AP Photo/Haraz N. Ghanbari

A quantitative interaction is an interaction in which the effect of one independent variable is strengthened or weakened at one or more levels of the other independent variable, but the direction of the initial effect does not change.

A qualitative interaction is a particular type of quantitative interaction of two (or more) independent variables in which one independent variable reverses its effect depending on the level of the other independent variable.

MASTERING THE CONCEPT

14.2: Researchers often describe interactions with one of two terms—quantitative or qualitative. In a quantitative interaction, the effect of one independent variable is strengthened or weakened at one or more levels of the other independent variable, but the direction of the initial effect does not change. In a qualitative interaction, one independent variable actually reverses its effect depending on the level of the other independent variable.

EXAMPLE 14.2

The grapefruit juice example is a helpful illustration of a quantitative interaction. Lipitor and Zocor lead to elevations of some liver enzymes in combination with water, but the absorption levels are even higher with grapefruit juice. This effect is not seen with a placebo, which has an equal effect regardless of beverage. Let’s invent some numbers to demonstrate this. The numbers in the cells in Table 14-4 don’t represent actual absorption levels; rather, they are numbers that are easy for us to work with in our understanding of interactions. For this exercise, we will consider every difference between numbers to be statistically significant. (Of course, if we really conducted this study, we would conduct the two-way ANOVA to determine exactly which effects were statistically significant.)

First, we consider main effects; then we consider the overall pattern that constitutes the interaction. If there is a significant interaction, we ignore any significant main effects. The significant interaction supersedes any significant main effects.

Table 14-4 includes mean absorption levels for the six cells of the study. It also includes numbers in the margins of the table, to the right of and below the cells; these numbers are also means, but are for every participant in a given row or in a given column. Each of these is called a marginal mean, the mean of a row or a column in a table that shows the cells of a study with a two-way ANOVA design. In Table 14-4, for example, the mean across from the row for grapefruit juice, 41, is the mean absorption level of every participant who was assigned to drink grapefruit juice, regardless of the assigned medication. The mean below the column for placebo, 3, is the mean absorption level of every participant who took the placebo, regardless of the assigned beverage. (Although we wouldn’t expect any absorption with the placebo, we gave it a small value, 3, to facilitate the explanation of interactions.)

The easiest way to understand the main effects is to make a smaller table for each, with only the appropriate marginal means. Separate tables let us focus on one main effect at a time without being distracted by the means in the cells. For the main effect of beverage, we construct a table with two cells, as shown in Table 14-5. The table makes it easy to see that the absorption level was higher, on average, for grapefruit juice than for water.

Let’s now consider the second main effect, that for medication. As before, we construct a table (such as Table 14-6) that shows only the means for medication, as if beverage were not included in the study. We kept the means for beverage in rows and for medication in columns, just as they were in the original table. You may, however, arrange them either way, using whichever order makes sense to you. Table 14-6 shows that the absorption levels for Lipitor and Zocor were higher, on average, than for placebo, which led to almost no absorption. Both results would need to be verified with a hypothesis test, but we seem to have two main effects: (1) a main effect of beverage (grapefruit juice leads to higher absorption, on average, than water does), and (2) a main effect of medication (Lipitor and Zocor lead to higher absorption, on average, than placebo does).

But that’s not the whole story. Grapefruit juice, for example, does not lead to higher absorption, on average, among placebo users. Here’s where the interaction comes in. Now we ignore the marginal means and get back to the means in the cells themselves, seen again in Table 14-7. Here we can see the overall pattern by framing it in two different ways. We can start by considering beverage. Does grapefruit juice boost mean absorption levels as compared to water? It depends. It depends on the level of the other independent variable, medication; specifically, it depends on whether the patient is taking one of the two medications or a placebo. We can also frame the question by starting with medication. Do Lipitor and Zocor boost mean absorption levels as compared to a placebo? It depends. They do anyway, even just in conjunction with water, but they do so to a far greater degree in conjunction with grapefruit juice. This is a quantitative interaction because the strength of the effect varies under certain conditions, but the direction does not.

People sometimes perceive an interaction where there is none. If Lipitor, Zocor, and a placebo all had higher mean absorption rates in conjunction with grapefruit juice (versus water), there would be no interaction. Lipitor and Zocor would always lead to a particular increase in average absorption levels versus a placebo—this would occur in the presence of any beverage. And grapefruit juice would always lead to a particular increase in average absorption levels versus water—this would occur in the presence of any medication. On the other hand, there is an interaction in the example we have been considering because grapefruit juice has a special effect with the two medications that it does not have with the placebo. The tendency to see an interaction when there is none can be diminished by constructing a bar graph, as in Figure 14-2.

The bar graph helps us to see the overall pattern, but one more step is necessary: to connect each set of bars with a line. We have two choices that match the two ways we framed the interaction in words above: (1) As in Figure 14-3, we could connect the bars for the first independent variable, medication. We would connect the three medications for the grapefruit condition, and then we would connect the three medications for the water condition. (2) Alternatively, as in Figure 14-4, we could connect the bars for the two beverages. We would connect the two beverages for Lipitor, for Zocor, and for placebo.

In Figure 14-4, notice that the lines do not intersect, but they’re not all parallel to one another either. If the lines were extended far enough, eventually the lines connecting the two bars for each medication would intersect the line connecting the bar for placebo. Perfectly parallel lines indicate the likely absence of an interaction, but we almost never see perfectly parallel lines emerging from real-life data sets; real-life data are usually messy. Nonparallel lines may indicate a statistically significant interaction, but we have to conduct an ANOVA to be sure. Only if the lines are significantly different from parallel can we reject the null hypothesis that there is no interaction; and we only want to interpret an interaction if we reject the null hypothesis.

Some social scientists refer to an interaction as a significant difference in differences. In the context of the grapefruit juice study, the mean difference between grapefruit juice and water is larger when participants are taking one of the medications than when they are taking the placebo. In fact, for those taking the placebo, there is no mean difference between grapefruit juice and water. This is an example of a significant difference between differences. This interaction is represented graphically whenever the lines connecting the bars are significantly different from parallel.

However, if grapefruit juice also led to an increase in mean absorption levels among those taking the placebo, the graph would look like the one in Figure 14-5. In this case, the mean absorption levels of Lipitor and Zocor do increase with grapefruit juice, but so does the mean absorption level of the placebo, and there is likely no interaction. Grapefruit juice has the same effect, regardless of the level of the other independent variable of medication. When in doubt about whether there is an interaction or just two main effects that add up to a greater effect, draw a graph and connect the bars with lines. However, don’t just draw lines and skip the step of creating the bar graph. Zacks and Tversky (1997) reported that some people view lines as indicating a continuous variable, with a participant in one of their experiments viewing a line graph and stating: “The more male a person is, the taller he/she is” (p. 148). A bar graph would likely have led that participant to perceive, accurately, that men are, on average, taller than women.

EXAMPLE 14.3

Do you think that, on average, people make better decisions when they consciously focus on the decision? Or do they make better decisions when the decision-making process is unconscious (i.e., making the decision after being distracted by other tasks)? Researchers in the Netherlands conducted a series of studies in which participants were asked to decide between two options following either conscious or unconscious thinking about the choice. The studies were analyzed with two-way ANOVAs (Dijksterhuis, Bos, Nordgren, & van Baaren, 2006).

In one study, participants were asked to choose one of four cars. One car was objectively the best of the four, and one was objectively the worst. Some participants made a less complex decision; they were told 4 characteristics of each car. Some participants made a more complex decision; they were told 14 characteristics of each car. After learning about the cars, half the participants in each group were randomly assigned to think consciously about the cars for 4 minutes before making a decision. Half were randomly assigned to distract themselves for 4 minutes by solving anagrams before making a decision. The research design, with two independent variables, is shown in Table 14-8. The first independent variable is complexity, with two levels: less complex (4 attributes) and more complex (14 attributes). The second independent variable is type of decision making, with two levels: conscious thought and unconscious thought (distraction).

The researchers calculated a score for each participant that reflected his or her ability to differentiate between the objectively best and objectively worst cars in the group. This score represents the dependent variable, and higher numbers indicate a better ability to differentiate between the best and worst cars. Let’s look at Table 14-9, which presents cell means and marginal means for this experiment. Note that the means are approximate and that the marginal means are created by assuming the same number of participants in each cell. As we consider these findings, we will assume that all differences are statistically significant. (In a real research situation, we would conduct an ANOVA to determine statistical significance.)

Because there was an overall pattern—an interaction—the researchers did not pay attention to the main effects in this study; an interaction trumps any main effects. However, let’s examine the main effects to get some practice. We’ll create tables for each of the two main effects so that we can examine them independently (Tables 14-10 and 14-11). The marginal means indicate that when the type of decision making is ignored entirely, people make better decisions, on average, in less complex situations than in more complex situations. Further, the marginal means also suggest that, when complexity of decision is ignored, people make better decisions, on average, when the decision-making process is unconscious than when it is conscious.

However, if there is also a significant interaction, these main effects don’t tell the whole story. The overall pattern of cell means renders this information misleading, even inaccurate, under certain conditions. The interaction demonstrates that the effect of the decision-making method depends on the complexity of the decision. Conscious decision making tends to be better than unconscious decision making in less complex situations, but unconscious decision making tends to be better than conscious decision making in more complex situations. This reversal of direction is what makes this a qualitative interaction. It’s not just the strength of the effect that changes, but the actual direction!

A bar graph, shown in Figure 14-6, makes the pattern of the data far clearer. We can actually see the qualitative interaction.

As we do with a quantitative interaction, we add lines to determine whether they are parallel (no matter how long the lines are) or intersect (or would do so if extended far enough), as in Figure 14-7. Here we see that the lines intersect without even having to extend them beyond the graph. This is likely an interaction. Type of decision making has an effect on differentiation between best and worst cars, but it depends on the complexity of the decision. Those making a less complex decision tend to make better choices if they use conscious thought. Those making a more complex decision tend to make better choices if they use unconscious thought. We would, as usual, verify this finding by conducting a hypothesis test before rejecting the null hypothesis that there is no interaction.

The qualitative interaction of decision-making method and complexity of situation was not likely to have been predicted by common sense. In such instances, we should be cautious before generalizing the findings. In this case, the research was carefully conducted and the researchers replicated their findings across several situations. For example, the researchers found similar effects in a real-life context when the less complex situation was shopping at a department store that sold clothing and kitchen products, and the more complex situation was purchasing furniture at IKEA. Such an intriguing finding would not have been possible without the inclusion of two independent variables in one study, which required that researchers use a two-way ANOVA capable of testing for an interaction.

So what do these findings mean for us as we approach the decisions we face every day? Which sunblock should we buy to best protect against UV rays? Should we go to graduate school or get a job following graduation? Should we consciously consider characteristics of sunblocks but “sleep on” graduate school–related factors? Research would suggest that the answer to the last question is yes (Dijksterhuis et al., 2006). Yet if the history of social science research is any indication, there are other factors not included in these studies that likely could affect the quality of our decisions. And so the research process continues.

CHECK YOUR LEARNING

Clarifying the Concepts

• 14-5 What is the difference between a quantitative interaction and a qualitative interaction?
• 14-6 Why are main effects ignored when there is an interaction? (We often say they are trumped by an interaction.)

Calculating the Statistics

• 14-7 Data are presented here for two hypothetical independent variables (IVs) and their combinations.
  IV 1, level A; IV 2, level A: 2, 1, 1, 3
  IV 1, level B; IV 2, level A: 5, 4, 3, 4
  IV 1, level A; IV 2, level B: 2, 3, 3, 3
  IV 1, level B; IV 2, level B: 3, 2, 2, 3
  
  1. Figure out how many cells are in this study’s table, and draw a grid to represent them.
  2. Calculate cell means and write them in the cells of the grid.
  3. Calculate marginal means and write them in the margins of the grid.
  4. Draw a bar graph of these data.

Applying the Concepts

• 14-8 For each of the following situations involving a real-life interaction, (i) state the independent variables, (ii) state the likely dependent variable, (iii) construct a table showing the cells, and (iv) explain whether the interaction is qualitative or quantitative.
  
  1. Caroline and Mira are both really smart and do equally well in their psychology class, but something happens to Caroline when she goes to their philosophy class. She just can’t keep up, whereas Mira does even better.
  2. Our college baseball team has had a great few years. The team plays especially well at home versus away when playing teams in its own conference. However, it plays especially well at away games (versus home games) when playing teams from another conference.
  3. Caffeinated drinks get me wired and make it somewhat difficult for me to sleep. So does working out in the evenings. When I do both, I’m so wired that I might as well stay up all night.