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Let's say I have an experiment where there are 3 conditions, A, B and C. If using a Latin square design, participants would be subjected to all 3 conditions in different orders. However, if the cost of running the experiment for each condition is expensive and time consuming, reducing this would be ideal. Is there an alternative design, for instance, where one could have 3 participants perform the following conditions: AB, BC, AC, would statistical analysis on this still be valid?

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This would be a randomized incomplete block design, with participants as blocks. You would need more than 3 participants in the event that treatment order matters, e.g. A-B vs B-A. If participants are ganged together for a random treatment, e.g. a training course, you might want to consider a split-plot design.

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