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