Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Transcription

1 nalysis of Variance and Design of Experiment-I MODULE V LECTURE - 9 FCTORIL EXPERIMENTS Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

2 Sums of squares Suppose n 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 symbol within a square bracket. For example, [ab] denotes the total t yield from the plots receiving i the treatment t t combination (ab). In a factorial experiment, the factorial effect totals t are [ ] = [ ab] [ b] + [ a] [ ] [ab] = treatment total, i.e. sum of r observations in which both the factors and B are at second level. [a] = treatment total, i.e., sum of r observations in which factor is at second level and factor B is at first level. [b] = treatment total, i.e., sum of r observations in which factor is at first level and factor B is at second level. [] = treatment total, i.e., sum of r observations in which both the factors and B are at first level. Thus r [ ] = yi( ab) yi( b) + yi( a) yi() = i= ' y (say). where is a vector of + and - and y is a vector denoting the responses from ab, b, a and. Similarly, other effects can also be found.

3 3 The sum of squares due to a particular effect is obtained as [ Total yield]. Total number of observations In a factorial experiment in an RBD, the sum of squares due to is ' ( y). SS = r Ina n factorial experiment in an RBD, the divisor will be r. n. If latin square design is used based on n x n Latin square, then r is replaced by n.

4 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 presented. It has following steps:. First write the treatment combinations in the standard order in the column at the beginning of table, called as treatment column.. Find the total yield for each treatment. Write this as second column of the table, called as yield column. 3. Obtain columns (), (),,(n) successively (i) obtain column () from yield column a) upper half is obtained by adding yields 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 (), (3),, (n) are obtained from preceding ones in the same manner as used for getting () from the yield columns. 4. This process of finding columns is repeated n times in n factorial experiment. [ ] column( n) 5. Sum of squares due to interaction =. Total number of observations

5 5 Example: Yates procedure re for a factorial experiment Treatment combinations Yield (total from all r replicates) () () () a b ab () (a) (b) (ab) () + (a) (b) + (ab) (a) - () (ab) - (b) () + (a) + (b)+ (ab) = [M ] -() + (a) - (b) + (ab) = [] -() - (a) + (b) + (ab) = [B] () - (a) - (b) +(ab)=[b] Note: The columns are repeatedly obtain times due to factorial experiment. Now SS = SSB = SSB = [ ] 4r [ B ] 4r [ B] 4r.

6 6 Example: Yates procedure for a 3 factorial experiment Treatment Yield (total from all r replicates) () () (3) (4) (5) (6) () u = () + ( a) v = u+ u w = v+ v [ M ] a ( a ) u = ( b) + ( ab) v = u3 + u w 4 = v3 + v [ ] 4 b (b) u3 = () c + ( ac) v3 = u5 + u6 w3 = v5 + v [ B ] 6 ab ( ab ) u4 = ( bc) + ( abc) v4 = u7 + u8 w4 = v7 + v [ B ] 8 c ( ac ) u5 = ( a) () v5 = u u7 w5 = v v [ C ] ac (ac ) u6 = ( ab) ( b) v6 = u4 u3 w6 = v4 v [ C ] 3 bc ( bc ) u7 = ( ac) ( c) v7 = v6 u5 w7 = v6 v [ BC ] 5 abc ( abc ) u8 = ( abc) ( bc) v8 = u8 u7 w8 = v8 v [ BC ] 7 The sum of squares are obtained as follows when the design is RBD: SS( Effect) = [ Effect] r. 3 For the analysis of n factorial experiment, the analysis of variance involves the partitioning i of treatment t t 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 7 For example: In 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 - Treatments 4 =3 B C Error 3(r - ) [ ] /4r [ B] /4r [ B] /4 r Total 4r - The decision rule is to reject the concerned null hypothesis when the related F - statistic F > F (,3( r )). effect α