16 graeco latin squares 323

7/29/2019 16 Graeco Latin Squares 323

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Stat 323 16-Graeco Latin Squares 1

GRAECO-LATIN SQUARE DESIGN

A Graeco-Latin square involves blocking in three directions; i.e., tries to balance three

nuisance factors.

Examples:

1. Crop yields from five different seed varieties planted in a field where both the N-Sdirection and the E-W direction appear to have different soil qualities and sections

have been allowed to be fallow for five different amounts of time.

2. Run times for three different search algorithms in Monte Carlo simulations using

three different computers, three different languages, and three different operating

systems.

Example The crop yield example could be extended to include Greek letters representing

the fallow times.

5 x 5 Graeco-Latin

Square

E-W Location

1 2 3 4 5

N-S

L

oc

1 A B C D E

2 E A B C D

3 D E A B C

4 C D E A B

5 B C D E A

Rearranging the rows and columns as well as randomly assigning fallow times to the Greek

letters:

5 x 5 Graeco-Latin

Square

E-W Location

1 2 3 4 5

N-S

Lo

c

1 B = 2.0 C = 1.6 A = 1.0 E = 1.2 D = 2.2

2 E = 0.0 A = 1.8 D = 1.6 C = 2.6 B = 2.7

3 D = 1.5 E = 2.2 C = 2.2 B = 2.3 A = 3.3

4 C = 1.4 D = 1.0 B = 3.4 A = 2.0 E = 2.7

5 A = 1.7 B = 2.8 E = 2.0 D = 3.2 C = 2.6


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General Linear Model: Yield versus Variety, N-S, E-W, Fallow

Factor Type Levels Values

Variety fixed 5 A, B, C, D, E

N-S fixed 5 1, 2, 3, 4, 5E-W fixed 5 1, 2, 3, 4, 5

Fallow fixed 5 1, 2, 3, 4, 5

Analysis of Variance for Yield, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

Variety 4 2.8200 2.8200 0.7050 4.81 0.028

N-S 4 2.6560 2.6560 0.6640 4.53 0.033

E-W 4 5.1400 5.1400 1.2850 8.77 0.005

Fallow 4 3.5120 3.5120 0.8780 5.99 0.016

Error 8 1.1720 1.1720 0.1465

Total 24 15.3000

S = 0.382753 R-Sq = 92.34% R-Sq(adj) = 77.02%

Unusual Observations for Yield

Obs Yield Fit SE Fit Residual St Resid

13 2.20000 2.68000 0.31563 -0.48000 -2.22 R

R denotes an observation with a large standardized residual.


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Expected Mean Squares, using Adjusted SS

Expected Mean

Square for

Source Each Term

1 Variety (5) + Q[1]

2 N-S (5) + Q[2]

3 E-W (5) + Q[3]

4 Fallow (5) + Q[4]5 Error (5)

Error Terms for Tests, using Adjusted SS

Synthesis

Source Error DF Error MS of Error MS

1 Variety 8.00 0.1465 (5)

2 N-S 8.00 0.1465 (5)

3 E-W 8.00 0.1465 (5)

4 Fallow 8.00 0.1465 (5)

Variance Components, using Adjusted SS

Estimated

Source Value

Error 0.1465

Least Squares Means for Yield

Variety Mean SE Mean

A 1.960 0.1712

B 2.640 0.1712

C 2.080 0.1712

D 1.900 0.1712

E 1.620 0.1712

Tukey 95.0% Simultaneous Confidence Intervals

Response Variable Yield

All Pairwise Comparisons among Levels of Variety

Variety = A subtracted from:

Variety Lower Center Upper ---------+---------+---------+-------

B -0.157 0.6800 1.5170 (--------*-------)

C -0.717 0.1200 0.9570 (-------*--------)

D -0.897 -0.0600 0.7770 (-------*--------)

E -1.177 -0.3400 0.4970 (--------*-------)

---------+---------+---------+-------

-1.0 0.0 1.0

Variety = B subtracted from:


C -1.397 -0.560 0.2770 (-------*--------)

D -1.577 -0.740 0.0970 (--------*-------)

E -1.857 -1.020 -0.1830 (--------*-------)

---------+---------+---------+-------

-1.0 0.0 1.0


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Variety = C subtracted from:


D -1.017 -0.1800 0.6570 (-------*--------)

E -1.297 -0.4600 0.3770 (-------*--------)

---------+---------+---------+-------

-1.0 0.0 1.0

Variety = D subtracted from:


E -1.117 -0.2800 0.5570 (-------*--------)

---------+---------+---------+-------

-1.0 0.0 1.0


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Read 4.3, Problems 4.28, 29

16 graeco latin squares 323

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