latin square designs. selected latin squares 3 x 34 x 4 a b ca b c da b c da b c da b c d b c ab a d...
TRANSCRIPT
Latin Square Designs
Latin Square Designs Selected Latin Squares
3 x 3 4 x 4A B C A B C D A B C D A B C D A B C DB C A B A D C B C D AB D A C B A D CC A B C D B AC D A B C A D B C D A B
D C A B D A B C D C B AD C B A
5 x 5 6 x 6A B C D E A B C D E FB A E C D B F D C A EC D A E B C D E F B AD E B A C D A F E C BE 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 DB C D AC D A BD 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:
Week 1 2 3 4 5 1 5.83 6.22 7.67 9.43 6.57 D P F C R 2 4.80 7.56 10.34 5.82 9.86 P D C R F
Drivers 3 7.43 11.29 7.01 10.48 9.27 F C R D P 4 6.60 9.54 11.11 10.84 15.05 R F D P C 5 11.24 6.34 11.30 12.58 16.04 C R P F D
The Model for a Latin Experiment
kijjikkijy
i = 1,2,…, t j = 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 treatmentri = the effect of the ith row
eij(k) = random error
k = 1,2,…, t
gj = the effect of the jth column
No interaction between rows, columns and treatments
ijjiijy • 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
ijjiijy Source S.S. d.f. M.S. F p-value
Treat SSTr t-1 MSTr MSTr /MSE
Rows SSRow t-1 MSRow MSRow /MSE
Cols SSCol t-1 MSCol MSCol /MSE
Error SSE (t-1)(t-2) MSE
Total SST t2 - 1
The Anova Table for Example
ijjiijy Source S.S. d.f. M.S. F p-value
Week 51.17887 4 12.79472 16.06 0.0001
Driver 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
Using SPSS for a Latin Square experiment
Rows Cols Trts 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
Tests of Between-Subjects Effects
Dependent Variable: COST
191.530a 12 15.961 20.028 .000
2120.050 1 2120.050 2660.273 .000
69.447 4 17.362 21.786 .000
51.179 4 12.795 16.055 .000
70.904 4 17.726 22.243 .000
9.563 12 .797
2321.143 25
201.093 24
SourceCorrected Model
Intercept
DRIVER
WEEK
CAR
Error
Total
Corrected Total
Type IIISum of
Squares dfMean
Square F Sig.
R Squared = .952 (Adjusted R Squared = .905)a.
The ANOVA table produced by SPSS
Latin Square -17
Advantages and Disadvantages
Advantages:• Allows for control of two extraneous sources of
variation.• Analysis is quite simple.
Disadvantages:• Requires t2 experimental units to study t treatments.• Best suited for t in range: 5 t 10.• The effect of each treatment on the response must be
approximately the same across rows and columns.• Implementation problems.• Missing data causes major analysis problems.
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:
Appetite Category 1 2 3 4 5 6 1 62.1 84.3 61.5 66.3 73.0 104.7 A B C D E F 2 86.2 91.9 69.2 64.5 80.8 83.9 B F D C A E
Initial 3 63.9 71.1 69.6 90.4 100.7 93.2 Weight C D E F B A
Category 4 68.9 77.2 97.3 72.1 81.7 114.7 D A F E C B 5 73.8 73.3 78.6 101.9 111.5 95.3 E C A B F D 6 101.8 83.8 110.6 87.9 93.5 103.8 F E B A D C
The Anova Table for Example
ijjiijy Source 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
Total 8210.0499 35
Diet SS partioned into main effects for Source and Level of Protein
ijjiijy Source 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
Total 8210.0499 35
Experimental Design
Of interest: to compare t treatments (the treatment combinations of one or several factors)
The Completely Randomized Design
Treats1 2 3 … t
Experimental units randomly assigned to treatments
The Model for a CR Experiment
ijiijy
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 treatmenteij = random error
The Anova Table for a CR Experiment
ijjiijy Source S.S. d.f. M.S. F p-value
Treat SSTr t-1 MST MST /MSE
Error SSE t(n-1) MSE
Randomized Block Design
Blocks
All treats appear once in each block
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
1
2
3
⁞
t
The Model for a RB Experiment
ijjiijy
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
eij = random error
bj = the effect of the jth blockNo interaction between blocks and treatments
ijjiijy • 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
ijjiijy Source S.S. d.f. M.S. F p-value
Treat SST t-1 MST MST /MSE
Block SSB n-1 MSB MSB /MSE
Error SSE (t-1)(b-1) MSE
The Latin square Design
All treats appear once in each row and each column
Columns
Row
s1
2
3
⁞
t2
3
t
1
13
2
The Model for a Latin Experiment
kijjikkijy
i = 1,2,…, t j = 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 treatmentri = the effect of the ith row
eij(k) = random error
k = 1,2,…, t
gj = the effect of the jth column
No interaction between rows, columns and treatments
ijjiijy • 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
ijjiijy Source S.S. d.f. M.S. F p-value
Treat SSTr t-1 MSTr MSTr /MSE
Rows SSRow t-1 MSRow MSRow /MSE
Cols SSCol t-1 MSCol MSCol /MSE
Error SSE (t-1)(t-2) MSE
Total SST t2 - 1