(i) Give a definition for connectedness in a three-way cross-classification as- suming that all interactions are zero.

(ii) Consider the three-way classification with factors A, B, C, where each factor has four levels. Denote a design point by (i, j, k) where i, j, k = 1, 2, 3, 4 indicate the levels of the factors A, B, C, respectively. Suppose we have the following design points:

Show that the design is connected. (iii) Write out and solve the NE for the design in (ii). (iv) Write out the model for the design in (ii) in matrix notation and show that the sum of squares for any factor is independent of the order of the factors in the model.

Transcribed Image Text:

(1, 1, 1). (1.2.2), (1,3,3), (1,4,4) (2,1,4), (2,2,1), (2,3.2), (2,4,3) (3, 1, 3). (3,2,4), (3,3. 1). (3,4,2) (4.1.2), (4.2,3), (4,3,4), (4.4.1)