Every achievement originates from the seed of determination

Every achievement originates from the seed of determination. 1

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

Crossed vs. Nested • 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. 3

1 Suppliers Batches Obs’ns { 1 2 3 2 4 Y 111 Y 121 Y 131 Y 141 Y 112 Y 122 Y 132 Y 142 Y 113 Y 123 Y 133 Y 143 1 2 3 3 4 Y 211 Y 221 Y 231 Y 241 Y 212 Y 222 Y 232 Y 242 Y 213 Y 223 Y 233 Y 243 1 2 3 4 Y 311 Y 321 Y 331 Y 341 Y 312 Y 322 Y 332 Y 342 Y 313 Y 323 Y 333 Y 343 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

MODEL Yijk = ti bj(i) ijk i = 1, . . . , a j = 1, . . . , b k= 1, . . . , n (the #of levels of the major factor) (the # of levels of the minor factor for each level of the major factor) (the # of replicates per (i, j) combination) Note: n= nij if unequal replicates for combinations. 5

• : the grand mean • ti : the difference between the ith level mean of the major factor (A) and the grand mean (main effect of factor A) • bj(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) 6

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

Yijk = Y • • • + (Yi • • - Y • • • ) + (Yij • - Yi • • )+ (Yijk - Yij • ) The parameter estimates are: • is estimated by Y • • • ; • ti is estimated by (Yi • • - Y • • • ); • bj(i) is estimated by (Yij • - Yi • • ). 8

(Yijk - Y • • • ) n. m. Yi • • - Y • • • i j k + n Yij • - Yi • • i j (Yijk - Yij • i j k OR, 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). 9

Purity Data Supplier 1 Batch 1 2 3 -2 -2 1 -1 -3 0 0 -4 1 0 -9 1 Batch totals yij. Supplier totals yi. . Supplier 2 -5 4 1 Supplier 3 2 3 4 1 0 -1 4 -2 4 0 -3 -1 5 -4 2 3 4 0 2 -2 1 3 0 3 4 0 -1 2 2 -2 2 0 2 2 1 6 -3 5 6 0 2 6 4 1 14 10

SSA 2 2 2 =4 • 3[(-5/12 -13/36) + (4/12 -13/36) + (14/12 -13/36) ] =15. 06 SSB/A 2 2 2 =3[(0/3 -(-5/12)) +((-9/3)-(-5/12)) +((-1/3)-(-5/12)) +(5/3 -(-5/12)) 2 2 2 +. . … +((-4/3)-4/12) +(6/3 -4/12) +((-3/3)-4/12) +(5/3 -4/12) ] =69. 92 SSE 2 2 2 = (1 -0) + (-1 -0) + (0 -0) + (-2+3) + (-3+3) +(-4+3) +… 2 2 2. . . . +(3 -2) + (2 -2) +(1 -2) = 63. 33 TSS =15. 06+69. 92+63. 33 = 148. 31 11

Anova Table (A: fixed, B: random) 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 12

In Minitab: Stat>>Anova>>General linear model and type model as “supplier batches(supplier)”: General Linear Model: purity versus suppliers, batches Factor supplier batches(supplier) Type fixed random Levels 3 12 Values 1 2 3 4 Analysis of Variance for purity, using Adjusted SS for Tests Source supplier batches(supplier) Error Total DF 2 9 24 35 Seq SS 15. 056 69. 917 63. 333 148. 306 Adj SS 15. 056 69. 917 63. 333 Adj MS 7. 528 7. 769 2. 639 F 0. 97 2. 94 P 0. 416 0. 017 13

Term Coef SE Coef T P 0. 3611 0. 2707 1. 33 0. 195 1 -0. 7778 0. 3829 -2. 03 0. 053 2 -0. 0278 0. 3829 -0. 07 0. 943 Constant supplier (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 14

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 15