For most real life situations I have come accross number of active effects is quite low. I need a situation where a large number (i.e. more than half) of the contrasts are active for illustrating a method of analysisng unreplicated factorial experiments.
You might consider using fractional factorial designs provided you are prepared to acept that no information will be available on a proportion of the interactions. Such designs are markedly more efficient than full factorial designs, but depend on you being able to specify which set of effects you can accept not having information on.
Dear Prof. Christopher Woodruff , Thank you for the answer. But I need a real actually observed data with large number of significant effects forillustrating some theory developed.
Thanks once again.
Dear prof. Kulkarni
As you know there is a principle in the theory of experimental design which is sparsity. If you want the data which have the significant effect more than 2^n-2 you should consider small designs like 2^3. This kind of designs can be find in standard book for experimental design like montgomery.
I think that in your case, at least initially, you could begin with a fractional factorial design of resolution IV. So, you'll be sure of main effects and you'll know what groups of 2 factors interactions are significant. For example, with a 2^(8-3) design, x1*x3 and x2*x6 are aliased. Then, if this group is significant, and if you know x2*x6 cannot have a effect, you can deduced that it is x1*x3. If no assuptions of this type are possible, you'll need to do additional experiments. These new experiments could be selected by D-optimality approach.
Dear Kulkarni,
Please see it link below.
https://stats.stackexchange.com/questions/349501/why-these-the-results-in-factorial-2k-experiment-analysis-with-r-are-different-o
All effects are significant.
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