Balanced Latin Square

Question

For making a good Between-Object user study, this is suggested to use a Latin Square to give all the different conditions, different order of representation of those conditions. However, when the number of conditions grow, the number of representation grows exponentially so a Balanced Latin Square is replaced. My question here is that, for the case of balanced one, does it have to cover all the conditions equally? For example, if we have 4 conditions, is it wrong we just test it with 6 participants? or the number of participants has to be a coefficient of 4?

since the formula to make the Balanced Latin Square is 1, 2, n, 3, n-1, 4, n-2 if we put n=4, it will become:

1-2-4-3
2-3-1-4
3-4-2-1
4-1-3-2

here, it is a Balanced Latin Square for 4 different representation conducted with 4 participant. my question is that if we add two more participant it will become like:

1-2-4-3
2-3-1-4
3-4-2-1
4-1-3-2

1-2-4-3
2-3-1-4

is this a correct Balanced Latin square or it should be for a coefficient of 4 -number of participant? like:

    1-2-4-3
    2-3-1-4
    3-4-2-1
    4-1-3-2

    1-2-4-3
    2-3-1-4
    3-4-2-1
    4-1-3-2

Answer 1

What you have aren't balanced latin squares mainly because they aren't even squares.:) You need to have the same number of rows as columns. So a balanced latin square of order 6 would be

$$\begin{matrix} 1&2&3&6&4&5\\ 2&3&4&1&5&6\\ 3&4&5&2&6&1\\ 4&5&6&3&1&2\\ 5&6&1&4&2&3\\ 6&1&2&5&3&4\\ \end{matrix} $$

In other words you need to have the same number of participants as condition to be able use the balanced latin square approach. Or rather you need to have the same number of participant groups. So really the number of participants must be a multiple of the number of conditions/treatments what have you.

Edit A nice brief exposition can be found here. It also includes an explanation of how to deal with the situation that the number of conditions you have is odd.

Answer 2

For example, if we have 4 conditions, is it wrong we just test it with 6 participants?

There's nothing to stop an experimenter from performing an experiment according to the scheme:

$$ \begin{array}{|cccc|} \hline \color{blue}{1} & \color{blue}{2} & 4 & 3 \\ 2 & 3 & 1 & 4 \\ 3 & \color{green}{4} & \color{green}{2} & 1 \\ 4 & 1 & 3 & 2 \\ \hline \color{blue}{1} & \color{blue}{2} & 4 & 3 \\ 2 & 3 & 1 & 4 \\ \hline \end{array} $$ The only problem is that it loses some of the desirable properties of using a balanced Latin square. In comparison:

• Each number occurs in each column an equal number of times in the Latin square. In the modified matrix, e.g., $1$ and $2$ occur in the first column twice, whereas $3$ and $4$ occur once.

• Each pair $(i,j)$ occurs as consecutive elements in a row exactly once in the balanced Latin square. In the modified matrix, e.g., $1$ occurs before $2$ twice, whereas $4$ occurs before $2$ once.

In a sense, this introduces two imbalances into the experiment.