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.

Suppliers Batches Y 111 Y 112 Y 113 Y 121 Y 122 Y 123 Y 131 Y 132 Y 133 Y 211 Y 212 Y 213 Y 221 Y 222 Y 223 Y 231 Y 232 Y 233 Y 311 Y 312 Y 313 Y 321 Y 322 Y 323 Y 331 Y 332 Y 333 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 Y 141 Y 142 Y Y 241 Y 242 Y 243 Y 341 Y 342 Y 343

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= n ij if unequal replicates for combinations. Y ijk =  i  (i)j  (ij)k

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 Y ijk = Y + (Y i - Y ) + (Y ij - Y i )+ (Y ijk - Y ij )  is estimated by Y ;  i  is estimated by (Y i - Y );  (i)j is estimated by (Y ij - Y i ). The parameter estimates are:

6 TSS = SSA + SSB/A + SSW Error and, in terms of degrees of freedom, M. m. n-1 = (M-1) + M(m-1) + M. m. (n-1). OR,  (Y ijk - Y )   n. m.  Y i - Y   I j k + n  Y ij - Y i   i j  (Y ijk - Y ij   i j k

7 Purity Data Batch Batch totalsy ij Supplier totalsy i Supplier 1Supplier 2Supplier 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)) … +((-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 = TSS = =

9 SourceSSQDFMSQF (P) A (suppliers) (0.42) B/A (batches) (0.02) Error Total Anova Table

10 General Linear Model: purity versus suppliers, batches Factor Type Levels Values supplier fixed batches(supplier) random Analysis of Variance for purity, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P supplier batches(supplier) Error Total In Minitab: Stat>>Anova>>General linear model and type model as “supplier batches(supplier)”:

11 Term Coef SE Coef T P Constant supplier (supplier)batches

12 Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 supplier (3) (2) + Q[1] 2 batches(supplier) (3) (2)