Some Possible Results from a Factorial ANOVA

Factorial designs are popular in social science research for a variety of reasons. One reason is that they allow you to test for several different types of effects in a single computation. The nature of these effects are illustrated later in this chapter.

First, however, it is important to note one drawback associated with factorial designs: they sometimes produce results that can be difficult to interpret compared to the results produced in a one-way ANOVA. Fortunately, this task of interpretation can be made much easier if you first prepare a figure that plots the results of the factorial study. This first section shows how this can be done.

Figure 10.2 presents one type of figure that is often used to illustrate the results of a factorial study. Notice that, with this figure, scores on the criterion variable (level of aggression displayed by the children) are plotted on the vertical axis. Remember that groups that appear higher on this vertical axis display higher mean levels of aggression.

The three levels of predictor variable A (amount of sugar consumed) are plotted on the horizontal axis. The point at the left represents group A1 (who received 0 grams of sugar), the middle point represents group A2 (the 20-gram group), and the point at the right represents group A3 (the 40-gram group).

The two levels of predictor variable B (participant sex) are identified by drawing two different lines in the body of the figure itself. Specifically, the mean scores on aggression displayed by the males (level B1) are illustrated with small circles connected by a solid line, while the mean scores on aggression displayed by the females (level B2) are depicted by small triangles connected by a dashed line.

In summary, the important points to remember when interpreting the figures in this chapter are as follows:

• The possible scores on the criterion variable are represented on the vertical axis.

• The levels of predictor A are represented as points on the horizontal axis.

• The levels of predictor B are represented by drawing different lines within the figure itself.

With this foundation, you are ready to learn about the different types of effects that can be observed in a factorial design, and how these effects appear when they are plotted in this type of figure.

Significant Main Effects

When a predictor variable (or independent variable) in a factorial design displays a significant main effect, it means that there is a difference between at least two levels of that predictor variable with respect to mean scores on the criterion variable. In a one-way analysis of variance, there is essentially just one main effect: the main effect for the study’s independent variable. In a factorial design, however, there is one main effect possible for each predictor variable examined in the study.

For example, the preceding study on aggression included two predictor variables: amount of sugar consumed and participant sex. This means that, in analyzing data from this investigation, it is possible to obtain any of the following outcomes related to main effects:

• a significant main effect for just predictor A (amount of sugar consumed);

• a significant main effect for just predictor B (participant sex);

• a significant main effect for both predictor A and B;

• no significant main effects for either predictor A or B.

This section describes these effects and illustrates how the effects appear when plotted in a figure.

A Significant Main Effect for Predictor A

Figure 10.2 shows a possible main effect for predictor A: “Amount of Sugar Consumed.” (To save space, we henceforth refer to this as the “sugar consumption” variable.) Notice that a relatively low mean level of aggression was displayed by participants in the 0-gram condition of the sugar consumption. When you look above the heading “Level A1,” you see that both the boys (represented with a small circle) and the girls (represented with a small triangle) display relatively low aggression scores. However, a somewhat higher level of aggression was exhibited by participants in the 20-gram condition: when you look above “Level A2,” you can see that both boys and girls display a somewhat higher aggression level. Finally, an even higher level of aggression was exhibited by participants in the 40-gram condition: when you look above “Level A3,” you can see that both boys and girls in this group display relatively high levels of aggression. In short, this trend shows that there is a main effect for the sugar consumption variable.

This leads to an important point. When a figure representing the results of a factorial study displays a significant main effect for predictor variable A, it demonstrates both of the following characteristics:

• The lines for the various groups are parallel.

• At least one line segment displays a relatively steep angle.

The first of the two conditions, that the lines should be parallel, ensures that the two predictor variables do not exhibit an interaction. This is important, because you normally will not interpret a significant main effect for a predictor variable if that predictor is involved in an interaction. In Figure 10.2, you can see that the lines for the two groups in the present study (the solid line for the boys and the dashed line for the girls) are parallel. This suggests that there probably is not an interaction between sex and sugar consumption in the present study. (The concept of interaction is explained in greater detail later in the section titled “A Significant Interaction.”)

The second condition, that at least one line segment should display a relatively steep angle, can be understood by again referring to Figure 10.2. Notice that the line segment that begins at Level A1 (the 0-gram condition) and extends to Level A2 (the 20-gram condition) is not horizontal; instead, it displays an upward angle. This is because aggression scores for the 20-gram group were higher than aggression scores for the 0-gram group. When you obtain a significant effect for the predictor A variable, you should expect to see this type of angle. Similarly, you can see that the line segment that begins at A2 and continues to A3 also displays an upward angle, also consistent with a significant effect for the sugar consumption.

Remember that these guidelines are merely intended to help you understand what a main effect looks like when it is plotted as in Figure 10.2. To determine whether this main effect is statistically significant, it is necessary to review the results of the analysis of variance, to be discussed later.

A Significant Main Effect for Predictor B

You would expect to see a different pattern if the main effect for the other predictor variable (predictor B) were significant. Earlier, you learned that predictor A was represented by plotting three points on the horizontal axis. In contrast, you learned that predictor B was represented by drawing different lines within the body of the figure itself: one line for each level of predictor B. In the present study, predictor B is participant sex, so a solid line is used to represent mean scores for boys and a dashed line is used to represent mean scores for girls.

When predictor B is represented in a figure by plotting separate lines for its various levels, a significant main effect for that variable is evident when the figure displays both of the following:

• The lines for the various groups are parallel.

• At least two of the lines are relatively far apart.

For example, a main effect for predictor B in the current study is depicted by Figure 10.3. Consistent with the two preceding points, the lines in Figure 10.3 are parallel to one another (indicating that there is no interaction between sex and sugar consumption), and separated from one another. Regarding this separation, notice that (in general) boys tend to score higher on the measure of aggression compared to girls. Furthermore, notice that this tends to be true regardless of how much sugar participants consume. Figure 10.3 shows the general trend that you would expect when there is a main effect for predictor B only:

A Significant Main Effect for Both Predictor Variables

It is possible to obtain significant effects for both predictor A and predictor B in the same investigation. When there is a significant effect for both predictor variables, you should encounter each of the following:

• The lines for the various groups are parallel (indicating no interaction).

• At least one line segment displays a relatively steep angle (indicating a main effect for predictor A).

• At least two of the lines are relatively separated from each other (indicating a main effect for predictor B).

Figure 10.4 shows what the figure might look like under these circumstances.

No Main Effects

Figure 10.5 shows what a figure might look like if there were main effects for neither predictor A nor predictor B. Notice that the lines are parallel (indicating no interaction), none of the line segments displays a relatively steep angle (indicating no main effect for predictor A), and that the lines are not separated (indicating no main effect for predictor B).

A Significant Interaction

An interaction can be defined in a number of ways. For example, with respect to experimental research (in which you manipulate independent variables), the following definition could be used:

an interaction is a condition in which the effect of one independent variable on the dependent variable is different at different levels of a second independent variable.

On the other hand, when conducting nonexperimental research (in which you are simply measuring naturally occurring variables rather than manipulating independent variables), it could be defined in this way:

an interaction is a condition in which the relationship between one predictor variable and the criterion is different at different levels of a second predictor variable.

These definitions are admittedly somewhat abstract at first glance. Once again, the concept of interaction is much easier to grasp when visually displayed. For example, Figure 10.6 displays a significant interaction between sugar consumption and participant sex in the present study. Notice that the lines for the two groups are no longer parallel: the line for the male participants now displays a somewhat steeper angle compared to the line for the females. This is the key characteristic of a figure that displays a significant interaction: lines that are not parallel.

Notice how the relationships portrayed in Figure 10.6 are consistent with the definition for interaction: the relationship between one predictor variable (sugar consumption) and the criterion (aggression) is different at different levels of a second predictor variable (participant sex). More specifically, the figure shows that the relationship between sugar consumption and aggression is relatively strong for the male participants: consuming larger quantities of sugar results in dramatic increases in aggression among the male participants. (Notice that the boys who consumed 40 grams of sugar displayed much higher levels of aggression than the boys who consume 0 or 20 grams.) In contrast, the relationship between sugar consumption and aggression is relatively weak among the female participants. Notice that the line for the females is fairly “flat”; that is, there is little difference in aggression between the girls who consumed 0 grams versus 20 grams versus 40 grams of sugar.

Figure 10.6 shows why you would normally not interpret main effects when an interaction is significant. To understand why, consider this: would it make sense to say that there is a main effect for sugar consumption in the study illustrated by Figure 10.6? Probably not. It is clear that sugar consumption does seem to have an effect on aggression among boys, but the figure suggests that sugar consumption probably does not have any meaningful effect on aggression among girls. To say that there is a main effect for sugar consumption might mislead readers into believing that sugar causes aggression in all children in much the same way (which it apparently does not). Whether sugar fosters aggression appears to depend on the child’s sex. In this situation, it would make more sense to, instead, do the following:

• Note that there was a significant interaction between sugar consumption and participant sex.

• Prepare a figure (like Figure 10.6) that illustrates the nature of the interaction.

• Test for simple effects.

Testing for simple effects is similar to testing for main effects but is done one group at a time. In the preceding analysis, for example, you would test for simple effects by first dividing your data into two groups. Data from the male participants would be separated from data from the female participants. With this done, you would perform an analysis to determine whether sugar consumption has a simple effect on aggression among just the male participants (and you would probably find that this simple effect is significant). Finally, you would perform a separate test to determine whether sugar consumption has a simple effect on aggression among just the female participants (and you would probably find that this simple effect is not significant). A later section shows how SAS can perform these tests for simple effects.

To summarize, an interaction means that the relationship between one predictor variable and the criterion is different at different levels of a second predictor variable. When an interaction is significant, you should interpret your results in terms of simple effects rather than main effects.