I have to prepare design for my experiment where i have 12 treatments (Food Supplement) which i need to feed 100 subjects and then record observation. So i was thinking of a design where i can block subjects and days (2 day study). Days, because only 6 supplements will be tried by 1 subject at a time. What kind of design i should use?
if i use Latin square design, i can block subjects variability like 12 (treatments)*12(subjects) but not days i think
This sounds a lot like a balanced incomplete block design to me. Have a look at this Discovery presentation and see if it makes sense.
This is a really interesting problem. You have 12 treatments, 6 of which can be administered simultaneously, and 100 participants, correct? The study is over 2 days, so you would have 200 total runs for this.
The problem here is that the number of treatments must be exactly equal to 6. This is like a mixture design problem because it creates linear dependency between the terms because the sum of those columns will be completely confounded with the intercept. I tried doing this by enumerating all combinations of 12 choose 6 (924 possible combos), and including those as covariates in an optimal design to choose some optimal subset of 200 runs of the possible 924. I also included a subject block. Even after removing the intercept, it stil says there's linear dependency and bombs out.
What would probably make your life a lot easier is to relax the constraint of having exactly 6 supplements per subject & day. For example, you could make the constraint "no more than 6" and then use an ordinary linear constraint with an optimal design. I'm generating one now as an example, but it is taking a very long time to return a design. I will post it when it finishes so you can see what I mean. You will still have the ability to estimate all main effects and 2-factor interactions.
Here's what I came up with using my suggested approach. I had to use fixed blocks instead of random because the random blocks were taking forever to finish a single random start (you can still analyze the block as a random effect, which is what I would suggest). I used 1000 random starts, which took 10-15 minutes to generate. The number of supplements in a given run range from 1-6, but over half of the runs have 6 treatments. Here's what that distribution looks like:
The design diagnostics look pretty good. There is very low correlation and the power for all effects is effectively 100% for a 1-1 signal to noise ratio.
There were quite a few artifacts from the random starting design left where some values were not exactly 0 or 1 but were very close. These can just be rounded to the closest value. I then made the 12 supplement factors categorical rather than continuous. I am attaching the resulting table.
One thing I did not address is the day block. You can add that in the dialog if desired, but I honestly think it makes very little difference if you just create it your self alternating Day 1, Day 2 all the way down the table.
To see the DOE dialog, run the DOE Dialog script and hit the cancel button immediately (or it will take 10 or so minutes to go through all the random starts). Then scroll down and press "Back".