Latin Square Designs Latin Square Designs Selected Latin

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Latin Square Designs

Latin Square Designs

Latin Square Designs Selected Latin Squares 3 x 3 4 x 4 ABCD BCA

Latin Square Designs Selected Latin Squares 3 x 3 4 x 4 ABCD BCA BADC CAB CDBA DCAB 5 x 5 ABCDE BAECD CDAEB DEBAC ECDBA ABCD BCDA CDAB DABC 6 x 6 ABCDEF BFDCAE CDEFBA DAFECB ECABFD FEBADC ABCD BDAC CADB DCBA ABCD BADC CDAB DCBA

A Latin Square

A Latin Square

Definition • A Latin square is a square array of objects (letters A, B,

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. ABCD BCDA CDAB DABC

In a Latin square You have three factors: • Treatments (t) (letters A, B,

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,

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

• 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 =

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 th ri = the effect of the i row between rows, columns and gj = the effect of the jth column treatments eij(k) = random error

 • A Latin Square experiment is assumed to be a three-factor experiment. •

• 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 Treat Rows Cols Error Total

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

The Anova Table for Example S. S. d. f. M. S. F p-value Week

The Anova Table for Example S. S. d. f. M. S. F p-value Week Driver 51. 17887 4 12. 79472 16. 06 0. 0001 69. 44663 4 17. 36166 21. 79 0. 0000 Car 70. 90402 4 17. 72601 22. 24 0. 0000 Error 9. 56315 12 0. 79693 Total 201. 09267 24 Source

Using SPSS for a Latin Square experiment Rows Cols Trts Y

Using SPSS for a Latin Square experiment Rows Cols Trts Y

Select Analyze->General Linear Model->Univariate

Select Analyze->General Linear Model->Univariate

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

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

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

The ANOVA table produced by SPSS

The ANOVA table produced by SPSS

Example 2 In this Experiment the we are again interested in how weight gain

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 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

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

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

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

The Anova Table for Example S. S. d. f. M. S. F p-value Inwt

The Anova Table for Example S. S. d. f. M. S. F p-value Inwt 1767. 0836 5 353. 41673 111. 1 0. 0000 App 2195. 4331 5 439. 08662 138. 03 0. 0000 Diet 4183. 9132 5 836. 78263 263. 06 0. 0000 Error 63. 61999 20 3. 181 8210. 0499 35 Source Total

Diet SS partioned into main effects for Source and Level of Protein S. S.

Diet SS partioned into main effects for Source and Level of Protein S. S. d. f. M. S. F p-value Inwt 1767. 0836 5 353. 41673 111. 1 0. 0000 App 2195. 4331 5 439. 08662 138. 03 0. 0000 Source 631. 22173 2 315. 61087 99. 22 0. 0000 Level 2611. 2097 1 2611. 2097 820. 88 0. 0000 SL 941. 48172 2 470. 74086 147. 99 0. 0000 Error 63. 61999 20 3. 181 8210. 0499 35 Source Total

Experimental Design Of interest: to compare t treatments (the treatment combinations of one or

Experimental Design Of interest: to compare t treatments (the treatment combinations of one or several factors)

The Completely Randomized Design 1 2 Treats 3 … t Experimental units randomly assigned

The Completely Randomized Design 1 2 Treats 3 … t Experimental units randomly assigned to treatments

The Model for a CR Experiment i = 1, 2, …, t j =

The Model for a CR Experiment i = 1, 2, …, t j = 1, 2, …, n yij = the observation in jth observation receiving the ith treatment m = overall mean ti = the effect of the ith treatment eij = random error

The Anova Table for a CR Experiment Source S. S. d. f. M. S.

The Anova Table for a CR Experiment Source S. S. d. f. M. S. F Treat Error SSTr SSE t-1 t(n-1) MST MSE MST /MSE p-value

Randomized Block Design Blocks 1 2 1 2 1 2 3 ⁞ t 3

Randomized Block Design Blocks 1 2 1 2 1 2 3 ⁞ t 3 ⁞ t 3 ⁞ t All treats appear once in each block

The Model for a RB Experiment i = 1, 2, …, t j =

The Model for a RB Experiment i = 1, 2, …, t j = 1, 2, …, b yij = the observation in jth block receiving the ith treatment m = overall mean ti = the effect of the ith treatment bj = the effect of the jth block eij = random error No interaction between blocks and treatments

 • A Randomized Block experiment is assumed to be a two-factor experiment. •

• A Randomized Block experiment is assumed to be a two-factor experiment. • The factors are blocks and treatments. • It is assumed that there is no interaction between blocks and treatments. • The degrees of freedom for the interaction is used to estimate error.

The Anova Table for a randomized Block Experiment Source S. S. d. f. M.

The Anova Table for a randomized Block Experiment Source S. S. d. f. M. S. F Treat Block Error SST SSB SSE t-1 n-1 (t-1)(b-1) MST MSB MSE MST /MSE MSB /MSE p-value

The Latin square Design Rows Columns 1 2 3 1 3 1 2 t

The Latin square Design Rows Columns 1 2 3 1 3 1 2 t ⁞ t All treats appear once in each row and each column

The Model for a Latin Experiment i = 1, 2, …, t j =

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, ri = the effect of the ith row columns and th treatments gj = the effect of the j column eij(k) = random error

 • A Latin Square experiment is assumed to be a three-factor experiment. •

• 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 Treat Rows Cols Error Total

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

Graeco-Latin Square Designs Mutually orthogonal Squares

Graeco-Latin Square Designs Mutually orthogonal Squares

Definition A Greaco-Latin square consists of two latin squares (one using the letters A,

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 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: There exists at most (t – 1) t x t Latin squares L

Note: There exists at most (t – 1) t x t Latin squares L 1, L 2, …, Lt-1 such that any pair are mutually orthogonal. e. g. It is possible that there exists a set of six 7 mutually orthogonal Latin squares L 1, L 2, L 3, L 4, L 5, L 6.

The Greaco-Latin Square Design - An Example A researcher is interested in determining the

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

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

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 i = 1, 2, …, t j =

The Model for a Greaco-Latin Experiment i = 1, 2, …, t j = 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

m = overall mean tk = the effect of the kth Latin treatment ll

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. •

• 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.

The Anova Table for a Greaco-Latin Square Experiment Source Latin Greek Rows Cols Error

The Anova Table for a Greaco-Latin Square Experiment Source Latin Greek Rows Cols Error Total S. S. d. f. M. S. F SSLa t-1 MSLa /MSE SSGr t-1 MSGr /MSE SSRow t-1 MSRow /MSE SSCol t-1 MSCol /MSE SSE (t-1)(t-3) MSE SST t 2 - 1 p-value

The Anova Table for Example Source S. S. d. f. Protein Lysine Cow Period

The Anova Table for Example Source S. S. d. f. Protein Lysine Cow Period Error Total 160242. 82 6 6 24 48 30718. 24 2124. 24 5831. 96 15544. 41 214461. 67 M. S. 26707. 1361 5119. 70748 354. 04082 971. 9932 647. 68367 F p-value 41. 23 7. 9 0. 55 1. 5 0. 0000 0. 0001 0. 7676 0. 2204

Next topic: Incomplete Block designs

Next topic: Incomplete Block designs