BLOCK DESIGNS WITH FACTORIAL STRUCTURE

Transcription

1 BLOCK DESIGNS WITH ACTORIAL STRUCTURE V.K. Gupta and Rajender Parsad I.A.S.R.I., Library Avenue, New Delhi The purpose of this talk is to expose the participants to the interesting work on block designs with factorial structure. or such designs it is possible to develop a one-to-one correspondence between the treatments of the block designs and the treatment combinations of a factorial experiment. Such a correspondence helps in analyzing block designs as confounded factorials and vice-versa. Some earlier attempts to identify such a correspondence can be traced back to the development of quasi-factorials or lattice designs. These designs admit simple factorial analysis even for mixed factorials. An account of these designs is available in textbooks (see e.g., Kempthorne, 95). We now describe an example to clarify the idea of correspondence between the block designs and factorial experiments. Consider a resolvable balanced incomplete block (BIB) design with blocks arranged as follows: B B B 3 B 4 B 5 B Let us associate with this design a -factorial experiments with two-factors as and, 0 and denoting the two levels of each of the two factors. Assign the four treatment combinations for the factorial experiment to the four treatments of the BIB design through the correspondence: 00 ; 0 ; 0 3; 4. This gives the following factorial design: Blocks or the BIB designs, the concurrence matrix is of the form NN ' = and the matrix M = NN'/rk = NN'/ 6.

2 Block Designs with actorial Structure The main effects and and the interaction can be estimated by using the following three mutually orthogonal observational contrasts: -(00) -(0) +(0) +() -(00) +(0) -(0) +() +(00) -(0) -(0) +() where (00), (0), (0) and () denote, respectively, the total of the observations from the experimental units receiving the treatment combinations 00, 0, 0 and. The vectors of coefficients s, s and s 3, corresponding to these contrasts, can be written as s = (- - ) = (- ) ( ) s = (- - ) = ( ) (- ), s3 = ( - - ) = (- ) (- ). It can be verified easily that these contrasts are eigen vectors of N N and hence of M. These contrasts are therefore basic contrasts. As the corresponding eigen values of M are each equal to /3 [ λ v / rk = 4/ ( 3x) = /3], it follows that the main effects and the two-factor interaction are estimated with the same loss of information /3. We shall see later that every BIB design with v = mxkxmn treatments can always be used as an m K m n factorial experiment and for such a design every effect is estimated with same relative loss of information ( r λ )/ rk [- λv/rk = ( rk - λv) / rk and using λ(v - ) = r(k - ), we get the result]. As a factorial experiment this class of designs has little utility because all the effects are confounded with the blocks to the same extent. Consider now the following group divisible partially balanced incomplete block (PBIB) design with groups (00, 0) and (0, ) and the parameters v = 4, b = 4, r =, k =, λ = 0, λ =, m =,n =. The plan of the design can be obtained by deleting the first two blocks from the plan of the BIB design just considered. Blocks The concurrence matrix of this design is 36

3 Block Designs with actorial Structure 0 NN = and the matrix M = NN / rk = NN / 4. It is easy to verify that s is an eigen vector of N N corresponding to the eigenvalue 0 and s and s 3 are the eigen vectors of N N corresponding to the eigenvalue, where s, s and s 3 are as defined above. Thus the block design provides a factorial experiment in which the main-effect is unconfounded and is therefore estimated with full efficiency, whereas the main effect and the interaction are each estimated with relative loss of information /. As a design for a factorial experiment this is superior to BIB design as this design not only estimates free from block effects but also requires less number of experimental units. Remark: Notice that the structures of N N given above are cyclic in terms of partitioned matrices, i.e. N N = { N N }, where { } denotes the cyclic pattern. The N and N matrices for the examples considered above are 3 0, and,, 3 0 respectively. Also row sums and column sums of N and N matrices are constant. It may be noted that such a structural form of N N helps in simplifying the analysis. Remark: The structure of the contrasts, as defined earlier, reveals that the higher order contrasts can be expressed as the Kronecker product of lower dimensional contrasts or eigen vectors. This property, which is true in general, is extremely helpful in determining the pattern of analysis. It is well known that the treatment contrasts of Kronecker product designs have such patterns. We now give some useful results related to block designs with factorial structure. A block design is said to have an orthogonal factorial structure if the adjusted sum of squares due to treatments in the block design can be partitioned orthogonally into sum of squares corresponding to main-effects and interactions of the factorial experiment. John and Smith (978) and Cotter, John and Smith (973) obtained sufficient conditions for a block design to have factorial structure. Mukerjee (979, 980, 98) obtained a necessary and sufficient condition on N N for the design to have factorial structure. The necessary and sufficient condition on N N is as follows: 363

4 Block Designs with actorial Structure The matrix N N can be expressed as a linear combination of Kronecker products of square matrices with same row and column totals. or the structures of John and Smith (97) and Cotter, John and Smith (973) as well as for designs with property (A), of Kurkjian and Zelen (963), N N and hence Ω matrix, Ω = ri NN / k + ( r / v) have structures that are linear combinations of Kronecker products of square matrices with same row and column totals or are circulant matrices having cyclic structure. We give below an illustration of a balanced confounded factorial experiment with three factors, viz., Nitrogen (40, 80 and 0 kg./hectare), Phosphorous (0, 40 and 80 kg./ hectare) and Potassium (0 and 40 kg./hectare). These 8 treatment combinations were arranged in 3 blocks of size 6 each. The experiment was conducted with 4 replications. To each of the blocks one control treatment was also added. There will now be 7 plots per block and experimental units in each replication. The analysis of the data was performed using PROC GLM of SAS. The main effects and interactions with single degree of freedom were obtained using contrast analysis. The comparison of the 8 treatment combinations with control was also made. The SAS commands and the output is given in the sequel. Options linesize=7; data ludh; input rep block N P K trt yield; cards;

5 Block Designs with actorial Structure

6 Block Designs with actorial Structure ; proc glm; class rep block trt; model yield = rep block(rep) trt; Contrast 'K' trt ; Contrast 'PL' trt ; Contrast 'PQ' trt ; Contrast 'NL' trt ; Contrast 'NQ' trt ; Contrast 'PLK' trt ; Contrast 'PQK' trt ; Contrast 'NLK' trt ; Contrast 'NQK' trt ; Contrast 'NLPL' trt ; Contrast 'NLPQ' trt ; Contrast 'NQPL' trt ; Contrast 'NQPQ' trt ; Contrast 'NLPLK' trt ; Contrast 'NLPQK' trt ; Contrast 'NQPLK' trt ; Contrast 'NQPQK' trt ; Contrast 'Con Vs Oth' trt -8 ; run; Output General Linear Models Procedure Class Level Information Class Levels Values REP BLOCK 3 3 TRT Number of observations in data set = 84 General Linear Models Procedure Dependent Variable: YIELD Sum of Mean Source D Squares Square Value Pr > Model Error Corrected Total

7 Block Designs with actorial Structure R-Square C.V. Root MSE YIELD Mean Source D Type I SS Mean Square Value Pr > REP BLOCK(REP) TRT Source D Type III SS Mean Square Value Pr > REP BLOCK(REP) TRT Contrast D Contrast SS Mean Square Value Pr > K PL PQ NL NQ PLK PQK NLK NQK NLPL NLPQ NQPL NQPQ NLPLK NLPQK NQPLK NQPQK Cont vs oth In the above we have discussed the procedure of bifurcation the treatment sum of squares into single df contrasts pertaining to various components of main effects and interactions. If, however, one is interested in obtaining the sum of squares due to main effects and interactions without partitioning them into single df contrasts. or this analyze the data from factorial experiments as per procedure of the design adopted and then bifurcate the treatment sum of squares into main effects and interactions. This can easily be done through contrast analysis. One has to define the set of contrasts for each of the main effects and interactions. Before describing the procedure of defining contrasts for main effects and interactions, we give some preliminaries. In general, let there be n-factors, say,,..., n th and i factor has si levels, n s i i= i =,..., n. The treatment combinations 367

8 Block Designs with actorial Structure in the lexico-graphic order are given by a a a a... n where denotes the symbolic i = 0,,..., s i '; =,,...,. The total number of factorial effects n direct product and ( ) i n (main effects and interactions) are. The set of main effects and interactions have a one-one correspondence with Ω, the set of all n-component non-null binary vectors. or example a typical p-factor interaction. g, g,..., g ( n ) p ( g g... g n, p n) p corresponds to the element x = x,..., x of Ω such that x = x =... = x = and = 0 for u g, g,..., g p. The treatment contrasts belonging to different interactions given by P x x x x x t, where P = P P... P n where x i i P = P if x = i i = ' si if x i = 0 g n g g p x u ( x x ) Ω x, x = n,..., where P i is a ( s i ) si matrix of complete set of linearly independent contrasts of order 0 0 s i and s i is a si vector of ones. or example, if s i = 4, then P i = 0. 3 To illustrate the above consider the case of a factorial experiment. Let the three factors be represented by 3. Then Ω, the set of non-null binary vectors is given by Ω = { 00, 00,,00, 0, 0, 0, }. Main effects and the interactions,,,, and are represented by,,,,,,. The matrices for these three factors are P i 3 3, P = ; P = and P 3 = The coefficient matrices of the treatment contrasts for the above set of main effects and interactions are given by are 368

9 Block Designs with actorial Structure actorial Effect Coefficient Matrix : 00 P = P 4 5 : 00 P = 3 P 5 3 : 00 P = 3 4 P3 : 0 P = P P 5 3 : 0 P = P 4 P3 3 : 0 P = 3 P P3 3 : P = P P P3 or sum of squares of these contrasts and testing of hypothesis, please see Contrast Analysis given in Module (Volume I) in the lecture on undamental of Design of Experiments. References Cotter, S.C., John, J.A., and Smith T.M.. (973). Multifactor experiments in nonorthogonal designs. Journal of Royal Statistical Society, B35, John, J.A. and Smith, T.M.., (97). Two factor experiments in non-orthogonal designs. Journal of Royal Statistical Society, B34, Kurkjian, B. and Zelen, M. (963). Application of calculas of factorial arrangements I: block and direct product designs. Biometrika, 50, Mukherjee, R. (979). Inter-effect orthogonality in factorial experiments. Calcutta Statistical Association Bulletin, 8, Mukherjee, R. (980). urther results on analysis of factorial experiments. Calcutta Statistical Association Bulletin, 9, -6. Mukherjee, R. (98). Construction of effect-wise orthogonal factorial designs. Journal of Statistical Planning and Inference, 5, Nigam, A.K., Puri, P.D. and Gupta, V.K. (988). Characterizations and analysis of block designs. Wiley Eastern Ltd. 369