Using the Nested Control Co-design Strategy for Designing ...
Nested Design
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Transcript of Nested Design
Nested Design
Comparison of 2 Shampoo Formulations on Hair Combability
M.L. Garcia and J. Diaz (1976). “Combability Measurements on Human Hair,” Journal of the Society of Cosmetic Chemists, Vol.27 pp.379-398
Data Description
• 2 Formulations of Shampoo (A and B)• 16 Hair Swatches Prepared (8 for A, 8 for B)• 5 Test Runs Made per Swatch (Replicates)• Combability Measurements made before and
after shampooing (we consider only AFTER here)
• Shampoo Treated as FIXED Factor, Swatch RANDOM
• N=2x8x5=80 Measurements
Statistical Model
( )
th th th
th1 2
( )
1,2 1,...,8 1,...,5
where:
Combability of k rep of j swatch of i shampoo formulation
Overall mean combability score
Effect of i shampoo formulation 0
ijk i j i ijk
ijk
i
j i
Y i j k
Y
α
th th 2( )
2
Effect of j swatch within i formulation ~ 0,
Random error term ~ 0,
j i b
ijk ijk
N
ε N
Note: For Computational purposes, we will label the swatches 1,…,16 and not 1,…,8 separately for each shampoo
Data
form swatch rep comb form swatch rep comb form swatch rep comb form swatch rep comb1 1 1 183 1 5 1 145 2 9 1 255 2 13 1 651 1 2 133 1 5 2 113 2 9 2 110 2 13 2 601 1 3 190 1 5 3 98 2 9 3 195 2 13 3 1051 1 4 153 1 5 4 138 2 9 4 93 2 13 4 901 1 5 173 1 5 5 140 2 9 5 213 2 13 5 1001 2 1 173 1 6 1 73 2 10 1 118 2 14 1 951 2 2 173 1 6 2 123 2 10 2 200 2 14 2 651 2 3 115 1 6 3 100 2 10 3 145 2 14 3 451 2 4 198 1 6 4 75 2 10 4 155 2 14 4 651 2 5 150 1 6 5 115 2 10 5 108 2 14 5 601 3 1 80 1 7 1 123 2 11 1 133 2 15 1 1451 3 2 75 1 7 2 138 2 11 2 155 2 15 2 1751 3 3 68 1 7 3 100 2 11 3 145 2 15 3 1251 3 4 70 1 7 4 253 2 11 4 240 2 15 4 1451 3 5 58 1 7 5 93 2 11 5 230 2 15 5 1801 4 1 115 1 8 1 38 2 12 1 150 2 16 1 601 4 2 125 1 8 2 55 2 12 2 130 2 16 2 551 4 3 120 1 8 3 35 2 12 3 110 2 16 3 601 4 4 125 1 8 4 38 2 12 4 185 2 16 4 451 4 5 148 1 8 5 53 2 12 5 105 2 16 5 65
Summary Statistics
Form A Form BSwatch Mean SD Swatch Mean SD
1 166.4 23.30 9 173.2 69.242 161.8 31.19 10 145.2 36.133 70.2 8.26 11 180.6 50.394 126.6 12.66 12 136.0 32.675 126.8 20.32 13 84.0 20.436 97.2 22.74 14 66.0 18.177 141.4 64.93 15 154.0 23.028 43.8 9.42 16 57.0 7.58
Form MeanA 116.775B 124.500
Overall 120.638
Sums of Squares
2 2
22
2 2 1
( )
2 2
Formulation 2 1 1
8(5) (116.775 120.6375) (124.5 120.6375) 1193.5
8(5)5
2 1
Swatch Nested Within Formulation 2(8 1) 14
( ) 5 (166.4 116.75) ... (43.8 116.775) (
F
ii
b
S F
df
SSF
E MSF
df
SSS F
2 2
2 2
2 2 2 2
173.2 124.5) ... (57.0 124.5)
5(13171.08 16568.64) 148698.6
( ) 5
Error (Rep Nested within swatch) 2(8)(5 1) 64
5 1 23 30 9 42 69 24 7 58 4 6977 9 11041 7 72078
b
E
E MSS F
df
SSE ( - ) . ... . . ... . ( . . )
2
4
Total 2(8)(5) 1 79
( ) 221970.5Total
.
E MSE
df
TSS SSF SSS F SSE
ANOVA – Tests of Factor Effects
Source df SS MS F P-valueFormulation 1 1193.5 1193.513 1193.5/10621.3=0.112 0.7390Swatch(Form) 14 148698.6 10621.33 10621.3/1126.2=9.43 0.0000Error 64 72078.4 1126.225Total 79 221970.5
H0: HA: ≠ T.S.: FForm = 0.112 (P=0.7390)
Conclude no formulation differences on combability
H0: b2 = 0 HA: b
2 > 0 T.S. FSwatch(F) = 9.43 (P=.0000)
Conclude Swatch effects vary with respect to combability