Question

Consider a factorial design model with 2 levels of Factor A, 3 levels of Factor B, and 2 observations at each combination of factor levels. Write this model in matrix notation as Y = X8+ €. Compute the matrices X'X and X'Y and use them to derive the least-squares estimators of all appropriate parameters from the normal equations XXB = X'Y Note: Do not write the matrices for a general factorial design model. Consider the particular factorial design described here.

Answer 1

let the model by be y = Bot Bit + B, X₂ + € (main effect model) where x, = -1,1 , levels of factor A & x₂ = -1,0,1 levels of factor B. let the observations at the given treatment combination be denoted as below observations trealment combination Y,,,Y, 2 Y21, 922 Уз. Уз Yui , Yuz Ysl Ysa Yoi, 962 Tť T T - - - ETOO - TO - then y = 14 Y Y Yui Isı Yol Y 219 22 9321 Yuz Y sa Yoa) -1 L-1 -1 0 -1 1 1 -1 1 0 1 1 1 1 1 0 1 1 1 -1 1 1 0 XX = 5 12 o Lo 0 0 1 12 0 o 28 vol. 12 - 10 Lo /12 0 o 78 V 8 = (xxj'x'y , XY = w EY 1E1FU Yu, + Is it y 1 t Yu 2 + 4s 21 Yor Y Zr Y 31 9 29 22 932 | 1-4, + Y3 - Yy1 + Yor-Y 12 + Y32-4 42 + 462

OWE Ti EMNE. Me (1 ч. - ч Т ух. У. + 9. – Ч. + 9. 9, 13, 94, 78