1

Nested (Hierarchical) Designs

In certain experiments the levels of one factor (eg. Factor B) are similar but not identical for different levels of another factor (eg. Factor A). Such an arrangement is called a nested or hierarchical design, with the levels of factor B nested under the levels of factor A.

2

1 2 3

1 2 3 1 2 3 1 2 3

Suppliers

Batches

Y111

Y112

Y113

Y121

Y122

Y123

Y131

Y132

Y133

Y211

Y212

Y213

Y221

Y222

Y223

Y231

Y232

Y233

Y311

Y312

Y313

Y321

Y322

Y323

Y331

Y332

Y333

Obs’ns{Consider a company that purchases its raw material from three different suppliers. The company wishes to determine if the purity of the raw material is the same from each supplier. There are 4 batches of raw material available from each supplier, and three samples are taken from each batch to measure their purity.

4 4

Y141

Y142

Y143

4

Y241

Y242

Y243

Y341

Y342

Y343

3

MODEL:

i = 1, ..., M (the #of levels of the major factor)j = 1, ..., m (the # of levels of the minor factor

for each level of the major factor)k= 1, ..., n (the # of replicates per (i,j) combination)

Note: n= nij if unequal replicates for combinations.

Yijk = i(i)j(ij)k

4

the grand meanithe difference between the ith

level mean of the major factor (A) and the grand mean (main effect of factor A)

(i)j the difference between the jth

level mean of the minor factor (B) nested within the ith level of factor A and the grand mean (main effect of factor B/A)

5

Yijk = Y•••+ (Yi•• - Y•••) + (Yij• - Yi••)+ (Yijk - Yij•)Yijk = Y•••+ (Yi•• - Y•••) + (Yij• - Yi••)+ (Yijk - Yij•)

is estimated by Y•••;

iis estimated by (Yi•• - Y•••);

(i)j is estimated by (Yij• - Yi••).

The parameter estimates are:

6

TSS = SSA + SSB/A + SSWError

and, in terms of degrees of freedom,

M.m.n-1 = (M-1) + M(m-1) + M.m.(n-1).

OR,

(Yijk - Y•••)n.m.Yi•• - Y•••

I j k

+ nYij• - Yi••

i j

(Yijk - Yij•

i j k

7

Purity Data

Batch 1 2 3 4 1 2 3 4 1 2 3 4

1 -2 -2 1 1 0 -1 0 2 -2 1 3

-1 -3 0 4 -2 4 0 3 4 0 -1 2

0 -4 1 0 -3 2 -2 2 0 2 2 1

Batch totals yij. 0 -9 -1 5 -4 6 -3 5 6 0 2 6

Supplier totals yi.. -5 4 14

Supplier 1 Supplier 2 Supplier 3

8

SSA =4 3[(-5/12-13/36) 2 + (4/12-13/36)

2 + (14/12-13/36)

2]

=15.06

SSB/A =3[(0/3-(-5/12)) 2+((-9/3)-(-5/12))

2+((-1/3)-(-5/12))

2+(5/3-(-5/12))

2

+....… +((-4/3)-4/12)

2+(6/3-4/12)

2+((-3/3)-4/12)

2+(5/3-4/12)

2]

=69.92

SSW = (1-0) 2 + (-1-0)

2 + (0-0)

2 + (-2+3)

2 + (-3+3)

2 +(-4+3)

2 +…

....... +(3-2) 2 + (2-2)

2 +(1-2)

2

= 63.33

TSS =15.06+69.92+63.33 = 148.31

•

9

Source SSQ DF MSQ F (P)

A (suppliers) 15.06 2 7.53 0.97 (0.42)

B/A (batches) 69.92 9 7.77 2.94 (0.02)

Error 63.33 24 2.64

Total 148.31 35

Anova Table

10

General Linear Model: purity versus suppliers, batches

Factor Type Levels Values supplier fixed 3 1 2 3batches(supplier) random 12 1 2 3 4 1 2 3 4 1 2 3 4

Analysis of Variance for purity, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F Psupplier 2 15.056 15.056 7.528 0.97 0.416batches(supplier) 9 69.917 69.917 7.769 2.94 0.017Error 24 63.333 63.333 2.639Total 35 148.306

In Minitab: Stat>>Anova>>General linear model and type model as “supplier batches(supplier)”:

11

Term Coef SE Coef T P

Constant 0.3611 0.2707 1.33 0.195

supplier

1 -0.7778 0.3829 -2.03 0.053

2 -0.0278 0.3829 -0.07 0.943

(supplier)batches

1 1 0.4167 0.8122 0.51 0.613

1 2 -2.5833 0.8122 -3.18 0.004

1 3 0.0833 0.8122 0.10 0.919

2 1 -1.6667 0.8122 -2.05 0.051

2 2 1.6667 0.8122 2.05 0.051

2 3 -1.3333 0.8122 -1.64 0.114

3 1 0.8333 0.8122 1.03 0.315

3 2 -1.1667 0.8122 -1.44 0.164

3 3 -0.5000 0.8122 -0.62 0.544

12

Expected Mean Squares, using Adjusted SS

Source Expected Mean Square for Each Term

1 supplier (3) + 3.0000(2) + Q[1]

2 batches(supplier) (3) + 3.0000(2)