APA Dictionary of Psychology

a type of study design in which multiple conditions or treatments are administered to the same participants over time. It is a form of Latin square that must fulfill three criteria: Each treatment must occur once with each participant, each treatment must occur the same number of times for each time period or trial, and each treatment must precede and follow every other treatment an equal number of times. For example, consider the following balanced Latin square for an experimental design involving four treatments (A, B, C, and D) and four people:
\begin{matrix} &&\mathrm{Treatment \; Order} \\ &&1\;\;\;\;\;\; 2 \;\;\;\;\;\;\;3\;\;\;\;\;\;\;4\\ \mathrm{Participant \;\; 1} &&A \;\;\;\;\;\;B\;\;\;\;\;\;D\;\;\;\;\;\;C\\ \mathrm{Participant \;\; 2} &&B \;\;\;\;\;\;C\;\;\;\;\;\;A\;\;\;\;\;\;D\\ \mathrm{Participant \;\; 3} &&C \;\;\;\;\;\;D\;\;\;\;\;\;B\;\;\;\;\;\;A\\ \mathrm{Participant \;\; 4} &&D \;\;\;\;\;\;A\;\;\;\;\;\;C\;\;\;\;\;\;B\\ \end{matrix}
As shown, one person receives the treatments in the sequence A, B, D, and then C; a second person receives them in the sequence B, C, A, and D; a third person receives them in the sequence C, D, B, and A; and a fourth person receives them in the sequence D, A, C, and B.