(2) using as few different levels as are required to estimate the response curve with one extra for
 assessing the adequacy of the response curve (or one for luckl).
These requirements are widely applicable and should be ignored only to use particular levels of
importance.

For the actual choice of levels it will often be at least approximately correct to use equally spaced levels
(possibly on the log scale). If the pattern of the response curve is expected to be strongly skewed then the
levels should be closer together where the response is changing rapidly and further apart in areas of lesser
change.

53 Incomplete Factorial Structure

When the set of treatments is to be a subset of a factorial structure (that is several factors are varied in the
set of treatments, but less than all the possible combinations are to included) then the consideration of the
precision of comparisons from different subsets is very important. Precision increases according to the total
number of combinations providing information about each comparison. This is the power of hidden
replication. For a very simple example consider two alternative subsets each of four combinations from
three two-level factors.


 Factor
Subset(l) A B C

Treatment 1 0 0 0
Treatment 2 1 0 0
Treatment 3 1 1 0
Treatment 4 1 1 1


 Factor
Subset(2) A B C

Treatment 1 0 0 0
Treatment 2 1 1 0
Treatment 3 1 0 1
Treatment 4 0 1 1


The precision of the estimate of the difference between levels 0 and 1 of factor A (or B or C) is more than
twice as good for subset (2). That is the variance of the estimate of the difference using (2) is less than 50%
of that using (1). The advantage derives from the use in (2) of all four combinations for estimating the
difference as compared with using only two combinations in (1) plus the non-independence of the three
estimates of differences in (1).

In general I believe the choice of treatments for an incomplete factorial structure has to reflect a balance
between the objective of comparing, and being seen to compare, particular treatment combinations and that
of estimating effects precisely. In a situation where previous experimentation has established that factors A,
B and C act almost completely independently and where each main effect is believed to be substantial then
the benefits for presentation of subset (1), may outweigh the consideration of precision. For example a
sequence of treatments following the expected adoption sequence of farmers may be more understandable
for farmers.

The statistical theory on simple subsets of factorial structures which are efficient for the estimation of main
effects and two-factor interactions is well established. Nice fractions containing four, eight or sixteen
combinations from structures with three, four, five or six two-level factors are easily found. Some
examples are shown:

3 factors: 4 combinations
 (000,011,101,110) or the complement (100,010,001,111)
4 factors: 8 combinations
 (0000,0011,0101,0110,1001,1010,1100,1111) or complement
4 factors: 4 combinations
 (0000,0110,1001,1111) or (0010,0101,1001,1110