1. 1 Every achievement originates from the seed of determination.

2. 2 Nested (Hierarchical) Designs By Kelly Fan, Cal. State Univ. East Bay

3. Crossed vs. Nested 3 Factor A is called crossed with factor B if the b levels of factor B are IDENTICAL for all levels of factor A 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, and factor B is nested under factor A.

4. 4 123 1 2 3 1 23 123 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 143 4 Y 241 Y 242 Y 243 Y 341 Y 342 Y 343

5. 5 MODEL i = 1,..., a (the #of levels of the major factor) j = 1,..., b(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  j(i)  ijk

6. 6  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)  j(i)  the difference between the jth level mean of the minor factor (B) nested and the grand mean within the ith level of factor A (main effect of factor B/A)

7. Assumption:  ijk follows N(0,  2 ) for all i, j, k, and they are independent. Additional restrictions/assumptions: Fixed effect Random effect Mixed effect 7

8. 8 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 );  j(i) is estimated by (Y ij - Y i ). The parameter estimates are:

9. 9 TSS = SSA + SSB/A + SSE and, in terms of degrees of freedom, a. b. n-1 = (a-1) + a(b-1) + a. b. (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

10. 10 Purity Data Batch123 412 12 1-2-2 110-1 02-21 3 -30 4-240 340-1 2 0-41 0-32-2 2022 1 Batch totalsy ij. 0-9-1 5-46-3 5602 6 Supplier totalsy i.. -5414 Supplier 1Supplier 2Supplier 3

11. 11 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 SSE = (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

12. 12 SourceSSQDFMSQF (P) A (suppliers)15.0627.530.97 (0.42) B/A (batches)69.9297.772.94 (0.02) Error63.33242.64 Total148.3135 Anova Table (A: fixed, B: random)

13. 13 General Linear Model: purity versus suppliers, batches Factor Type Levels Values supplier fixed 3 1 2 3 batches(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 P supplier 2 15.056 15.056 7.528 0.97 0.416 batches(supplier) 9 69.917 69.917 7.769 2.94 0.017 Error 24 63.333 63.333 2.639 Total 35 148.306 In Minitab: Stat>>Anova>>General linear model and type model as “supplier batches(supplier)”:

14. 14 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

15. 15 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) Note. Restricted and unrestricted models are the same for nested designs