Latin Square Designs

Latin Square Designs Selected Latin Squares 3 x 3 4 x 4 A B C A B C D A B C D A B C D A B C D B C A B A D C B C D A B D A C B A D C C A B C D B A C D A B C A D B C D A B D C A B D A B C D C B A D C B A   5 x 5 6 x 6 A B C D E A B C D E F B A E C D B F D C A E C D A E B C D E F B A D E B A C D A F E C B E C D B A E C A B F D F E B A D C

A Latin Square

Definition A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column. Example - 4 x 4 Latin Square. A B C D B C D A C D A B D A B C  

In a Latin square You have three factors: Treatments (t) (letters A, B, C, …) Rows (t) Columns (t) The number of treatments = the number of rows = the number of colums = t. The row-column treatments are represented by cells in a t x t array. The treatments are assigned to row-column combinations using a Latin-square arrangement  

Example A courier company is interested in deciding between five brands (D,P,F,C and R) of car for its next purchase of fleet cars. The brands are all comparable in purchase price. The company wants to carry out a study that will enable them to compare the brands with respect to operating costs. For this purpose they select five drivers (Rows). In addition the study will be carried out over a five week period (Columns = weeks).

Each week a driver is assigned to a car using randomization and a Latin Square Design. The average cost per mile is recorded at the end of each week and is tabulated below:

The Model for a Latin Experiment i = 1,2,…, t j = 1,2,…, t k = 1,2,…, t yij(k) = the observation in ith row and the jth column receiving the kth treatment m = overall mean tk = the effect of the ith treatment No interaction between rows, columns and treatments ri = the effect of the ith row gj = the effect of the jth column eij(k) = random error

A Latin Square experiment is assumed to be a three-factor experiment. The factors are rows, columns and treatments. It is assumed that there is no interaction between rows, columns and treatments. The degrees of freedom for the interactions is used to estimate error.

The Anova Table for a Latin Square Experiment Source S.S. d.f. M.S. F p-value Treat SSTr t-1 MSTr MSTr /MSE Rows SSRow MSRow MSRow /MSE Cols SSCol MSCol MSCol /MSE Error SSE (t-1)(t-2) MSE Total SST t2 - 1

The Anova Table for Example Source S.S. d.f. M.S. F p-value Week 51.17887 4 12.79472 16.06 0.0001 Driver 69.44663 17.36166 21.79 0.0000 Car 70.90402 17.72601 22.24 Error 9.56315 12 0.79693 Total 201.09267 24

Using SPSS for a Latin Square experiment Trts Rows Cols Y

Select Analyze->General Linear Model->Univariate

Select the dependent variable and the three factors – Rows, Cols, Treats Select Model

Identify a model that has only main effects for Rows, Cols, Treats

The ANOVA table produced by SPSS

Example 2 In this Experiment the we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low). There are a total of t = 3 X 2 = 6 treatment combinations of the two factors. Beef -High Protein Cereal-High Protein Pork-High Protein Beef -Low Protein Cereal-Low Protein and Pork-Low Protein

In this example we will consider using a Latin Square design Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories. A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories. A Latin square is then used to assign the 6 diets to the 36 test animals in the study.

In the latin square the letter A represents the high protein-cereal diet B represents the high protein-pork diet C represents the low protein-beef Diet D represents the low protein-cereal diet E represents the low protein-pork diet and F represents the high protein-beef diet.

The weight gain after a fixed period is measured for each of the test animals and is tabulated below:

The Anova Table for Example Source S.S. d.f. M.S. F p-value Inwt 1767.0836 5 353.41673 111.1 0.0000 App 2195.4331 439.08662 138.03 Diet 4183.9132 836.78263 263.06 Error 63.61999 20 3.181 Total 8210.0499 35

Diet SS partioned into main effects for Source and Level of Protein d.f. M.S. F p-value Inwt 1767.0836 5 353.41673 111.1 0.0000 App 2195.4331 439.08662 138.03 631.22173 2 315.61087 99.22 Level 2611.2097 1 820.88 SL 941.48172 470.74086 147.99 Error 63.61999 20 3.181 Total 8210.0499 35

Graeco-Latin Square Designs Mutually orthogonal Squares

Definition A Greaco-Latin square consists of two latin squares (one using the letters A, B, C, … the other using greek letters a, b, c, …) such that when the two latin square are supper imposed on each other the letters of one square appear once and only once with the letters of the other square. The two Latin squares are called mutually orthogonal. Example: a 7 x 7 Greaco-Latin Square Aa Be Cb Df Ec Fg Gd Bb Cf Dc Eg Fd Ga Ae Cc Dg Ed Fa Ge Ab Bf Dd Ea Fe Gb Af Bc Cg Ee Fb Gf Ac Bg Cd Da Ff Gc Ag Bd Ca De Eb Gg Ad Ba Ce Db Ef Fc

Note: At most (t –1) t x t Latin squares L1, L2, …, Lt-1 such that any pair are mutually orthogonal. It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L1, L2, L3, L4, L5, L6 .

The Greaco-Latin Square Design - An Example A researcher is interested in determining the effect of two factors the percentage of Lysine in the diet and percentage of Protein in the diet have on Milk Production in cows. Previous similar experiments suggest that interaction between the two factors is negligible.

For this reason it is decided to use a Greaco-Latin square design to experimentally determine the two effects of the two factors (Lysine and Protein). Seven levels of each factor is selected 0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E), 0.5(F), and 0.6(G)% for Lysine and 2(a), 4(b), 6(c), 8(d), 10(e), 12(f) and 14(g)% for Protein ). Seven animals (cows) are selected at random for the experiment which is to be carried out over seven three-month periods.

A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (Lysine and Protein) to a period and a cow. The data is tabulated on below:

The Model for a Greaco-Latin Experiment j = 1,2,…, t i = 1,2,…, t k = 1,2,…, t l = 1,2,…, t yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth Greek treatment

tk = the effect of the kth Latin treatment m = overall mean tk = the effect of the kth Latin treatment ll = the effect of the lth Greek treatment ri = the effect of the ith row gj = the effect of the jth column eij(k) = random error No interaction between rows, columns, Latin treatments and Greek treatments

A Greaco-Latin Square experiment is assumed to be a four-factor experiment. The factors are rows, columns, Latin treatments and Greek treatments. It is assumed that there is no interaction between rows, columns, Latin treatments and Greek treatments. The degrees of freedom for the interactions is used to estimate error.

Greaco-Latin Square Experiment The Anova Table for a Greaco-Latin Square Experiment Source S.S. d.f. M.S. F p-value Latin SSLa t-1 MSLa MSLa /MSE Greek SSGr MSGr MSGr /MSE Rows SSRow MSRow MSRow /MSE Cols SSCol MSCol MSCol /MSE Error SSE (t-1)(t-3) MSE Total SST t2 - 1

The Anova Table for Example Source S.S. d.f. M.S. F p-value Protein 160242.82 6 26707.1361 41.23 0.0000 Lysine 30718.24 5119.70748 7.9 0.0001 Cow 2124.24 354.04082 0.55 0.7676 Period 5831.96 971.9932 1.5 0.2204 Error 15544.41 24 647.68367 Total 214461.67 48