The means for the two levels of factor C are -50 and +50, the SS for the Main Effect of C is 80000 and the
reduced effects are


+13 +83 -89 +129
-86 +12 -198 +136
-82 +22 -156 +81
+35 +2 -48 +145


The means for the two levels of factor D are -76 and +76, the SS for the Main Effect of D is 184862 and
the reduced effects are


+89 +7 -13 +53
 -10 -64 -122 +60
 -6 -54 -80 +5
+111 -74 +28 +69


Notice that these reduced effects are now generally much less than the treatment effects with which we
started. We can observe how rapidly they diminish at each stage by summing the squares of the reduced
effects. After subtracting the effects of the four main effects the remaining sum of squares is:

(892 +72 +132 + +282 +692)x 2 = 132694 on 10 df.

compared with the total treatment SS of 543798 on 14 dfand the error SS of 135209 on 14 df.

At this stage we might decide that the reduced SS is now sufficiently close to what would be expected,
based on the error mean square, (the F ratio is

 (132694/10) / (135209/14) = 1.37)

that we should not examine the interaction effects. However we shall continue a little further, if only for
illustrative purposes.

To, calculate, and adjust for, a two-factor interaction effect, we must first calculate the means for the four
combinations of levels of the two factors. Consider the AxB interaction, for which the four combinations
are the four rows of the table of treatment effects. The four means are +34, -34, -34 and +34 (we should not
be surprised that the numbers are all the same since there is only one df for this interaction effect). The
reduced effects are:


+55 -27 -47 +19
+24 -30 -88 +94
+28 -20 -46 +39
+77 -108 -6 +35