Thus for example (R, — R;) with P at P, may differ from (R, — R.) with P at 
P., but the experiment is quite unable to detect such an effect. 
  
  
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Let us now suppose that instead of making the observations according to the 
plan of Figure 1, we carry out the experiment at all the combinations of the two 
levels of P, Q, R, thus: 
p (QR; Pr, Qi, R.; Pr, Q:, Ry; Pr; Q., R.; P... Q; R,: 
P. Q,, Rz PQ. R1: PS Q R,, 
making a total of eight observations, the same total as in the classical plan, but 
arranged in accordance with the factorial scheme. The eight observations now 
occupy the eight corners of the cube in Figure 2. To estimate the effect of changing 
P from P, to P, we can in this case compare the average of the four readings 
corresponding to P = P,, with the average of the four readings made at P — P, 
i.e. the average of the P, plane of the cube with the average of the P. plane. It can 
rightly be argued, however, that since the other factors Q and R have been chang- 
ed in the meanwhile, the difference is not entirely due to changing the level of P. 
This indeed is true. But the variations of Q and R are symmetrical in the two 
planes. The disturbing influences of these variations, therefore, cancel out unless 
either or both factors interact with P. Similar considerations apply to the other 
two factors. We have therefore to examine the possible interactions before we 
proceed to take the averages. 
Let us not forget here that the classical design did not throw any light on 
these interactions. An important advantage of the factorial design is that it makes 
it possible to estimate all of them. To obtain the interaction between, say P and 
R, we average over Q the front plane with the back plane. Thus, averaging 
(P,Q R) and (P,Q.R) we obtain '/ ((P,Q,R,) + (P, Q. Ra)h which we 
denote by (P, R, : Q). The remaining averages will be (P, R. : Qy; (P. R,.: Q) 
and (P, R, : Q). The averages may be arranged in a Table as follows: 
(P, R4) (P. Ri) 
Q 
(P. R.) (P. R.) 
The top line gives (P; — P.) at R = R,, and the bottom line gives (P, — P) 
when R = R.. Since each number in the table is the average of two results, we 
can estimate the experimental error. We can therefore determine whether or not 
(P, — Pj) at R = R, is significantly different from (P; — P,) at R = R, If 
20