Introduction to Factorial Designs

The preceding chapter described a simple experiment in which you manipulated a single independent variable, type of rewards. Because there was a single independent variable in that study, it was analyzed using a one-way ANOVA.

But imagine that there are actually two independent variables that you want to manipulate. In this situation, you might think that it would be necessary to conduct two separate experiments, one for each independent variable; this would be incorrect. In many cases, you can manipulate both independent variables in a single study.

The research design used in such a study is called a factorial design. In a factorial design, two or more independent variables are manipulated in a single study so that the treatment conditions represent all possible combinations of the various levels of the independent variables.

In theory, a factorial design might include any number of independent variables. In practice, however, it generally becomes impractical to use many more than two or three. This chapter illustrates factorial designs that include just two independent variables and thus can be analyzed using a two-way ANOVA. More specifically, this chapter deals with studies that include two predictor variables, both measured on a nominal scale, as well as a single criterion variable assessed on an interval or ratio scale.

The Aggression Study

To illustrate the concept of factorial design, imagine that you are interested in conducting a study that investigates aggression in eight-year-old children. Here, aggression is defined as any verbal or behavioral act performed with the intention of harm. You want to test two hypotheses:

• Boys will display higher levels of aggression than girls.

• The amount of sugar consumed will have a positive effect on levels of aggression.

You perform a single investigation to test these two hypotheses. The hypothesis that you are most interested in is the second hypothesis, that consuming sugar will cause children to behave more aggressively. You will test this hypothesis by manipulating the amount of sugar that a group of school children consume at lunchtime. Each day for two weeks, one group of children will receive a lunch that contains no sugar at all (0 grams of sugar group). A second group will receive a lunch that contains a moderate amount of sugar (20 grams), and a third group will receive a lunch that contains a large amount of sugar (40 grams). Each child will then be observed after lunch, and a pair of judges will tabulate the number of aggressive acts that the child commits. The total number of aggressive acts committed by each child over the two-week period will serve as the dependent variable in the study.

You begin with a sample of 60 children: 30 boys and 30 girls. The children are randomly assigned to treatment conditions in the following way:

• 20 children are assigned to the “0 grams of sugar” treatment condition;

• 20 children are assigned to the “20 grams of sugar” treatment condition;

• 20 children are assigned to the “40 grams of sugar” treatment condition.

In making these assignments, you ensure that there are equal numbers of boys and girls in each treatment condition. For example, you verify that, of the 20 children in the “0 grams” group, 10 are boys and 10 are girls.

The Factorial Design Matrix

The factorial design of this study is illustrated in Figure 10.1. You can see that this design is represented by a matrix that consists of two rows (running horizontally) and three columns (running vertically):

When an experimental design is represented in a matrix such as this, the matrix is easier to understand if you focus on just one aspect at a time. For example, consider just the three vertical columns of Figure 10.1. The three columns are headed “Predictor A: Amount of Sugar Consumed” so obviously, these columns represent the various levels of sugar consumption (independent variable). The first column represents the 20 participants in Level A1 (i.e., the participants who received 0 grams of sugar), the second column represents the 20 participants in Level A2 (i.e., 20 grams), and the last column represents the 20 participants in Level A3 (i.e., 40 grams).

Now consider just the two horizontal rows of Figure 10.1. These rows are headed “Predictor B: Participant Sex.” The first row is headed “Level B1: Males,” and this row represents the 30 male participants. The second row is headed “Level B2: Females,” and represents the 30 female participants.

It is common to refer to a factorial design as a “r × c” design, in which “r” represents the number of rows in the matrix, and “c” represents the number of columns. The present study is an example of a 2 × 3 factorial design because it has two rows and three columns. If it included four levels of sugar consumption rather than three, it would be referred to as a 2 × 4 factorial design.

You can see that this matrix consists of six different cells. A cell is the location in the matrix where the row for one independent variable intersects with the column for a second independent variable. For example, look at the cell where the row named B1 (males) intersects with the column headed A1 (0 grams). The entry “10 Participants” appears in this cell, which means that there are 10 participants assigned to this particular combination of “treatments” under the two independent variables. More specifically, it means that there are 10 participants who are both (a) male and (b) given 0 grams of sugar. (“Treatments” appears in quotation marks in the preceding sentence because “participant sex” is obviously not a true independent variable that is manipulated by the researcher; it is merely a predictor variable.)

Now look at the cell in which the row labeled B2 (females) intersects with the column headed A2 (20 grams). Again, the cell contains the entry “10 Participants,” which means that there is a different group of 10 children who experienced the treatments of (a) being female and (b) receiving 20 grams of sugar. You can see that there is a separate group of 10 children assigned to each of the six cells of the matrix. No participant appears in more than one cell.

Earlier, it was said that a factorial design involves two or more independent variables being manipulated so that the treatment conditions represent all possible combinations of the various levels of the independent variables. Figure 10.1 illustrates this concept. You can see that the six cells of the figure represent every possible combination of sex and amount of sugar consumed: males are observed under every level of sugar consumption; the same is true for females.