Restricted Randomized Two-level Fractional Factorial Designs using Gray Code

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1 Restricted Randomized Two-level Fractional Factorial Designs using Gray Code Dr. Veena Budhraja Department of Statistics, Sri Venkateswara College, University of Delhi, Delhi Mrs. Puja Thapliyal Assistant Professor, Department of Statistics., University of Delhi ABSTRACT Traditionally, randomization is used to generate sequence of run orders. But in situations like where the factors of interest are correlated with trend or where it is expensive to frequently change the levels of factors in the design, randomization may not give the desired results. Hence one is interested in obtaining the run order in which the factor effects are trend free and the design has minimum level changes. In other words a systematic run order with trend free effects and with minimum number of level changes is preferred. This paper provides a different approach to construct n and n-p fractional factorial designs satisfying both design criteria of trend freeness and minimum level change using GRAY CODE. The designs are further compared with designs given by Coster and Cheng(1988) using the uniformity measures. Keywords: Systematic designs, uniformity, fractional factorial designs. 1. ITRODUCTIO The choice of a good design in order to run an experiment is an important issue in experimental design. There are not only the statistical considerations ( the proposed model, the experimenter s objective etc.) but also practical considerations ( financial limitations, time considerations etc. ) that can influence their choice. Traditionally, the usual advice given to the experimenter is to randomize the run order of the experiments. But sometimes randomizing the run order does not achieve the desired effect of neutralizing the influence of unknown factors or it may lead to designs that may have factor effects aliased with the time trend or the run order may lead to designs that have excessive changes in the factor levels. Therefore it is essential to sequence the runs such that a) Factor effects are orthogonal to the time trend. b) Factors with expensive or difficult to vary levels are minimally varied during an experimentation, The problem of obtaining run order satisfying the properties a) and b) was studied by Cox(1951), Hill(1960). Wilcoxon (1966) dealt with problem (a) whereas Dic kinson (1974) considered (b) for 4 and 5 factorial design. Joiner and Campbell (1976) gave example to emphasize the importance of a) and b). Cheng (1985) and Coster and Cheng (1988) described Generalized Foldover Scheme (GFS) to select run order to achi eve requirements a) and b). Sometimes in practical situations, it is easier for experimenter to choose the factors and interactions of interest than to decide on the defining contrast at the design stage. However, most of the methods (Cheng (1985) and Coster and Cheng (1988)) involve finding defining contrasts under the constraints first and then selecting an appropriate run order. For this reason the methods are not easy to use for experimenter whowant to arrange both experiments and appropriate run orders. The paper presents a method to construct run orders of n and n-p fractional factorial designs with factor effects orthogonal to time trend and minimal level change during an experimentation, using Gray codes. 1 Dr. Veena Budhraja, Mrs. Puja Thapliyal

2 The method of constructionis different from the one given by Coster and Cheng (1988) in the sense that the former first give the defining relationship to construct the run order whereas in the method the run orders are generated first and the defining relationship are determined from the run orders. We further compared the constructed designs with the designs given by Coster and Cheng (1988) using the different uniformity measures e.g. Wrap around Discrepancy, Central discrepancy.. PRELIMIARIES Definition 1: A binary Gray code of length n is a sequence s 0, s 1, s,, s n -1 of the n distinct n-bit strings (or words) of 0 s and 1 s, with the property that the Hamming distance between successive code words is one (where Hamming distance between s i and s i+1 is the number of positions where they differ). Definition : Time Count of a specific effect is calculated as the inner product of the column that corresponds to this effect and the row number. The effects with zero time count are said to have linear trend free (LTF) property. 3. MIIMUM LEVEL CHAGE n, n, FACTORIAL DESIGS One of the most important criteria in an experimental process is the number of level changes for factors if cost/time element is considered in the experiment. The minimum number of factor level changes is obtained when the level of only one factor is changed between two consecutive runs. Thus, in a n design, a minimum number of level changes equals n -1. Such type of designs can easily be obtained using Gray Codes. ( Gilbert(1958) ). METHOD OF COSTRUCTIO Step 1 Let (1) = {0,1} and (1) ={1,0} be the mirror image of (1). Step For, (i) = where (i) denotes the mirror image of (i) and and are null vector and unit vector of order (i-1) 1. Step 3 Assign the factors. to the n columns in (n). Step 4 A n factorial experiment with n-1 + n = n -1 number of level changes which is minimum possible. Let us call these designs as Gray designs. Remark 1.The method given above can be extended easily to obtain q n ; (q>)minimum level change Gray designs.. This technique can be generalised to construct n factorial experiment with n-1 + n- + n = n -1 number of level changes, which is the minimum possible. ILLUSTRATIO 1 In order to construct 3 factorial designs we begin with S (1) = {0,1} then its mirror image (1) ={1,0}. Appending a column of -1 = zeroes and ones to ( (1) (1)) we get () = The mirror image of () is Dr. Veena Budhraja, Mrs. Puja Thapliyal

3 () = Appending a column of = 4 zeroes and 4 ones we get (3), a 3 factorial design given in Table 1. Table 1: 3 factorial design A B C C LC TC The run order obtained in table 1 is 1, a, ab, b, bc, abc, ac, c. We observe that the number of level change be made linear trend free (LTF) using following mapping. Let denotes the mapped factor. Step 3 For the first half of, Replace first zeroes by 0, next zeroes by 1 and next ges for the factors A, B and C are in the order, 1, 0 respectively. The total number of level changes is 7 which is the minimum possible. 4. TRED FREE n FULL FACTORIAL DESIG The method described above can also be used to construct linear trend free full factorial design. METHOD OF COSTRUCTIO Step1 Construct n factorial design with minimum number of level changes using the above method Step The time count (TC) for the n th factor A n is non-zero which has one level change. This factor can zeroes by 0. For the second half of the A n containing n-1 ones, replace first ones by1, next ones by 0 and next ones by 1. We may note that the number of level changes for the new mapped factor will always be 5. Thus, using the method described above, a n factorial design is obtained with all main effects linear trend free and number of level changes is n +3. ILLUSTRATIO Consider the design in Illustration 1. In this run order, the main effect of all factors except 3 rd factor C are linear trend free as the time counts for the factors A and B are zero. To make the factor C linear trend free, we use the 3 Dr. Veena Budhraja, Mrs. Puja Thapliyal

4 mapping as described in Step 3 of the above method. The first half of C contains zeroes and second half contains ones. Thus, the first half of it using Step 3, is replaced by and the second half is replaced by Let C denotes the mapped factor, Then, the new run order is 1, ac, abc, b, bc, ab, a, c. The number of level changes for C is 5 which is also linear trend free and is displayed in last column of Table 1.Thus, a 3 factorial design in which all the main effects are linear trend free and the number of level change is 11, is constructed. 5. TRED FREE n-p FRACTIOAL FACTORIAL DESIG The above procedure yields n factorial design with minimum number of level changes in which (n-1) main effects are linear trend free. We observe here that the interactions involving these (n-1) factors are also linear trend free. ow we construct n-p fractional factorial design with all main effects linear trend free. The procedure to construct such designs is given below. METHOD OF COSTRUCTIO Step 1 Consider a n-p, full factorial design constructed using the above method. The design has (n-p-1) factors that are with LTF property and the (n-p) th factor, a mapped factor which is made LTF through mapping. StepAssign the p factors to the interaction of the first (n-p-1) factors that are having LTF property i.e. a new factor is generated by the interaction involving only unmapped factors, then LC in the new factor will be equal to the sum of the LC of the factors in the interaction i.e. for = ; i,j = 1,...(n-p-1), i j and k = n-p+1,...,n. LC(A k)= LC(A i) + LC(A j) Step 3We get a n-p fractional factorial design in which all main effects are linear trend free. Remark 1. These designs may not be minimum level change designs.. If the interactions, involving mapped factor is assigned to a new factor then LC for the new factor can not be obtained directly. ILLUSTRATIO 3 To construct 4-1 fractional factorial designs, consider the 3 design constructed in Illustration with mapped factor denoted by C. The table below gives the time counts of all two factor interactions along with number of level changes. We observe that the two factor interaction AB is linear trend free (TC = 0). Assigning D= AB, we get a 4-1 resolution III fractional factorial design with defining relation I= ABD, with all main effects linear trend free. Further, the total number of level changes(lc) for the design is 17, where LC(D)= LC(A)+ LC(B) = 6. 4 Dr. Veena Budhraja, Mrs. Puja Thapliyal

5 Table 3: 4-1 Resolution III fractional factorial design A B C D=AB AC BC LC TC Table 4 lists the n-p (4 n 9, 1 p 5) fractional factorial designs with total LC. The designs obtained may not be minimum level change designs but all main effects in the designs generated are having LTF property. Table 4: n-p (4 n 9, 1 p 4) Fractional factorial designs. Design Unmapped Factors/ Mapped Factor Assignment LC for Generated Factor A,B/C D=AB 6 A,B,C/D E=ABC 14 A,B,C,D/ E F=ABCD 30 A,B,C/ D E=ABC,F=BC 14, 6 A,B,C,D,E/ F G=ABCDE 56 A,B,C,D/E G=ABC,F=BCD 8, 14 A,B,C/D G=AB, F=BC, E=AC A,B,C,D,E,F/G H=ABCDEF 16 1, 6, 10 A,B,C,D,E/ F G=ABCD,H=BCDE 60, 30 A,B,C,D,F/E G=ABD, H=BCD, F=ACD 6, 14, A,B,C,D,E,F/ G H=CDEF, J=ABEF 30, 10 A,B,C,D,E/ F G=CDE, J=ABDE H=ABC, 14, 56,54 A,B,C,D/E F=ABC,G=BCD,H=A CD, J=ABD 8,14,,6 5 Dr. Veena Budhraja, Mrs. Puja Thapliyal

6 The resolution of the design depends on the assignment of the factors, which depends on the experimenter s requirement. 6. UIFORMITY In order to compare GRAY design with design of Coster& Cheng(1988) we calculated the measure of uniformity CD and WD. Using analytical formula given byhickernell(1998a,b) and Fang et.al.(00) for both the designs and are given in Table 5.Hickernell (1998b) gave analytical formula for the CD and WD as n k ( CD ( )) 1 n u j kj 05. ukj n u j i ki 0. 5 u ji 0. 5 uki u ji k n 1 k ( WD( )) n 3 u j i ki uji uki uji We compare our designs with the Coster and Cheng(1988) designs. The uniformity measures for both the designs are tabulated in Table 5. Table 5: Uniformity measures CD and WD values for the designs. Gray Designs CC Designs Design CD WD CD WD From Table 5 we observe that the uniformity measure for both Gray and CC designs are same. Hence under the assumption that errors are independent with zero mean and constant variance, both the designs are equally efficient in terms of uniformity. Conclusion An article provides a different approach to construct minimum level change factorial designs and trend free factorial designs, using gray codes. The proposed designs are proven to be equally efficient in terms of uniformity, with designs of Coster and Cheng (1988) designs as evident from Table 5. REFERECES 6 Dr. Veena Budhraja, Mrs. Puja Thapliyal

7 [1] Coster, D.C., and Cheng, C.S Minimum cost trend free run orders of fractional Factorial design. The Annals of Statistics, 16, [] Cox, D.R Some systematic experimental designs. Biometrika, 38, [3] Cheng, C.S Run orders of factorial designs: Proceedings of the Berkeley conference in Honor of Jerzy eyman and Jack Kiefer,Vol.II, pp Lucien M.Le.Cam and R.A. Olishen eds. Wadsworth, Inc. [4] Cheng, C.S. and Steinberg, D.M Trend robust two level factorial designs. Biometrika, 78, [5] Dickinson, A.W Some run orders requiring a minimum numbers of Factor level changes for and main effect plans. Technometrics, 16, [6] Daniel, C. and Wilcoxon, F Factorial p-q plans robust against linear and quadratic trends. Technometrics, 8, [7] Fang, K.T, Ma. C.X., and Mukerjee, R. 00. Uniformity in fractional factorial.in:monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, eds. By Fang KT, Hickernall FJ, iederreiter H, Springer-Verlag Berlin. [8] Hickernell, F.J A generalized discrepancy and Quadrature error bound. Math. comput., 67, 1, [9] Hill, H.H Experimental Designs to adjust for time trends. Technometrics,, [10] Gilbert, E Gray Codes and Paths on the n-cube. Bell System Tech. J 37, [11] Golay, M.J.E otes on Digital Coding. Proceedings IEEE, 37, 657. [1] Joiner, B.L. and Campbell, C Designing experiments when run order is important. Technometrics, 18, [13] Kiefer, J Construction and optimality of generalized youden design. In J..Srivastava, Ed, A survey of statistical design and linear models, orth Holland, Amsterdam. 7 Dr. Veena Budhraja, Mrs. Puja Thapliyal