Of course all thi discussion has ignored the possibility of interaction. Fractions of the complete factorial
 set constructed in the way outlined also provide the best possible information about interaction [Note that
 design 3 has two observations each for (i) A but not B, (ii) B but not A, (iii) both A and B, and (iv) both A
 and B, and that interference from C and D is zero]. Designs 1 and 2 in contrast provide no information on
 interaction.

 If we are considering not the choice of a subset of treatments but different subsets for different blocks
 within a farm or for different farms, then the same principles apply. All other things being equal, we would
 prefer not to repeat subsets, but to use subsets not involving the same treatments. Thus, if design 3 were
 used in one block (or one farm), the ideal subset for a second block (or farm) would be:


 Design 5


 FP+A
 FP+B
 FP+C
 FP+D
 FP+A+B+C
 FP+A+B+D
 FP+A+C+D
 FP+B+C+D


 The combination of one block of Design 3 and one block of design 5 produces a classical confounded
 design with the four-factor interaction confounded. This would be a very good design if the circumstances
 were to be exactly appropriate. However, just like any other recipe design, it should be used only when the
 conditions of proper blocking, total resources and relevance of questions are suitable. If blocks of size 5 or
 6 or 10 or 12 are clearly more suitable, then we should construct designs for those block sizes, and the
 arguments for numbers of treatments per farm are identical.

 Of course there are questions about how designs such as 3,4 or 5 will be perceived by the farmer (perhaps
 also by the researcher's colleagues). This may lead to some modifications in designs and some explanation
 of designs in terms of capacity to examine changes, both individually and in combination. There is
 considerable further scope for developing designs and explanation of these principles.

 The analysis of designs involving subsets of factorial structures allocated either to different farms or to
 different blocks within a farm can be completed on any computer program that can handle multiple
 regression analysis. This requires the definition of variables to represent the effects of interest together with
 dummy variables representing block and site differences.

 To illustrate the analysis, we consider an experiment at three sites at each of which duplicate plots of a
 (different) subset of treatment combinations are used to comprise the experiment for that site. The three site
 experiments have six, six and nine treatments respectively; the first uses design 4, the last design 5 plus an
 FP treatment, and the second a design like design 4 chosen to complement the other two by including
 almost all the factorial combinations in the total experiment.

 The designs and the yield data (artificial) are given in Table 2. The data format for any multiple regression
 program is shown in Table 3. The results (from GENSTAT) are shown in Table 4.

 This is the basic analysis which provides estimates of A, B, C, and D main effects. Interaction effects can
 be estimated by constructing additional columns in Table 3 from the columns representing the relevant
Factors. Note that the first five columns after the yield column are the dummy factors for blocks and sites.