Constructing a Latin Square

In the last three or four years, a puzzle craze called Su Doku has rolled across college campuses, offices, and homes, intriguing tens of thousands of people. College students who should be studying are often found engrossed instead in Su Doku puzzles, which require the arrangement of numbers in rows and columns. Interestingly, Su Doku is very reminiscent of the topic of this website section, Latin Squares. If you are already familiar with Su Doku, then you will find Latin Square designs quite congenial. If you are not into Su Doku, then you will appreciate being introduced to both.

Like Su Doku, Latin squares are special arrangements of letters or numbers. They are useful in within-subjects designs as systems of counterbalancing to control for confounding due to sequence effects. In this section, we will discuss two methods of developing Latin Squares. First, however, we will present a brief review of within-subjects designs and counterbalancing.

Because each participant undergoes all conditions in within-subjects designs (see Chapter 11), there are no group differences due to sampling error. Thus, groups are equivalent at the start of the study. Also, these designs are particularly sensitive to the effects of independent variables. These advantages lead many researchers to prefer within-subjects over between-subjects designs.

The disadvantage of within-subjects designs is the potential for confounding due to sequence effects (see Chapter 8 ). These arise from participants’ growing experience with the procedures (practice effects) and the influence of prior conditions on later responses (carryover effects).

The best control for sequence effects is to vary the order of presentation of the conditions. In a random order of presentation, the order of presentation of conditions is randomly arranged and participants are randomly assigned to sequences. In a counterbalanced order of presentation the order is systematically arranged and participants are randomly assigned to the conditions.

Counterbalancing can be complete or partial. In complete counterbalancing,

all possible orders of conditions occur,
each participant is exposed to all conditions of the experiment,
each condition is presented an equal number of times,
each condition is presented an equal number of times in each position,
each condition precedes and follows each other condition an equal number of times.

To determine how many different orders of presentation (sequences of conditions) are needed for complete counterbalancing, we calculate X! (X factorial), in which X is the number of conditions. A factorial is calculated by multiplying the number of conditions by all integers smaller than the number. In a study with two conditions, there are only two orders of presentation (X! = 2 x 1 = 2). With three conditions, there are six orders of presentation (3 x 2 x 1 = 6). With four conditions, there are 24 orders (4 X 3 X 2 X 1 = 24), and if the experiment has six conditions, there are 720 orders of presentation (6 X 5 X 4 X 3 X 2 X 1 = 720). Obviously, for more than three or four conditions, complete counterbalancing is not feasible. Partial counterbalancing offers the best solution.

In partial counterbalancing,

a subset of all possible orders of conditions is used.
each participant is exposed to all conditions of the experiment,
each condition is presented an equal number of times,
each condition is presented an equal number of times in each position,

To create a partially counterbalanced order we can randomly select some of the possible orders of presentation, and randomly assign participants to these orders. Alternatively, we could use a Latin square design, a more formalized partial counterbalancing procedure.

Creating a Latin Square

Latin squares are named after an ancient Roman puzzle that required arranging letters or numbers so that each occurs only once in each row and once in each column. They are efficient in research because several variables can be examined in a single study, but without the inordinate time and effort needed for testing all possible orders of conditions (Mason, Gunst, & Hess, 1989).

Use of Latin Square assumes that, for each number of conditions used in studies, there is a theoretical "population" of all possible Latin Square arrangements. The task for the researcher, then, is to select one or more of those possible squares with as little bias as possible – hopefully in such manner that every theoretical square has the same probability of being selected as any other one. Thus there are two related objectives in calculating a Latin Square for a study: (1) to select a subset from all possible orders of the conditions of the study and (2) to select the subset randomly, as an unbiased choice from all possible orders.

Let us suppose that we have a within-subjects study with four conditions, A, B, C, and D. Each participant is to undergo each of the four conditions. Complete counterbalancing would require 24 experimental conditions (4!). By creating a Latin Square we can select an unbiased subset of the 24 conditions, and run our study with good control over sequence effects. The square is laid out in rows and columns, the number of which equals the number of levels or factors. A four-factor study will have four columns and four rows.

A relatively easy way of creating a Latin Square for this four-factor study would proceed as follows:

Begin by setting out the first row and first column of a standard square, that is, one that has its first row and first column arranged in "the standard order," (1,2,3,4, or A, B, C, D).
A B C D
B
C
D
To generate the second row, place the first letter (A) of the previous row in the last position of the second row. Then shift all other letters forward one position:
A B C D
B C D A
C
D
To generate the last two rows, continue that process.
A B C D
B C D A
C D A B
D A B C

You now have a Standard Latin Square that meets the criteria that each condition must appear exactly once in each row and column--that is, each condition appears once in each position. This Latin Square can now be used in your study as a fairly unbiased arrangement of the conditions of the study. One or more of your participants can be randomly assigned to the rows, and that will determine the sequence of conditions that each participant undergoes.

However, this square is not completely a random choice from among all of the possible 4 X 4 squares, because we had decided to set the first row and column in the standard sequence (i.e., we constrained our choice). That is, we decided not to choose randomly from all possible squares, but to limit our selection to a choice from the subset of all possible standard squares. Thus we might want to introduce a greater degree of randomization into our choice.

Think of the numbers 1, 2, 3, and 4 as corresponding to the letters A, B, C, and D, respectively. Using a table of random numbers (see Appendix F), select a sequence for the integers 1, 2, 3, and 4. I did so and turned up the sequence 4, 2, 1, 3. (Your sequence will be randomly different from this one.) That random sequence corresponds to the follow order of letters.
D B A C
Construct the remaining three rows in the same manner as in steps 2 and 3 above: That is, you take the first letter in the first line and move it to the end of the list to create the order for the second line, and do the same with the order in the second line to create the third line, and so on. This process creates the following randomly selected Latin Square
D B A C
B A C D
A C D B
C D B A

You then randomly assign an equal number of participants to each row to determine the sequence in which they will undergo the experimental conditions.

You could generate more Latin Squares in the same manner, providing even more alternative sequences. Each different square would constitute a "block" and the data would then be analyzed not only by condition (A, B, C, D) but also by each block (lines 1, 2, 3, 4). The details of this analysis are complicated, but the references below will give you the specifics.

There are several ways of constructing Latin Squares, all with the same goals: to select one or more squares randomly from the population of Latin Squares that exists for any given number of conditions. For studies that have five or more conditions the procedure becomes more complex, including random permutations (i.e., transformations) of columns as well as rows, and additional steps in producing the final squares to be used in a study (Myer and Wells, 2003). It is beyond the scope of this section to discuss methods beyond this introductory treatment of Latin Squares. For more complete discussions, interested students should see the books listed in the References List.

Summary

The Latin Square design is a partially counterbalanced design that helps to control for sequencing effects in within-subjects designs. The squares are readily generated and are composed of rows and columns that equal the number of factors used in the study. The Advantages of using Latin Squares is that some control over sequencing effects is achieved and it is efficient compared with conducting a fully counterbalanced experimental design. A limitation is that while main effects of factors can be tested, interactions cannot be tested.

References

Ackoff, R. L. (1953). The design of social research. Chicago: University of Chicago Press.

Edwards, A. L. (1998). Experimental design. N.Y. : Addison-Wesley

Keppel, G. (2006). Introduction to design and analysis. New York: NY: Worth.

Mason, R. L., Gunst, R. F., & Hess, J. C. (1989). Statistical design and analysis of experiments. New York: NY: Wiley.

Myers, J. L,. & Well, A. D. (2003). Research design and statistical analysis. Mahwah, NJ: Erlbaum .