CLASSIFICATION OF LATIN SQUARES. Dr Nada Lakić

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1 Journal of Agricultural Sciences Vol. 47, No 1, 2002 Pages UDC: 311 Review articles CLASSIFICATION OF LATIN SQUARES Dr Nada Lakić Abstract: Efficacy and profitability of results and eventually the conclusions of an experiment were found to depend on the statistical model for organizing an experiment. No thoroughgoing studies have been reported to date in our statistical literature on Latin square designs, one of the three basic experimental designs. The objective of the study was to define the insufficiently known subsets of Latin square designs having special properties and classify them using a number of criteria. Key words: experimental design, Latin square designs, randomization. I n t r o d u c t i o n The term Latin square design has for the first time been used in solving problems pertaining to the movement and arrangement of figures on a chessboard, although Romans are known to have begun their games with problems focusing on squares and the distribution of certain numbers or elements on them. The oldest written reference on the use of Latin square dates back as far as 1723 and deals with the arrangement of 16 cards of a deck. However, the Latin square design has only recently attracted considerable attention of those involved in mathematics and statistics. The reason for the increased interest may be linked to the application of algebra of generalized binary systems, studying combinations and particularly finite geometries. Practical application of Latin squares when distributing treatments in an experiment has particularly encouraged further investigations. Initially, statistical methods were applied in planning field experiments. The advantage of using the square design in plot trials with the objective of eliminating the differences in soil fertility was registered very early. However, with the first application of the Latin square model the importance of impartial Dr Nada Lakić, Assistant Professor, Faculty of Agriculture, Belgrade-Zemun, Nemanjina 6, FR Yugoslavia

2 106 Nada Lakić estimation of error has not been fully understood. In addition, the accepted distributions of treatments were systematic, usually of a specially simple kind or a type believed to be able to eliminate completely the existing difference in soil fertility. Systematic treatment distribution when estimating experimental error had its disadvantages in practice and has thus been replaced by randomization. Definition of the Latin square Randomization in the application of the treatment to the units studied is considered necessary contributing to the appropriate estimation of experimental error. However, distribution of treatments at random is not always desirable in experimental work. The characteristics of experimental material and trial objectives often require partial restrictions of the distribution of treatments at random. Three major models of experimental designs have been developed depending on the properties of restricted randomization. One of them is the Latin square. Latin square 1 is the experimental plan in the square form whereby randomized distribution of treatments takes two courses, namely in rows and in columns. Each treatment analyzed is known to appear only once in each row and each column. In this way the effect of two sources of non-homogeneity of experimental material is being eliminated. The statistical model indicates that thanks to the Latin square plan the total value of experimental results (x ijk ) may be divided into five components: x ijk = μ + R i + K j + T k + e ijk. General average is indicated as μ, R i represents the component part characterizing all units of the i th row, K j stands for the effect of the j th column, T k is the result of the effect of the treatment k and e ijk is the experimental error, normally distributed with the average 0 and the constant variance. Therefore, from the experimental error using the analysis of variance in the case of the Latin square the variation stands out as the result of factors controlled by means of rows and columns. Types and classification of Latin squares Latin squares may be classified according to the: a) distribution of treatments, 1 The term Latin square was for the first time used by Euler, In a study on enumeration of different square distributions Euler used Latin letters as symbols and thus introduced the term Latin square. Later, in addition to letters, numbers have also been introduced to define treatments.

3 Classification of Latin squares 107 b) number of controlled variation sources c) number of dimensions, and d) number of replications. a) Mode of treatment distribution may be systematic, at random or at random with restrictions. In the initial phase, when analyzing the final results of the experiment, no attention at all was paid to the value of experimental error, and therefore treatments were systematically distributed. However, with the evolution of logistically based inductive conclusions drawn, it became obvious that systematic treatment distribution did not contribute to unbiased estimation of experimental error. Thus, it was impossible to draw reliable conclusions from the results obtained in the trial. Systematic distribution of treatments in experimental designs may be considered justifiable in cases offering favourable information on noncontrolled sources of variation or in cases when its conduction seems practical (e.g. different treatments correspond an order offered by a machine). The distribution of a treatment at random in a trial is known to ensure conditions under which continuous replications of treatments will unable the favourization of some treatments at the expense of others being influenced by different sources of variations either known or unknown. It is recommendable to use randomization even in cases when systematic distribution poses no serious problems, because using randomization the researcher may find himself protected from unexpected surprises. Nevertheless, in experiments with a small number of treatments randomization treatments may lead to systematic distribution. In that case restricted randomization may be applied. Based on restricted randomization different experimental designs have been developed and different types of Latin squares as well. When speaking of randomization in the case of Latin squares, randomization within rows and columns are of major importance along with the restrictions which indicate that each treatment appears only once in each row and column. In the group of Latin squares with restricted randomization of treatments, restricted randomization of treatments with rows and columns has been introduced. Based on the characteristics of these restrictions, the classification of Latin squares is as follows: 1. reduced and standard 2. coincidal and self-coincidal 3. complete and quasi-complete. Based on the distribution in the first row and first column, Latin squares have been classified as reduced and standard. Latin square is considered reduced if its first row and first column contains elements in the numerical (1,2,.n) or

4 108 Nada Lakić lexicographic order (A,B,C,.). On the other hand, it is considered standard if only its first row contains elements in the natural order. Some authors classify them into squares with standard and semi-standard form, whereby reduced Latin square is considred standard form and standard Latin square is considred semistandard form. This classification is considered important because reduced forms are used for randomization of Latin squares, whereas standard forms are needed in order to study the properties of orthogonal Latin squares. Based on the structure of all rows and colums, Latin squares may be classifed as coincidal and self-coincidal. Two Latin squares are known to be coincidal provided the order of the symbols in the rows of one square equals the order in the columns of the other square. Latin square is considred self-coincidal provided the same square is obtained changing columns and rows. With regard to the statistical balance, Latin squares may be classifed as complete and quasi-complete, wholly or only according to either rows or columns. The properties of the row complete Latin square refer to the fact that each treatment is being followed by all the other treatments only once in the rows. This may be achieved in the case of the square of even degree. In the case of uneven number of treatments, a pair of nxn Latin squares is used. Analogously, in the case of the column complete Latin square, each treatment is followed by all the other treatments only once in the columns. Latin square complete according to rows and colums at the same time, e.g. in which rows and colums of each treatment are followed by all the other treatments only once is known as complete Latin square. However, in case each unordered pair of treatments appears in neighbouring fields twice in rows or columns or in rows and columns, Latin square is known as row quasi-complete Latin square, column quasi-complete Latin square or wholly quasi-complete Latin square. Complete Latin squares are special cases of quasi-complete Latin squares. In the case of complete Latin squares the property of immediacy counts, e.g. the ordered pair (p,q) differs from the pair (q,p). This is of major importance in cases when treatments are applied to the same unit at different times, which may contribute to the changes in the conditions under which the experiment is carried out. b) According to the number of controlled systematic variation sources, square experimental designs may be classified as: Latin square, Graeco-Latin square and hyper Graeco-Latin square. A pair of orthogonal 2 Latin squares with different symbols is known as the Graeco-Latin square. This square enables the control of 2 Two Latin squares L 1 = IIa ij II and L 2 =IIb ij II of the n degree are orthogonal provided each arranged pair of symbols appears only once in the n 2 pairs (a ijj,b ij ), i = 1, 2,.,n j j=1,2,.., n.

5 Classification of Latin squares 109 three systematic sources of variations and treatment variation. A set of mutually orthogonal Latin squares is known as the hyper-graeco-latin square provided different symbols are used in each square. c) Based on the number of dimensions, the classification may be as follows: Latin square, Latin cube and m dimensional hyper-cube. Namely, Latin square is a two-dimensional model. However, by increasing the number of dimensions to three, Latin cube may be obtained, a model with n rows, n columns and n layers. An m-dimensional Latin-hyper-cube may be obtained by increasing the number of dimension to let s say m. Three-dimensional matrix nxnxn, containing n layers whereby each has n rows and n columns represents the Latin cube provided there are n different elements whereby each is replicated n 2 times, i.e. provided the distribution of elements is such that each appears in each of the n layers with the n frequency. This concept differs from the mathematical in the sense that there may be rows, columns or layers in which some of the elements do not appear and other elements are replicated. Based on this fact, the classification of Latin cubes may be as follows: 1. Regular or 3-regular each row, each column and each layer containts all the elements only once; 2. 2-regular each element appears once in each row and column but not in each layer; 3. 1-regular each element appears only once in each row but not in each column and in each layer; 4. 0-regular elements are replicated in all three directions. Two Latin cubes are orthogonal provided each ordered pair of elements 0, 1,..n-1 between the n 3 pairs of elements chosen from the corresponding fields of squares, appears n times. m-dimensional Latin hyper-cube of the n degree and r th class is nxnx xn dimensional matrix with n r different elements which appear with the frequency n m-r so that each element appears n m-r-1 times in each of the m sets from the n parallel (m-1)-dimensional linear subspaces, i.e. layers. Two Latin hyper-cube n degree and r class are considered orthogonal provided each element of either first or lattter appear n m-2r times with each of the element of the other. d) According to the number of replications, which in the case of the basic Latin square model equals the number of treatments, squares having greater and smaller number of replications than the number of treatments have been developed. A greater number of replications than the number of treatments characterizes Latin squares with partial replication, frequent and incomplete squares. In the case of modified, Latis squares and incomplete Latin squares the number of replications is smaller than the number of treatments. The property of Latin squares with partial replication reflects itself in the choice of the additionl n observings, whereby each row, column and treatment contains a total of n+1 treatment.

6 110 Nada Lakić The condition for the Latin square, whereby each treatment appears once in each row and each column may be substituted by the condition where each treatment may appear in each row and column in an identical number of times. Squares having such property are known as frequent Latin squares or F-squares. The Latin square with less than n 2 filled in fields is known as the incomplete or partial Latin square. An incomplete Latin square is the square with omitted row and column. Among the modifications of the Latin square is the Latin square with an added column and subtracted row, as well as the Latin square with added row and column. A modified Latin square is a rectangular distribution with n rows and t columns, classified into n sections, whereby each contains k continuous columns and thereby each treatment apperas only once in each row and once in each section. In any case, a modified Latin square may be generalized as the semi- Latin square which is a rectangular distribution of n rows and t columns forming groups of sections each containing 2 continuous columns, whereby each treatment appears once in each row and once in each pair of columns. Lattice square is the square whose size equals the square root of the number of treatments. In this square during one replication each treatment appears with all the other treatments at least once in each row or column. According to the number of replications lattice squares may be cassified as: simple or two replicates, triple replicates, four replicates, five replicates etc. The application of rectangular lattice is recommended for the number of treatments for which square lattice is inapplicable. In the case of a large number of treatments cubic lattice may be applied instead of square lattice. The magic square may be related to the Latin square. Magic square of the n degree is the square distribution of n 2 integers (usually different, but not necessarily), whereby the sum of each row, column and diagonal is the same. If there are n different integers among the n 2 elements, whereby each is replicated n times, the magic square is known as the diagonal Latin square. The Latin square is left semi-diagonal if it has different elements in the main right-to-left diagonal, right semi-diagonal if it has different elements in the main left-to-right diagonal. The Latin square being simultaneously left and right semi-diagonal is known as diagonal or double diagonal. Cross Latin square has all the identical elements on the right-to-left diagonal and left-to-right diagonal. The magic square is pandiagonal, perfect or diabolic provided the group of elements on each left-to-right diagonal and all the diagonals parallel to it has the sum equal to the sum of elements of each row, column and right-to-left diagonal and diagonals parallel to it. The magic square in which both sums and products of the elements in all the rows, columns and the right-to-left diagonal are constant is known as the additivemultiple magic square. The new type of designs known as the new type of the magic square was introduced in 1955 and was later known as the Room s square. The Room s

7 Classification of Latin squares 111 design of the n=2m degree contains the square distribution with the 2m-1 field in each row and each column, thereby each field is either empty or contains an arranged pair of symbols chosen from the set of 2m elements. C o n c l u s i o n There is no perfect design for each and every experiment. Based on the sources of variation in an experimental material, objectives of the trial, procedure difficulties and data sampling and computing costs, an optimal model was determined. From the statistical standpoint, an experimental design having a smaller experimental error proved more efficient. However, the number of the degrees of freedom should be taken into account. The aim of the study was to define different types of Latin squares with the objective of indicating the most favourable experimental design in experimental work. In addition, it has been stated that different modifications may eliminate some of the disadvantages of the basic Latin square model or ensure homogeinity of experimental conditions. Latin square sets have also been classified into four subsets with the aim of encouraging the choice of the Lating square type most suitable for treatment distribution under the given experimental conditons. In addition to its basic purpose serving as the model for experimental design, in statistical model development Latin square is known to be the basis and generator for other experimental designs. R E F E R E N C E S 1. B a i l e y, R.A. (1984): Quasi-complete Latin squares: Construction and randomization, Journal of the Royal Statistical Society, B 46, C o c h r a n, W.G., C o x, G.M. (1957): Experimental designs, 2 nd Ed., Wiley, New York. 3. D é n e s, J., K e e d w e l l, A.D. (1974): Latin squares and their applications, English Universities, London. 4. F r e e m a n, G.H. (1979): Complete Latin squares and related experimental designs, Journal of the Royal Statistical Society, B 41, Gill, P.S., Shukla, G.K. (1985): Experimental designs and their efficiencies for spatially correlated observations in two dimensions, Communications in Statistics - Theory and Methods, 14, H e d a y a t, A., S e i d e n, E. (1970): F-square and orthogonal F-squares design: A generalization of Latin squares and orthogonal Latin square design, The Annals of Mathematical Statistics, 41, Preece, D.A., Freeman, G.H. (1983): Semi-Latin squares and related designs, Journal of the Royal Statistical Society, B 45, R o j a s, B., W h i t e, R.F. (1957): The modified Latin square, Journal of the Royal Statistical Society, B 19, S m a l l, C. (1988): Magic squares over fields, American Mathematical Monthly, 7, Y o u d e n, W.J., H u n t e r, J.S. (1955): Partialy replicated Latin squares, Biometrics, 11, Received March 5, 2002 Accepted April 8, 2002

8 112 Nada Lakić KLASIFIKACIJA LATINSKIH KVADRATA Nada Lakić R e z i m e Različiti modeli eksperimentalnih planova rezultirali su iz potrebe da se u što većoj meri objasni varijabilitet eksperimentalnih rezultata. Od tri osnovna plana eksperimenata latinski kvadrat je najefikasniji, s obzirom da omogućava da se metodom analize varijanse najviše smanji eksperimentalna greška. Od nastanka do danas razvijao se i sam koncept latinskog kvadrata. Skup latinskih kvadrata u radu je podeljen u podskupove prema načinu raspoređivanja tretmana, broju kontrolisanih izvora varijacija, broju dimenzija i broju ponavljanja. U radu su, takođe, definisane karakteristike kvadrata iz prezentiranih podskupova. Izložena klasifikacija treba da pomogne istraživačima u eksperimentalnim naukama, kao što je poljoprivreda, da pravilnim izborom plana eksperimenta lakše dođu do efikasnijih i ekonomičnijih rezultata i pouzdanijih zaključaka. Primljeno 5. marta Odobreno 8. aprila Dr Nada Lakić, docent, Poljoprivredni fakultet, Katedra za statistiku, Beograd-Zemun, Nemanjina 6, SR Jugoslavija