = Total of all the SS due to main and interaction effects. 7 For example: In 22 factorial experiment in an RBD with r replications, the division of degrees of freedom and the treatment sum of squares are as follows: Source Degrees of Sum of squares freedom Replications r -1 Treatments 4 – 1=31 = 3 2 A 1 [ A] /4r 2 B 1 [Br] /4 2 C 1 [ ABr] /4 Error 3(r -1) Total 4r -1 The decision rule is to reject the concerned null hypothesis when the related F - statistic FFeffect >−1−α (1, 3(r 1))..