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 Batches 1 2 3 4 4 2 1 3 2 3 4 1 Obs’ns{ Y131 Y132 Y133 Y111 Y112 Y113 Y121 Y122 Y123 Y221 Y222 Y223 Y311 Y312 Y313 Y321 Y322 Y323 Y331 Y332 Y333 Y141 Y142 Y143 Y231 Y232 Y233 Y341 Y342 Y343 Y241 Y242 Y243 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. 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). (Yijk - Y•••)n.m.Yi•• - Y••• Ijk + nYij• - Yi••  ij (Yijk - Yij• i j k OR,

7. Purity Data 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. Anova Table

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: 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)