y = q_0 + q_a * x_a + q_b * x_b + q_ab * x_a * x_b
Recall that the q's are called effects.
Substituting the four observations of y, denoted y_1, ...,y_4, in a 2**2 experiment yields:
y_1 = q_0 - q_a - q_b + q_ab = 15
y_2 = q_0 + q_a - q_b - q_ab = 45
y_3 = q_0 - q_a + q_b - q_ab = 25
y_4 = q_0 + q_a + q_b + q_ab = 75
Solving for q's yields:
q_0 = 1/4 ( y_1 + y_2 + y_3 + y_4 )
q_A = 1/4 (-y_1 + y_2 - y_3 + y_4 )
q_B = 1/4 (-y_1 - y_2 + y_3 + y_4 )
q_AB = 1/4 ( y_1 - y_2 - y_3 + y_4 )
An equivalent representation is a sign table:
Experiment | I A B AB y -----------+------------------------------- 1 | 1 -1 -1 1 15 2 | 1 1 -1 -1 45 3 | 1 -1 1 -1 25 4 | 1 1 1 1 75 -----------+------------------------------- Totals: | 160 80 40 20 Totals/4: | 40 20 10 5
Note: The last line contains the same q's we obtained before:
q_0 = 40
q_a = 20
q_b = 10
q_ab = 5