they differ significantly, we may infer the existence of an interaction. Otherwise, 
we may feel safe to conclude that the effects of P and R are simply additive: 
that the action of either is independent of the level of the other, within the 
ranges of level covered by the experiment. The other interactions can be similar- 
ly derived. The technique of the analysis of variance carries out this procedure 
most effectively, as demonstrated in Section 3.1. 
If we are satisfied that the effect of changing P is independent of both the 
level of R and the level of Q, we can then proceed to average the P, and P. 
planes: the difference between the averages being solely due to the change of the 
level of P. The second advantage of the factorial design follows at once from 
this. For we now assess the P effect from averages of two sets of four observations 
each, whereas in the classical design of Figure 1 we had only two observations 
in each set to estimate the effect. 
Should we find that one of the interactions, say RP, is significant while the 
other is not, we would have to study the effect of P at each of the two levels of 
the interacting factors, i. e. with R at R, and R, separately. In this case the effect 
of changing P from P, to P, would be obtained for each level of R as the diffe- 
rence between the averages of two pairs of observations. At R = R, these will be 
(P, Q., R.), (P, Q. R,) and (P; Q, Ry, (P; QR). The: paits of observations 
which give the effect of P when R = R, are obtained by writing R, for R, in the 
preceding set. The number of observations used in the estimation of the P effect 
is now two, the same as in the classical plan. But with the factorial plan we are 
able to assess the P effect at the two levels, whereas the classical design would 
not give the P effect at R = R,, because we would have no data at this level. 
If both interactions of P with the other two factors happened to be signifi- 
cant, we would have to compare individual observations, since we can no longer 
average on either R or Q. We can nevertheless still test the significance of the 
effects against the residual, as we did in discussing the experiment on the long 
term differential shrinkage of film in Section 3.1. But as was pointed out there, 
the residual includes in it the interaction of the three factors, and may conse- 
quently be inflated. The estimation of the P effect, at those combinations of the 
other two factors which are common between the two designs, is less accurate in 
the factorial experiment; but it should be noted that the classical design gives 
only half the possible combinations. To obtain the remaining effects in the clas- 
sical style we would need to make duplicate observations at the remaining four 
corners of the cube, which would become a factorial scheme with repeated ob- 
servations; and the difficulties arising from the use of the residual in place of the 
error would be cleared away at once. Instead of repeating the same observations 
as was just suggested, we can add a new factor which is not likely to interact 
with all the other factors, so that the new residual will not be inflated. By adding 
the new factor we would of course gain much new information. Thus, while 
repetition of observations has obvious advantages, and should therefore be strong- 
ly recommended whenever practicable, many of these advantages can often be 
obtained by increasing the number of factors experimented upon, which in the 
meanwhile serves well the purpose of learning about the additional factors. 
In fact, the experiments planned in accordance with factorial schemes are 
most efficient, in the sense that they give the maximum amount of information 
that can possibly be obtained from a certain number of observations. 
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