MUTUALLY ORTHOGONAL LATIN SQUARES AND THEIR USES

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1 MUTUALLY ORTHOGONAL LATIN SQUARES AND THEIR USES LOKESH DWIVEDI M.Sc. (Agricultural Statistics), Roll No. 449 I.A.S.R.I., Library Avenue, New Delhi 0 02 Chairperson: Dr. Cini Varghese Abstract: A Latin square arrangement is an arrangement of s symbols in s rows and s columns, such that every symbol occurs once in each row and each column. When two Latin squares of same order are superimposed on one another, in the resultant array if every ordered pair of symbols occurs exactly once, then the two Latin squares are said to be orthogonal. If in a set of Latin squares, any two Latin squares are orthogonal then the set is called Mutually Orthogonal Latin Squares (MOLS) of order s. Methods of constructing MOLS when v is prime or prime power are discussed here. Further, the use of MOLS for the construction of experimental designs like Balanced Incomplete Block Designs, Partially Balanced Incomplete Block Designs, Rectangular Lattice Designs, Change Over designs and Neighbour Balanced Designs is also discussed along with examples. Keywords: Balanced Incomplete Block Designs, Change over Designs, Graecolatin Squares, Hypergraecolatin Squares, Mutually Orthogonal Latin Squares, Neighbour Balanced Designs, Partially Balanced Incomplete Block Designs, Rectangular Lattice Designs. Introduction Heterogeneity in the experimental material is one of the important problems to be reckoned with the designing of scientific experiments. Occasionally, one can find a nuisance factor, which, though not of interest to the experimenters, does contribute significantly to the variability in the experimental material. For the experimental situation where there is only one nuisance factor, the block designs are being used. Various levels of this nuisance factor are used for blocking. When the heterogeneity present in the experimental material is in two directions, that is rows and columns, then double grouping is done. The purpose of double grouping is to eliminate from the errors all the differences among rows and equally all differences among columns. The experimental material should be arranged and the experiment should be conducted such that the differences among rows and columns represent major sources of variation. For example, in an animal experiment with the objective of comparing the effect of four feeds, let young calves be the experimental units with their body weight during a certain period as the variate under study. Let it be intended to eliminate the variation due to breeds and ages of the calves. So breed and age are the two factors that correspond to two sources of variation. The calves are, therefore, to come from four breeds and four age groups. The 6 calves required for the experiment should each belong to a different breed-age combination. There should be four calves belonging to each breed and each of these four calves should come from different age group.

2 The suitable design for the above situation is a row-column design or design for two-way elimination of heterogeneity. The simplest and most commonly used design with complete rows and complete columns both equal in such situation is a Latin square design. It is a design in which both rows and columns form a randomized block design, that is, treating rows as blocks forms a RBD and treating columns as blocks forms a RBD. A Latin square arrangement is an arrangement of s symbol in s 2 cells arranged in s rows and s columns such that each symbol occurs once in each row and in each column. This s is called the order of the Latin square. The following is a Latin square for 4 treatments: A B C D B C D A C D A B D A B C Application of Latin Squares: Latin squares have a wide variety of applications in many areas of science other than those in agricultural experiments. For instance, let us suppose that a scientist wants to test four different drugs (called A, B, C and D) on four volunteers. To make it a fair test, he decides that every volunteer has to be tested with a different drug each week, but no two volunteers are allowed the same drug at the same time. Saying that, each row represents a different volunteer and each column represents a different week, he can plan the whole experiment using the following Latin square: Week Week 2 Week 3 Week 4 Volunteer A B C D Volunteer2 C D A B Volunteer3 D C B A Volunteer4 B A D C Standard Form of a Latin Square: A Latin square is said to be in standard form if the symbols or letters in the first row and first column appear in natural order. The following Latin squares are in standard form: A B C D B C D A C D A B D A B C Semi Standard Form: A Latin square is said to be in semi standard form if the symbols or letters in the first row appear in the natural form. For example, the following Latin squares are in semi standard form: A B C D C D A B D C B A B A D C A B C D D C B A B A D C C D A B 2

3 Graecolatin Square: Two Latin squares of the same order s when superimposed on one another, if each pair of symbols in the resultant array occurs only once they are called orthogonal. Such a pair is often called a Graecolatin square, because traditionally Latin letters are used for the first square and Greek letters for the second square. Example.: Consider the following pair of Latin squares for s = 3: A B C C A B B C A α β γ β γ α γ α β When the two squares are superimposed, every pair of ordered Latin and Greek symbols occurs exactly once and hence the two Latin squares under consideration constitute a graecolatin square. A α B β C γ C β A γ B α B γ C α A β Mutually Orthogonal Latin Squares (MOLS): In a set of Latin squares of order s, if every pair of Latin squares are orthogonal then the set is called Mutually Orthogonal Latin Squares or MOLS of order s. It is also called hypergraecolatin squares. It is well known that the maximum number of MOLS possible of order s is (s-). Set of (s- ) MOLS is known as complete set of MOLS. Complete set of MOLS of order s exists when s is prime or prime power. A table of complete sets of MOLS for s 2 9 can be seen in Fisher and Yates (963). For example, a complete set of MOLS of order 4 is given below: I II III II superimposed on I III superimposed on II III superimposed on I Here, it can be observed that each pair of treatments occurs only once in each of the superimposed arrays and hence the above set of Latin squares is a complete set of MOLS. 2. Construction of MOLS (Das and Giri, 986) Let v = s. s 2. s p where each factor s i is either a prime or a prime power. The s i elements of the GF(s i ) (i =,2,,p) are used for forming combinations of elements of the p different fields as below: 3

4 Combine the elements from p different fields taking one from each field in all possible ways. There is evidently v such combinations. If v is a prime or prime power then p = and each such combinations is just an element of its field. Such combinations of the p field element are used as symbols for writing the Latin squares. Let the v combinations be written in a row and again in a column so as to obtain the summation table of all possible sums, two by two, of the row column combinations (mod s). This column will be called the principal column and the row, the principal row. By addition or multiplications of two combinations means addition or multiplication of each pair of corresponding element, (that is occurring in the same position) in the two combinations in the respective field. It can be easily seen that the summation table gives a Latin square. Next, each combination in the principal column is multiplied by a combination, say, (a,a 2,a 3,, a p ), where a i 0 or (i =,2,,p). The resultant column is the second principal column. Again another summation table is formed by using this second principal column and the first principal row. This table gives the second Latin square which is orthogonal to the one obtained earlier. Again a third principal column is obtained by multiplying the different elements by the first principal column by another multiplier, say, (b,b 2,b 3,...,b p ),where b i a i, or or 0 (i=,2,3,,p), i.e., the multiplier is so chosen that no element in any field is repeated in the different multiplier. The third Latin square is obtained by adding the third principal column and first principal row. This square is orthogonal to previous two and this process is continued till suitable multipliers are available. As each time, a new element is introduced in each field in the multiplier combinations, we can not get more then s- multipliers where s is the minimum factor of v. Each of these multipliers contains a different non-zero element of the field of s and so not more then s- can be taken without repeating an element in this field. If v is prime or prime power, each multiplier combination consists of only one element. We can therefore get (v-2) multipliers which are the different non zero elements in its field other than unity. Illustration: When v = s is prime number Suppose v = 3. Elements of G.F.(3) are 0,, 2. The row and column numbers in the first Latin square are kept in natural order. Then the contents of first Latin square are obtained by adding the corresponding entries of row and column (mod 3). Principal column 0 2 Principal row

5 The principal column in the second Latin square is obtained by multiplying the entries in the first principal column of the first Latin square by elements of G.F.(3). The contents of second Latin square are then obtained by adding the corresponding entries of row and column (mod 3). In this case, entries in the first principal column are multiplied by 2 as 0 and will not make any difference. Principal column 0 2 Principal row Therefore, a complete set of MOLS for v = 3: I II Illustration: When v is prime power Let v = 2 2 = 4. The elements in the Galois field G.F.(2 2 ) are 0,, α, α 2 (= α+) with α 2 +α+ = 0 as the minimal function and α a primitive element. The principal row and column of the first Latin square is taken in natural order. The first Latin square is then obtained by adding the corresponding entries of row and column (mod 2). Principal Principal row column 0 α α+ 0 α α+ 0 α α+ 0 α+ α α α+ 0 α+ α 0 Principal column for the second Latin square is obtained by multiplying the entries in the principal column of the first Latin square by α. The second Latin square (given below) is then obtained by adding the corresponding entries of row and column (mod 2). Principal column 0 α α+ Principal row 0 α α+ 0 α α+ α α+ 0 α+ α 0 0 α+ α Principal column of the third Latin square is obtained by multiplying the elements of the principal column of the first Latin square by α 2 = α+. The third Latin square is then obtained by adding the corresponding entries of row and column. 5

6 Principal column 0 α+ α Principal row 0 α α+ 0 α α+ α+ α 0 0 α+ α α α+ 0 Therefore, a complete set of MOLS for v = 4 is obtained using these 3 Latin squares and taking α = 2 and α+=3: I II III Uses of MOLS MOLS are widely used in the construction of various experimental designs like Balanced Incomplete Block (BIB) Designs, Partially Balanced Incomplete Block (PBIB) Designs, Rectangular Lattice (RL) Designs, Change Over Designs (CODs), Neighbour Balanced Designs (NBDs), etc. The method of construction of these designs using MOLS is explained in the following section. 3. Construction of BIB Designs An incomplete block design with v treatments distributed over b blocks, each of size k (< v) such that each treatment occurs in r blocks, no treatment occurs more than once in a block and each pair of treatments occurs together in λ blocks, is called a balanced incomplete block (BIB) design. A method of constructing BIB designs (Dey, 986) is explained below with illustration: Method of Construction: Let there be v = s 2 (s is a prime or power of a prime) treatments, numbered as, 2 s 2. Arrange these treatment numbers in the form of a s s square array in natural order, i.e., in a standard array. The contents of each of the s rows of this array are taken to form blocks giving a set of s blocks; another set of s blocks forming another complete replication is obtained by taking the contents of each of the s columns of this array. Next, consider (s ) MOLS and superimpose them one by one on the above standard array of treatments. The treatment numbers that fall on a particular symbol of a Latin square are taken to form a block. Thus s blocks corresponding to the s symbols of Latin square can be obtained from each MOLS. From the complete set of (s-) MOLS (if such a set exists), the design obtained is a resolvable BIB design with parameters v = s 2, b = s(s+), r = (s+), k = s and λ =. This series of BIB designs is also known as balanced (s+) lattice designs. 6

7 Example 3.: Let v = 3 2 =9, the standard array is as follows: Now, two (= s-) orthogonal Latin squares are taken and superimposed on this array as shown below: I A B2 C3 B4 C5 A6 C7 A8 B9 II A B2 C3 C4 A5 B6 B7 C8 A9 Considering all the four replications, (one from rows, one from columns and the other two from 2 orthogonal Latin squares) a resolvable BIB design with v = 9, b = 2, r = 4, k = 3 and λ = is obtained as given below: Blocks Rep I Rep II Rep III Rep IV Construction of PBIB (2) Designs An incomplete block design is said to be partially balanced with two associate classes if it satisfies the following requirements: (a) The experimental material is divided into b blocks of k units each, different treatments being applied to the units in the same blocks. (b) There are v (>k) treatments each of which occur in r blocks. (c) There can be established a relation of association between any two treatments satisfying the following conditions: Two treatments are either first associates or second associates. Each treatment has exactly n i i th associates (i=, 2). Given any two treatments which are i th associates, the number of treatments common to the j th associates of the first and k th associate of the second is p i jk and is independent of the pair of treatments we start with (i, j, k =,2). (d) Two treatments which are mutually i th associates occur together in exactly λ i blocks (i =, 2). L m Association Scheme: Let there be v = s 2 treatments. On these treatments, a Latin square association scheme with m (m 2) constraints is defined as follows: Let the v treatments be arranged in an s s array and assume that (m-2) MOLS of side s each exist, which are superimposed on the square array. Two treatments are first associates if they 7

8 occur in the same row or same column of the array or in positions occupied by the same letters in any of the Latin squares; second associates otherwise. Method of Construction of L m Designs (Dey, 986): Let there be v = s 2 treatments, numbered as, 2 s 2. Arrange these treatment numbers in the form of a s s square array in natural order, i.e., in a standard array. The contents of each of the s rows of this array are taken to form blocks giving a set of s blocks; another set of s blocks forming another complete replication is obtained by taking the contents of each of the s columns of this array. The resultant design is a simple lattice design which is a PBIB design with a Latin square association scheme, that is, a L 2 -PBIB design. The parameters of this design are v = s 2, b = 2s, r = 2, k = s, λ =, λ 2 = 0. Next, consider (m-2) MOLS and superimpose them one by one on the above standard array of treatments. The treatment numbers that fall on a particular symbol of the Latin square are taken to form a block. Thus s blocks corresponding to the s symbols of a Latin square can be obtained from each of the MOLS. If blocks are formed from rows, columns and using one MOLS, the resultant design is a triple lattice design which is a L 3 - PBIB (2) design with the parameters v = s 2, b = 3s, r = 3, k = s, λ =, λ 2 = 0. The quadruple lattice is an L 4 - PBIB (2) design with parameters v = s 2, b = 4s, r = 4, k = s, λ =, λ 2 = 0 can be obtained using rows, columns and 2 MOLS. In general the m-ple lattice is a L m - PBIB (2) design with parameters v = s 2, b = ms, r = m, k = s, λ = and λ 2 = 0. Example 3.2: Let v = 3 2 =9. Blocks Rep I Rep II Rep III This is a triple lattice design which is a PBIB (2) design following L 3 association scheme. If last one replication is deleted, the design obtained is a simple lattice design. Remark: In the above method, when s is prime or prime power, if blocks are formed from rows, columns and using a complete set of (s-) MOLS, the resultant design is a BIB design with parameters mentioned in section Construction of RL Designs Harshberger (947) developed rectangular lattices for s(s+) treatments in blocks of size s units. There are several methods of constructing rectangular lattice designs. One method of constructing a rectangular lattice design which is a resolvable design is given by Shrikhande (965), using a balanced lattice design for (s+) 2 treatments obtained through MOLS. 8

9 Method of Construction: A balanced lattice design for v = (s+) 2 can be obtained by using a complete set of s MOLS when v = (s+) 2 is prime or prime power, as explained in section 3.. Now, if in a balanced lattice, the first replication is omitted, and if all treatments that appear in any one selected block in the first replicate are omitted the resultant design is a rectangular lattice in which every block is of size s and in which no two treatments appear together more than once in the same block. The method fails when no balanced lattice exists, as with 36 treatments. 2 Parameters are v = s(s + ), b = (s + ), r = (s + ), k = s.. Example 3.3: Consider the following balanced lattice with v = (s+) 2 = 3 2 : Blocks From this design, deleting the first replication and if all treatments that appear in the third block of the first replication from all other replications, the following resolvable rectangular lattice design is obtained: Blocks Construction of CODs The experiments in which each experimental unit receives some or all of the treatments, one at a time, over a certain period of time are known as CODs. The peculiarity of a COD is that any treatment applied to a unit in a certain period influences the responses of the unit not only in the period of its application but also leaves residual effects in the following periods. These residual effects or carry-over effects may be of different magnitudes. Residuals, which persist only for one succeeding period, are called first order residual effects or simply, first residual effects. In general, the k th order residual effect is one, which persists up to k successive periods. This feature of residuals is the distinguishing feature of CODs. A COD is called uniform on periods if each treatment occurs in each period the same number of times. A design is called uniform on units if each treatment is applied to each experimental unit the same number of times. A design is called uniform if it is uniform on both periods and units. 9

10 The analysis is made easier if each treatment is preceded equally often by every other treatment. Designs having this property have been called combinatorially balanced with respect to first residual effects. A change-over design permitting the estimation of first order residual effects is called balanced if the variance of any elementary contrast among the direct effects is constant and the variance of any estimated elementary contrast among the residual effects is also constant. Method of Construction: A uniform class of CODs balanced for first as well as second order residual effects for v treatments in p (= v) periods and n = [v (v-)] experimental units can be obtained by juxtaposing MOLS with rows representing periods and columns, the experimental units. Even if t [ t (v-3)] row(s) of the above arrangement is(are) deleted, this remains to be a COD balanced for first as well as second order residual effects. Example 3.4: A COD obtained using MOLS for v = 4, p = 4 and n = 2 is given below: 2 Period 3 4 Experimental Unit Remark: It may be noted that the famous Williams squares CODs balanced for direct and first order residual effects for v = 3, p =3 and n= 6 can be obtained using MOLS of order v = 3. Example 3.5: A Williams square COD for v =3, p =3 and n = 6 is given below: Period 2 3 Experimental Unit Construction of NBDs In many experiments, especially in agriculture, the response on a given plot may be affected by treatments on neighboring plots as well as by the treatment applied to that plot. A block design with adjacent neighbouring effects is balanced if every treatment has every other treatment appearing constant number of times as a right and as a left neighbour. A block of treatments with border plots (plots at both ends of a block) is left - circular if the treatment in the left border is the same as the treatment in the right - end inner plot and right - circular if the treatment in the right border is the same as the treatment in the left - end inner plot. A circular block is a left and right circular block and a circular design is a design with all its blocks circular. 0

11 A method of construction of NBDs using MOLS has been given by Tomar et al. (2005). Method of Construction: Let v = s (s > 4) be a prime or prime power and the s treatments be denoted by,2,,s. Develop (s -) mutually orthogonal latin squares (MOLS) of order s by multiplying first principal row by the elements of GF(s), except and s, and adding corresponding entries in each cell. Juxtapose these MOLS so that we obtain an arrangement of s symbols in s(s-) rows and s columns. Deleting i last columns (i =,2,,s-4) and taking rows as blocks along with border plots, to make blocks circular, would result in an incomplete block design totally balanced for competition effects. This series of designs can be obtained for any k, 4 k s-. The parameters of the design are v = s, b = s(s-), r = (s )(s i) and k = (s i). Example 3.6: Let v = s = 5. Juxtaposing 4 MOLS of order 5, deleting last column (i =) and taking rows as blocks along with border plots would result in the following block design for competition effects with parameters v = 5, b = 20, r = 6 and k = 4: Here, the left and right most treatments in each block which are given in bold letters represents the left and right border plot treatments of that block, respectively. It can be seen that the design is totally balanced in the sense that all the treatment contrasts pertaining to direct and neighbour (left and right) effects are estimated with same variance. 4. Conclusions A complete set of MOLS exists only when the number of treatments is prime or prime power. Methods for constructing MOLS are discussed through examples. Further, MOLS are widely used for the construction of various experimental designs. The methods of constructing commonly used designs like BIB Designs, PBIB Designs, Rectangular Lattice Designs, Change Over designs and Neighbour Balanced designs have been explained here.

12 References Das, M.N. and Giri, N.C. (986). Design and analysis of experiments. Wiley Eastern Limited, New Delhi. Dey, A. (986). Theory of block designs. Wiley Eastern Ltd., New Delhi. Fisher, R.A. and Yates, F. (963). Statistical tables for biological, agricultural and medical research. 6 th edition, Longman Group Ltd., London. Harshbarger, B. (947). Rectangular lattices. Va. Agri. Expt. Stat. Memoir,, -26. Shrikhande, S.S. (965). On a class of partially balanced incomplete block designs. Math. Statist., 36, Tomar, J.S., Jaggi, Seema and Varghese, Cini (2005). On totally balanced block designs for competition effects. Jour. Applied Statistics, 32(),

Construction of lattice designs using MOLS

Malaya Journal of Matematik S(1)(2013) 11 16 Construction of lattice designs using MOLS Dr.R. Jaisankar a, and M. Pachamuthu b, a Department of Statistics, Bharathiar University, Coimbatore-641046, India.