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STAT:5201 Applied Statistic II

Latin Square Design (no replication)

Rocket Propellant Example

An experimenter is studying the effects of five different formulations of a rocket propellant on the burning rate of the propellant. Each formulation is mixed from a batch of raw material that is only large enough for five formulations to be tested. Also the formulations are prepared by several operntors, whose skill and experience may affect the burn rate.

Response: Burn rate. Factors: Formulation (5 levels), Material (5 levels), and Operator( 5 levels).

SAS statements for data input and Proc GLM:

data rocket; do material=l to 5;

do operator=l to 5; input formulation $ rate ©©; output;

end; end;

datalines; A 24 B 20 C 19 D 24 E 24 B 17 C 24 D 30 E 27 A 36 C 18 D 38 E 26 A 27 B 21 D 26 E 31 A 26 B 23 C 22 E 22 A 30 B 20 C 29 D 31

proc print data=rocket; run;

Obs material operator

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1

formulation

A B c D E B c D E A c D E A B D

1

rate

24 20 19 24 24 17 24 30 27 36 18 38 26 27 21 26

17 4 2 E 18 4 3 A 19 4 4 B 20 4 5 c 21 5 1 E 22 5 2 A 23 5 3 B 24 5 4 c 25 5 5 D

proc gplot data=rocket; plot rate* (formulation material operator);

run;

Rate vs. Formulation:

30

A

Rate vs. Material:

20 + +

c D

fonnulation

matenal

2

31 26 23 22 22 30 20 29 31

Rate vs. Operator:

ra:

30

JO

operator

SAS statements for Proc GLM:

proc glm data=rocket plot=diagnostics; class formulation material operator; model rate = formulation material operator/solution; lsmeans formulation/adjust=tukey pdiff; output out=diags r=residual p=predicted;

run;

The GLM Procedure Class Level Information

Class Levels Values formulation 5 A B C D E material operator

Dependent

Source Model Error Corrected

Source

5 5

Variable: rate

Total

formulation material operator

1 2 3 4 5 1 2 3 4 5

Sum of DF Squares 12 548.0000000 12 128.0000000 24 676.0000000

DF Type III SS

4 330.0000000 4 68.0000000 4 150.0000000

Mean Square 45.6666667 10.6666667

Mean Square

82.5000000 17.0000000 37.5000000

3

F Value Pr > F 4.28 0.0089

F Value Pr > FI 7.73 0.0025 ...e, 1.59 0.2391 3.52 0.0404

STAT:5201 Applied Statistic II

Latin Square Design (replicated)

Auto Carbon Monoxide (CO) Emission Example

Suppose the EPA is interested in recommending a certain brand of gasoline for cars with respect to CO emis-sion. There are three brands of gasoline they're interested in comparing. Since the amount of CO emitted depends on both the types of car (l:small/family sedan, 2:medium/van, 3:large/truck) and the speed the car is traveling in terms of types of routes (l:slow/25mph, 2:medium/45mph, 3:fast/65mph), these two factors, namely car and route, are taken into consideration during the design stage.

A Latin Square design is decided upon, but with only 3 treatment level:,; (a=3), there will be very few d.f. for error with a single square. So, four replications are done.

Response: emission. Factors: gas (3 levels), car (3 levels), and route ( 3 levels). Possible nesting of blocking factors will be described later.

Here are the 4 Latin squares with rows as cars and columns as routes:

A c B B A c c A B c c B A c B A A B c A B A c A c B B c A B

Here are the observed CO emissions:

45 28 24 42 30 26 47 31 25 68 42 23 66 39 25 64 40 27 100 58 38 98 60 33 105 63 35

SAS statements for data input and Proc GLM:

data coemission; do car = 1 to 3;

do route = 1 to 3; do square = 1 to 4;

input gas $ emission ©©; output;

end; end;

end; datalines;

B A c B A c

48 25 65 28

104 34

30 41 65

A 45 B 42 C 47 C 48 C 28 A 30 A 31 B 25 B 24 C 26 B 25 A 30 C 68 C 66 A 64 A 65 B 42 B 39 B 40 C 28 A 23 A 25 C 27 B 41 B 100 A 98 B 105 B 104 A 58 C 60 C 63 A 34 C 38 B 33 A 35 C 65

run;

1

SAS statements for SAME CARS AND SAME ROUTES:

/*same cars and same routes*/ proc glm data=coemission;

class gas car route square; model emission=gas car route square; lsmeans gas;

run;

The GLM Procedure Class Level Information

Class Levels Values gas 3 A B C car 3 1 2 3 route 3 1 2 3 square 4 1 2 3 4

Dependent Variable: emission Sum of

Source DF Squares Model 9 16960.16667 Error 26 3070.83333 Corrected Total 35 20031.00000

Dependent Variable: emission Source DF Type III SS

gas 292.666667 car 6667.166667 route 9968.666667 square 31.666667

Least Squares Means

Mean Square F Value Pr > F 1884.46296 15.96 <.0001

118.10897

Mean Square F Value Pr > F

146.333333 1.24 0.3062 3333.583333 28.22 <.0001 4984.333333 42.20 <.0001

10.555556 0.09 0.9652

gas

A B c

emission LS MEAN

44.8333333 51.6666667 47.0000000

Uo1rJ Co.rs

o r · '.,if f, cl/J-e

2

SAS statements for DIFFERENT CARS AND SAME ROUTES:

/*different cars and same routes*/ proc glm data=coemission;

class gas car route square; model emission=gas car(square) route square; lsmeans gas;

run;

The GLM Procedure

Class Level Information Class Levels Values

gas car route square

Dependent

Source Model Error

Variable:

Corrected Total

Dependent Variable: gas car(square) route square

The GLM Procedure Least Squares Means

3 3 3 4

A B C 1 2 3 1 2 3 1 2 3 4

emission

DF 15 20 35

emission 2

Sum of Squares

16976.33333 3054.66667

20031.00000

292.666667 6683.333333 9968.666667

31.666667

Mean Square F 1131. 75556

152.73333

146.333333 835.416667

4984.333333 10.555556

Value Pr > F 7.41 <.0001

0.96 0.4005 5.47 0.0010

32.63 <.0001 0.07 0.9757

emission 3 ' Ccd'S gas LS MEAN /,un l<..Q_ A 44.8333333 B 51.6666667 (3-F)4 J .+. (f <[; Im c 47.0000000 Car

3

sgu.qf-<1.

,,

SAS statements for SAME CARS AND DIFFERENT ROUTES:

/*same cars and different routes.*/ proc glm data=coemission;

class gas car route square; model emission=gas car route(square) square; lsmeans gas;

run;

The GLM Procedure

Class Level Information

Class Levels Values

gas 3 A B C car 3 1 2 3 route 3 1 2 3 square 4 1 2 3 4

The GLM Procedure

Dependent Variable: emission

Sum of Source DF Squares Model 15 18072.83333 Error 20 1958.16667 Corrected Total 35 20031.00000

Source DF Type III SS gas 2 292.66667 car 2 6667.16667 route(square) 11081. 33333 square 31. 66667

Least Squares Means

Mean Square F Value Pr > F 1204.85556 12.31 <.0001

97.90833

Mean Square F Value Pr > F 146.33333 1.49 0.2483

3333.58333 34.05 <.0001 1385.16667 14.15 <.0001

10.55556 0 .11 0.9545

' j \oJ.e S lwit JlUL emission uc;/"--12 gas LS MEAN so A 44.8333333 br (I J '' B 51.6666667 l5 c 47.0000000 ro e

4

SAS statements for DIFFERENT CARS AND DIFFERENT ROUTES:

/*different cars and different routes.*/ proc glm data=coemission;

class gas car route square; model emission=gas car(square) route(square) square; lsmeans gas;

run;

The GLM Procedure

Class Level Information

Class Levels Values gas 3 A B C car 3 1 2 3 route 3 1 2 3 square 4 1 2 3

Dependent Variable: emission

Source DF Model 21 Error 14 Corrected Total 35

Dependent Variable: emission

Source gas car(square) route(square) square

Least Squares Means

emission gas LSMEAN

A 44.8333333 B 51.6666667 c 47.0000000

DF

4

Sum of Squares Mean Square

18089.00000 861.38095 1942.00000 138.71429

20031.00000

Type III SS Mean Square 292.66667 146.33333

6683.33333. 835.41667 11081. 33333 1385.16667

31.66667 10.55556

Ca\S

5

F Value Pr > F 6.21 0.0005

F Value Pr > F 1.05 0.3743 6.02 0.0018 9.99 0.0001 0.08 0.9719

CfoJ-.es 0/1'1/J {,(_Q___

Medical Example of a Latin Square

6

7

Second Example of a Latin Square

Canadian Medical Association Journal, October 5, 2010, 182(14).

8