I love Latin (squares)

Below is a conversation between myself and professor Raymond Klein, my former supervisor, about Latin squares and the design of an experiment he is working on:

Ray: There are 3 conditions (ABC). Each participant will draw a different scene in each condition. There are 6 orders of the conditions so we will need to run multiples of 6 participants to have order completely counterbalanced. My intuition says that if we also have 3 scenes (123) and want these completely counterbalanced with the 6 orders then we would need 6×6 = 36 participants. Is that correct?

Me: Yes.
ABC 123
ACB 132
BAC 213
BCA 231
CAB 312
CBA 321

Assign each to each = 36

Preserving the complete counterbalancing of order of conditions (which is essential) can we run any multiple of 6 and use a Latin square to ensure at least that scenes are balanced across order and across conditions?

Tricky. In order to preserve complete counterbalancing of order of conditions and ensure scenes are balanced across order and condition you need to run a minimum of 18 Ss. Look at a counterbalanced group of conditions:

Ss 01: ABC
Ss 02: ACB
Ss 03: BAC
Ss 04: BCA
Ss 05: CAB
Ss 06: CBA

You will notice that each letter appears twice in each position. Looking at the first position, if we assigned Ss 01 to A1 and Ss 02 to A2 we would need a third Ss to assign to A3. If we did this for all our letters we would end up with 9 Ss and an unbalanced set of conditions. If you did 12 Ss, you would have A appearing in the first position four times. Again, not possible to equally assign to three scenes. If you did 18 however, then each letter appears in each position six times which will allow us to equally assign scene to condition while maintaining a counterbalanced set of conditions. The next set of 18 will complete all 36 Ss needed for a fully counterbalanced design so Latin squares are somewhat moot (with the exception of my manuall generation of 12 unique 3-dimensional squares and organizing them to avoid untoward effects of order). The fact that I find this more fun than playing a new Nintendo game will almost certainly be of interest to my neurologist. Below is your fully counterbalanced design. The first 16 Ss are counterbalanced condition with partially counterbalanced scenes.

Ss 01: A1 B2 C3
Ss 02: B3 C1 A2
Ss 03: C2 A3 B1
Ss 04: A2 C1 B3
Ss 05: B1 A3 C2
Ss 06: C3 B2 A1
Ss 07: A3 B1 C2
Ss 08: B2 C3 A1
Ss 09: C1 A2 B3
Ss 10: A1 B3 C2
Ss 11: B3 A2 C1
Ss 12: C2 B1 A3
Ss 13: A2 B3 C1
Ss 14: B1 C2 A3
Ss 15: C3 A1 B2
Ss 16: A3 C2 B1
Ss 17: B3 A1 C2
Ss 18: C1 B3 A2
Ss 19: A1 B3 C2
Ss 20: B2 C1 A3
Ss 21: C3 A2 B1
Ss 22: A2 C3 B1
Ss 23: B3 A1 C2
Ss 24: C1 B2 A3
Ss 25: A3 B2 C1
Ss 26: B1 C3 A2
Ss 27: C2 A1 B3
Ss 28: A1 C2 B3
Ss 29: B2 A3 C1
Ss 30: C3 B1 A2
Ss 31: A2 B1 C3
Ss 32: B3 C2 A1
Ss 33: C1 A3 B2
Ss 34: A3 C1 B2
Ss 35: B1 A2 C3
Ss 36: C3 B2 A1

I think you have a unique talent and predilection for squares.

Read more about counterbalanced measures design. Any questions?

One Response to I love Latin (squares)
  1. [...] Repeated measures within subject. Expose participants (Ss) to a series of reading comprehension tests set in downstyle. Record their time to read. Then have them answer comprehension questions. Record their time to answer and comprehension accuracy. Repast this with passages set in upstyle. Be sure to counterbalance the order of passages and styles between and within Ss through use of a Latin square to remove any untoward effects of order (for more reading on my fondness of Latin squares see my post “I love Latin (squares) ”). [...]

Leave a Reply

You must be logged in to post a comment. Click here to log in.

Trackback URL http://readthetype.com/latinsquares/trackback/