Nested (Hierarchical) Designs

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
Suppliers
1
2
3
Batches
4
4
1
2
3
2
3
4
1
Obsns
Y131 Y132 Y133
Y111 Y112 Y113
Y121 Y122 Y123
Y221 Y222 Y223
Y231 Y232 Y233
Y311 Y312 Y313
Y321 Y322 Y323
Y331 Y332 Y333
Y141 Y142 Y143
Y241 Y242 Y243
Y341 Y342 Y343
Y211 Y212 Y213

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.
3
MODEL
Yijk ?????i????(i)j????(ij)k
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.
4

• ?? the grand mean
• ?i? the 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)
The parameter estimates are

• ??is estimated by Y
• ?i?is estimated by (Yi - Y)
• ?(i)j is estimated by (Yij - Yi).

6
??????(Yijk - Y)????n.m.???Yi - Y?? I
j k
n?????Yij - Yi ?? i j
???????(Yijk - Yij?? i j k
OR,
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
(-23) 2 (-33) 2 (-43) 2 ....... (3-2)
2 (2-2) 2 (1-2) 2 63.33
TSS 15.0669.9263.33 148.31
9
Anova Table
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
10
General Linear Model purity versus suppliers,
batchesFactor Type Levels
Values supplier fixed 3 1 2
3batches(supplier) random 12 1 2 3 4 1 2 3
4 1 2 3 4Analysis of Variance for purity, using
Adjusted SS for TestsSource 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 StatgtgtAnovagtgtGeneral 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)batche
s 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) Q1 2
batches(supplier) (3) 3.0000(2)