A Latin Square - Rocket Propellant *
A Latin Square is a one-factor design that extends blocking to two nuisance
factors. For example, suppose we want to study the effects of five different
formulations of rocket propellant. Suppose we also want to block (or
average over) two nuisance factors:
1) Batch of raw material
2) Operator.
This is a 5 x 5 Latin Square and looks like this:
Raw
material
batch
1
2
3
4
5
1
A
B
C
D
E
2
B
C
D
E
A
3
C
D
E
A
B
4
D
E
A
B
C
5
E
A
B
C
D
Operator
This arrangement is a square and with each of the five treatments (a
formulation of rocket propellant) represented by a Latin letter. Each letter
occurs once for each operator and once for each batch of raw material.
The Latin Square allows blocking of two nuisance factors. In this example
we block on batch of raw material (the row) and operator (the column).
We will use this example to illustrate how to build and analyze a Latin
square design using Design-Expert® software.
* Douglas Montgomery, Design and Analysis of Experiments, 4th edition,
John Wiley, Example 5-4 on page 196.
1-1
Rocket Propellant
Procedure to create a 5x5 Latin Square design:
1. From the “File” menu choose “New Design…”
2. On the “Factorial” tab choose “General Factorial” and “3” Categorical
Factors:
3. Name the first blocking factor (e.g. Row), enter the number of rows
“5” and the block level names:
1-2
Rocket Propellant
4. Name the second blocking factor (e.g. Column), enter the number of
rows “5” and the block level names:
5. Name the treatment factor (e.g. Formulation), enter the number of
rows “5” and the factor level names:
1-3
Rocket Propellant
6.
7.
8. Right click on the column heading “Std” and choose “Sort by
Standard Order”.
9. Right click on the column heading “Run” and choose “Re-order a
currently displayed”.
1-4
Rocket Propellant
10. Delete rows 26 – 125:
•
Left click on the button to the left of row 26.
•
While holding down the shift key, scroll down and left click the
button to the left of the last row, i.e. row 125.
•
Right click on a button to the left of any row in the selected area
and choose “Delete Row[s]”:
“Are you sure you want to delete these rows??”
“If you delete or ignore these rows you will have only
one level of factor/component 3. Do you want to
delete/ignore these rows?”
1-5
Rocket Propellant
11. Treatments (formulations) need to be assigned correctly. Type in the
Latin letters that represent the five treatments (formulations) in a
cyclic order. Starting at the top (row 1) type:
A, B, C, D, E
B, C, D, E, A
C, D, E, A, B
and finally E, A, B, C, D
1-6
D, E, A, B, C
Rocket Propellant
12. Right click on the column heading “Run” and choose “Randomize…”.
13. Save your design as “Example 5-4.dx6”.
14. Run the simulation:
•
right click on the response column heading
•
select "Simulate response"
•
choose the "Rocket 5x5 Latin Square.sim" file and click OK
15. Analyze the response:
•
Click on the response node “Burn rate”.
•
Click on the “Effects” button.
a. Right click on “A” and choose “Block”
b. Right click on “B” and choose “Block”
c. Right click on “C” and choose “Model”
1-7
Rocket Propellant
16. Click on the “ANOVA” button:
Block term includes A, B
Analysis of variance table [Partial sum of squares]
Sum of
Mean
F
Source
Squares
DF
Square
Value
Block
218.00
8
27.25
Model
330.00
4
82.50
7.73
C
330.00
4
82.50
7.73
Residual
128.00
12
10.67
Cor Total
676.00
24
17. Click on the “Diagnostics” button and check as usual:
Normal plot of residuals
99
N orm al % probability
95
90
80
70
50
30
20
10
5
Residuals vs. P redicted
1
3.00
Studentized R es iduals
-1.41
1.50
-0.51
0.40
1.30
2.21
Studentized R es iduals
0.00
-1.50
-3.00
17.60
21.60
25.60
29.60
33.60
Predicted
1-8
Prob > F
0.0025
0.0025
Rocket Propellant
18. Note the Box-Cox plot recommends an inverse transformation, this
makes sense for a rate.
D E S I G N -E X P E R T P l o t
B u rn ra t e
5 .6 9
L n (R e s id u a lS S )
Lam bda
C u r re n t = 1
B e st = -1
L o w C . I . = -2 . 8 3
H i g h C .I. = 0 .8 7
B o x - C o x P lo t fo r P o w e r T r a n s fo r m s
5 .3 9
R e c o m m e n d t ra n sf o r m :
I n v e rse
(L a m b d a = - 1 )
5 .0 9
4 .7 9
4 .4 9
-3
-2
-1
0
1
2
3
Lam bda
19. Reanalyze using the inverse transform:
Block term includes A, B
Analysis of variance table [Partial sum of squares]
Sum of
Source
Squares
Block
5.331E-004
Model
9.072E-004
C
9.072E-004
Residual 2.321E-004
Cor Total 1.672E-003
DF
8
4
4
12
24
1-9
Mean
Square
6.664E-005
2.268E-004
2.268E-004
1.934E-005
F
Value
Prob > F
11.73
11.73
0.0004
0.0004
Rocket Propellant
20. Click on the “Model Graphs” button:
Plot the treatment (Formulation) and average over the blocks (Rows
and Columns):
O n e F a c t o r P lo t
D E S I G N -E X P E R T P l o t
1 . 0 / (B u rn ra t e )
33
X = C : F o rm u l a t i o n
A c t u a l F a c t o rs
A : R o w = A v e ra g e
B : C o l u m n = A v e ra g e
B u r n r a te
29
25
21
17
A
B
C
C : F o r m u l a ti o n
1-10
D
E