Split-Plot Experiment
Top Shrinkage by Wool Fiber Treatment and
Number of Drying Revolutions
J. Lindberg (1953). “Relationship Between Various Surface Properties of Wool Fibers: Part
II: Frictional Properties,” Textile Research Journal, Vol. 23, pp. 225-237
Data Description
• Experiment to Compare 4 Wool Fiber Treatments at 7
Dry Cycle Lengths over 4 Experimental Runs (Blocks)
• Response: Top Shrinkage of Fiber
• Restriction on Randomization: Within Each block, each
treatment is assigned to whole plot, then measurements
made at each of 7 dry cycle times (split plots)
• Whole Plot Treatments: Untreated, Alcoholic Potash (15
Sec, 4Min, 15Min)
• Subplot Treatments: Dry Cycle Revolutions (200 to 1400
by 200)
• Blocks: 4 Experimental Runs (possibly different days)
Block Layout
Within each block,
randomize the 4
treatments to the 4
whole plots
Revs
200
400
600
800
1000
1200
1400
200
400
600
800
1000
1200
1400
200
400
600
800
1000
1200
1400
200
400
600
800
1000
1200
1400
Trt
A
A
A
A
A
A
A
B
B
B
B
B
B
B
C
C
C
C
C
C
C
D
D
D
D
D
D
D
Whole Plot
Subplot
Marginal Means
run
1
2
3
4
revs
200
400
600
800
1000
1200
1400
average
23.63
24.77
22.48
22.07
average
6.45
13.43
19.56
25.13
28.96
32.84
36.30
trt
1
2
3
4
average
31.30
23.42
21.18
17.05
No Clear Run Effects
As Alcoholic Potash increases
(TRT), Shrinkage Decreases
As #Revs increases, Shrinkage
Increases
Interaction Plot - Trt*Revs
50
45
40
35
Shrinkage
30
trt 1
trt 2
trt 3
trt 4
25
20
15
10
5
0
0
200
400
600
800
Dryer Revolutions
1000
1200
1400
1600
Analysis of Variance
Source
Treatments
Blocks
BlkxTrt (Error 1)
Revs
TrtxRevs
Error 2
Total
df
3
3
9
6
18
72
111
SS
3012.53
124.29
114.64
11051.78
269.51
99.16
14671.90
MS
1004.18
41.43
12.74
1841.96
14.97
1.38
F
78.84
P-Value
0.0000
1337.46
10.87
0.0000
0.0000
Note that there is a significant interaction (as well as main effects).
Thus the “profile” relating shrinkage to # of revolutions differs by
treatment
Decomposing the Revolution and Trt/Rev Interaction
Sum of Squares into Polynomial Effects
• Note that for Revolutions, we have thus far
treated these as “nominal” categories, however,
it is a continuous variable
• We can break down its sums of squares into
orthogonal polynomials representing linear,
quadratic, cubic, … components (6 in all)
• Graph appears to show at least linear and
quadratic terms.
• Similar breakdown can be done on
Treatment/Revolution interaction
Partitioning of SSRevs and SSTrtxRevs
Source
Revs
Revs Linear
Revs Quadratic
Revs Cubic
Revs Quartic
Revs Quintic
Revs Sextic
TrtxRevs
TxR Linear
TxR Quadratic
TxR Cubic
TxR Quartic
TxR Quintic
TxR Sextic
Error 2
Total
df
6
1
1
1
1
1
1
18
3
3
3
3
3
3
72
111
SS
11051.78
10846.83
198.88
2.87
1.43
0.12
1.65
269.51
154.37
104.77
7.41
0.77
0.87
1.32
99.16
14671.90
MS
1841.96
10846.83
198.88
2.87
1.43
0.12
1.65
14.97
51.46
34.92
2.47
0.26
0.29
0.44
1.38
F
1337.46
7875.93
144.40
2.08
1.04
0.09
1.20
10.87
37.36
25.36
1.79
0.19
0.21
0.32
P-Value
0.0000
0.0000
0.0000
0.1532
0.3113
0.7669
0.2773
0.0000
0.0000
0.0000
0.1559
0.9055
0.8892
0.8110
The Revolution Main effect and the Treatment/Revolution Interaction
is made up of significant linear and quadratic components
Observed/Fitted Values - Quadratic Model
50
45
40
35
trt 1
trt 2
trt 3
trt 4
trt1(fit)
trt(fit)
trt3(fit)
trt4(fit)
Shrinkage
30
25
20
15
10
5
0
0
200
400
600
800
Revolutions
1000
1200
1400
1600
Procedure for Obtaining Polynomial SS
1. Obtain coefficients for orthogonal polynomials from stat design or math source
Revs
Linear
Quadratic
Cubic
Quartic
Quintic
Sextic
200
-3
5
-1
3
-1
1
400
-2
0
1
-7
4
-6
600
-1
-3
1
1
-5
15
800
0
-4
0
6
0
-20
1000
1
-3
-1
1
5
15
1200
2
0
-1
-7
-4
-6
1400
3
5
1
3
1
1
2. Obtain Linear Combinatio n of Means Across Revs
L1  3 y 200  2 y 400  1y 600  0 y 800  1y1000  2 y1200  3 y1400
...
L6  1y 200  6 y 400  15 y 600  20 y 800  15 y1000  6 y1200  1y1400
Procedure for Obtaining Polynomial SS
3. Obtain the sums of squares for each contrast
(4)( 4) L12
Linear : SS1 
(3) 2  (2) 2  (1) 2  (0) 2  (1) 2  (2) 2  (3) 2
...


(4)( 4) L26
Sextic : SS 6 
(1) 2  (6) 2  (15) 2  (20) 2  (15) 2  (6) 2  (3) 2
Where the (4)(4)  16 in numerator represents the number of
observatio ns per rev level (4 blocks x 4 treatment s)


This process can also be extended to the TreatmentxRev interaction by:
1) apply it within treatments (there are only 4 samples per Trt/Rev combination)
2) Sum each polynomial component over treatments and subtract results from 3)
EXCEL Calculations of Polynomial SS
All Trts(Rev)
All Trts(Rev)
Trt1
Trt2
Trt3
Trt4
Trt1
Trt2
Trt3
Trt4
Sum(T1:T4)
Sum(T1:T4)-Rev
L
SS
L
L
L
L
SS
SS
SS
SS
SS
SS
linear quadratic
137.78
-32.31
10846.83 198.88
161.33
-72.93
141.65
-18.80
132.28
-19.48
115.85
-18.05
3717.97
253.24
2866.39
16.83
2499.53
18.06
1917.32
15.51
11001.20 303.65
154.37
104.77
cubic
1.04
2.87
3.90
-0.27
0.30
0.23
10.14
0.05
0.06
0.03
10.28
7.41
quartic
3.71
1.43
0.80
5.63
1.30
7.12
0.02
0.82
0.04
1.32
2.20
0.77
quintic
-0.80
0.12
-2.92
1.60
-2.93
1.05
0.41
0.12
0.41
0.05
0.99
0.87
Note the second and last rows (of the numeric values) give the polynomial
sums of squares for the factors Rev, and Trt x Rev, respectively.
sextic
-9.76
1.65
-22.93
-7.75
-9.93
1.55
2.28
0.26
0.43
0.01
2.97
1.32