How can we determine the number of replications of an experiment by factorial experimental design? Can anyone explain 4 x 3 factorial design in a simple manner?
Is it 4 factors with 3 treatments?
What exactly to you mean by replication in this case? The number of combinations that has to be tested? In other words, the number of trials that has to be performed to obtain sufficient coverage?
For 2k factorial design (that is, k factors with two treatments each), then you need to test 16 combinations. I guess that n treatments each at least is nk, given that you can only pick one treatment per factor leading to 81 combinations in total (an astounding number, if the the experiment concerns real subjects and not a simulation). There are techniques in combinatorial testing addressing how to reduce the number of test cases. Further, in experimentation there are fractional factorial design.
https://en.wikipedia.org/wiki/Fractional_factorial_design
http://www.combinatorialtesting.com/
Thank you. By replication, I mean the number of trials to be performed as well as the combinations to be estimated. It is 4 x 3 treatment
For full coverage, then you have 34 combinations. Depending on what you want to do, you may have to replicate each combinations as many time as required by the analysis you assume. If, for example, you want to achieve a level where the average can be used so that the central limit theorem is valid (you can disregard the distribution), then the recommendation is >30 trials for each combination that you want to study. To cover everything, then you need 81*30=2430 people, which is clearly too many and you probably do not need to cover everything anyway. So, you need to reduce it and I recommend that you read a book on experimental design concerning this. How to reduce the number is highly dependent on the factors and the treatments themselves.
My favorite is Wohlin, C 2012, Experimentation in software engineering, Springer, New York.
However, there are more general works. See, for example, the links in the previous posts.
This means that you have 2 factors, one at 4 levels and the other has 3 levels.
You can determine the number of experiments you would do by multiplying 3X4X n, where n is the number of replications. Please note that replications should be at least 2. The more you do replications, the more precise results you get.
You can refer to the link below:
http://www.socialresearchmethods.net/kb/expfact.php
Best of luck!
I recommend that you check chapters 3 and 5 in "Design and Analysis of Experiments"
by Douglas C. Montgomery (Author)
http://www.amazon.com/Design-Analysis-Experiments-Douglas-Montgomery/dp/1118146921
What types of variables are you dealing with?
If both are categorical, you will need 4x3xN number of samples. If the factor with 3 levels is continuous, you can create an Optimal Response Surface and use fewer runs (9).
The proper numer of replications must be calculated. For that you should before review the concept of Statistical Power.
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