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

Revs Trt200 A400 A600 A800 A1000 A1200 A1400 A200 B400 B600 B800 B1000 B1200 B1400 B200 C400 C600 C800 C1000 C1200 C1400 C200 D400 D600 D800 D1000 D1200 D1400 D

Whole Plot

SubplotWithin each block, randomize the 4 treatments to the 4 whole plots

Marginal Means

run average1 23.632 24.773 22.484 22.07

trt average1 31.302 23.423 21.184 17.05

revs average200 6.45400 13.43600 19.56800 25.131000 28.961200 32.841400 36.30

No Clear Run Effects

As Alcoholic Potash increases (TRT), Shrinkage Decreases

As #Revs increases, Shrinkage Increases

Interaction Plot - Trt*Revs

0

5

10

15

20

25

30

35

40

45

50

0 200 400 600 800 1000 1200 1400 1600

Dryer Revolutions

Sh

rin

ka

ge trt 1

trt 2

trt 3

trt 4

Analysis of Variance

Source df SS MS F P-ValueTreatments 3 3012.53 1004.18 78.84 0.0000Blocks 3 124.29 41.43BlkxTrt (Error 1) 9 114.64 12.74Revs 6 11051.78 1841.96 1337.46 0.0000TrtxRevs 18 269.51 14.97 10.87 0.0000Error 2 72 99.16 1.38Total 111 14671.90

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 SSTrtxRevsSource df SS MS F P-ValueRevs 6 11051.78 1841.96 1337.46 0.0000 Revs Linear 1 10846.83 10846.83 7875.93 0.0000 Revs Quadratic 1 198.88 198.88 144.40 0.0000 Revs Cubic 1 2.87 2.87 2.08 0.1532 Revs Quartic 1 1.43 1.43 1.04 0.3113 Revs Quintic 1 0.12 0.12 0.09 0.7669 Revs Sextic 1 1.65 1.65 1.20 0.2773TrtxRevs 18 269.51 14.97 10.87 0.0000 TxR Linear 3 154.37 51.46 37.36 0.0000 TxR Quadratic 3 104.77 34.92 25.36 0.0000 TxR Cubic 3 7.41 2.47 1.79 0.1559 TxR Quartic 3 0.77 0.26 0.19 0.9055 TxR Quintic 3 0.87 0.29 0.21 0.8892 TxR Sextic 3 1.32 0.44 0.32 0.8110Error 2 72 99.16 1.38Total 111 14671.90

The Revolution Main effect and the Treatment/Revolution Interaction is made up of significant linear and quadratic components

Observed/Fitted Values - Quadratic Model

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5

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15

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30

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0 200 400 600 800 1000 1200 1400 1600

Revolutions

Sh

rin

ka

ge

trt 1

trt 2

trt 3

trt 4

trt1(fit)

trt(fit)

trt3(fit)

trt4(fit)

Procedure for Obtaining Polynomial SS

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

1. Obtain coefficients for orthogonal polynomials from stat design or math source

1400120010008006004002006

1400120010008006004002001

1615201561

...

3210123

Revs Across Means ofn CombinatioLinear Obtain 2.

yyyyyyyL

yyyyyyyL

Procedure for Obtaining Polynomial SS

s) treatment4 x blocks (4 level revper nsobservatio

ofnumber therepresentsnumerator in 16(4)(4) theWhere

)3()6()15()20()15()6()1(

)4)(4( :Sextic

...

)3()2()1()0()1()2()3(

)4)(4( :Linear

contrasteach for squares of sums Obtain the 3.

2222222

26

6

2222222

21

1

LSS

LSS

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

linear quadratic cubic quartic quintic sexticAll Trts(Rev) L 137.78 -32.31 1.04 3.71 -0.80 -9.76All Trts(Rev) SS 10846.83 198.88 2.87 1.43 0.12 1.65Trt1 L 161.33 -72.93 3.90 0.80 -2.92 -22.93Trt2 L 141.65 -18.80 -0.27 5.63 1.60 -7.75Trt3 L 132.28 -19.48 0.30 1.30 -2.93 -9.93Trt4 L 115.85 -18.05 0.23 7.12 1.05 1.55Trt1 SS 3717.97 253.24 10.14 0.02 0.41 2.28Trt2 SS 2866.39 16.83 0.05 0.82 0.12 0.26Trt3 SS 2499.53 18.06 0.06 0.04 0.41 0.43Trt4 SS 1917.32 15.51 0.03 1.32 0.05 0.01Sum(T1:T4) SS 11001.20 303.65 10.28 2.20 0.99 2.97Sum(T1:T4)-Rev SS 154.37 104.77 7.41 0.77 0.87 1.32

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.