1. Factorial Design

2. Consider the example given in the text book
Consider the example given in the text book. It is a 2level, 2-factor experiment as given below in the table:
Factor A
Low
High
Factor B
-1,-1
+1, -1
-1, +1
+1,+1
Factor A
Low
High
Factor B 20 40 30 52

3. Main effect of Factor A is the difference between the average response at the low level of A and the average response at the high level of A.
Similarly the main effect of B is

4. The general regression equation is Y = β0 + β1X1+ β2X2+ β12X1X2+ε The estimates of β1 and β2 are one half the values of the corresponding main effect β1 = 21/2 = 10.5 β2 = 11/2 = 5.5 The interaction effect AB=1 β12= ½= 0.5 β0 =( )/4 =35.5 Now the regression model Y = β0 + β1X1+ β2X2+ β12X1X2+ε becomes Y = X1+5.5 X X1X2

7. If there is an interaction At the low level of Factor B, the effect of A is A= 50-20= 30 And at high level of Factor B, the effect of A is A =12- 40=-28 So the magnitude of interaction effect is the average difference in these two A effects, AB = (-28-30)/2 = -29 The general regression equation is Y = β0 + β1X1+ β2X2+ β12X1X2+ε

8. The interaction effect AB=-29 β12= -29/2=-14.5
Now the regression model
Y = β0 + β1X1+ β2X2+ β12X1X2+ε
becomes
Y = X1+5.5 X X1X2