1 Every achievement originates from the seed of determination.

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

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.

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

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  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)

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 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 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 Purity Data Batch Batch totalsy ij Supplier totalsy i Supplier 1Supplier 2Supplier 3

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)) … +((-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 = TSS = =

12 SourceSSQDFMSQF (P) A (suppliers) (0.42) B/A (batches) (0.02) Error Total Anova Table (A: fixed, B: random)

13 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)”:

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

15 Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 supplier (3) (2) + Q[1] 2 batches(supplier) (3) (2) Note. Restricted and unrestricted models are the same for nested designs