What would be a better alternative to two balanced latin squares (human subject experiment)?

Instead of running a certain balanced latin square twice (if you have m experimental conditions but 2 m subjects), what would be a better approach (to cover twice as many of the possible orderings and get the maximum protection against ordering effects etc.)?

...or in general terms, what is a better alternative to running a certain balanced latin square n times in a psychological research experiment with n m subjects (where n is an whole number)? Is there anything better than seeding the first columns (or rows) of other squares randomly?

A classical Latin Square confounds two factor interactions with main effects. If the second Latin Square differs from the first in how it combines the levels, you may be able to break some of the confounding. If the two are identical but run in different orders, you'll have an estimate of pure error. Which is better depends on what assumptions you are willing to make and what knowledge is more important to you.

Run order should be randomized if at all possible. Even if you use the same square twice, you should randomize each one separately. If you get the same run order both times something almost certainly went wrong with your randomization.

• +1 in any case, but there are a few things I don't follow about your answer: 1. I am specifically considering balanced Latin squares, whereas you say "classical Latin Square". To be clear, what I am talking about, for example, include {ABDC,BCAD,CDAB,DACB} but not {ABCD,DABC,CDAB,BCDA}. Which are you discussing?
– A.M.
Commented Jul 1, 2013 at 17:20
• 2. Your "If the two are identical but run in different orders" also makes me think we are talking about different things. Running {ABDC,BCAD,CDBA,DACB} would be no different than running {ABDC,BCAD,DACB,CDBA} since it is a different subject for each run of four conditions.
– A.M.
Commented Jul 1, 2013 at 17:22
• Note: my first comment should read "[...] include {ABDC,BCAD,CDBA,DACB} [...]". (It has been just over 5 minutes since I made that typo!) I corrected it in my second comment, though.
– A.M.
Commented Jul 1, 2013 at 17:28
• 3. I do not understand what you mean by "Run order should be randomized". Again, using the runs of groups of 4 conditions in the example in my comment "2." does not make any difference, though I would consider changing that order to be using a different square. ...and if you are talking about changing the runs of 4 conditions (not just the ordering of the runs), then you are definitely talking about using a different square.
– A.M.
Commented Jul 1, 2013 at 17:31
• 4. Following from that, your statement "Even if you use the same square twice, you should randomize each one separately." makes no sense to me. If you change any aspect of the square (the ordering of conditions within runs (for each subject) or even just the ordering of those runs, then you are no longer using the same square.
– A.M.
Commented Jul 1, 2013 at 17:33