# A Slow, Gentle Introduction to Factorial Experiments

Many people try to improve their moods by manipulating two factors--eating chocolate and drinking caffeinated beverages. To test whether these two factors improve mood, you could do two experiments on mood: one experiment to look at chocolate’s effect and another to look at caffeinated beverage’s effect.

## What is a factorial experiment?

Instead of doing two separate experiments, why not look at both chocolate’s and caffeine’s effects on mood in one experiment?  You could look at both factors at once by conducting a factorial experiment: an experiment in which you independently manipulate more than one factor at a time.

## What are 3 keys to doing a factorial experiment correctly?

To do a factorial experiment correctly, you must

1.      randomly assign each participant to amounts or types of

2.      at least two different factors (e.g., an amount of chocolate and an amount of caffeine)

3.      so that you are assigning participants to one of at least four different combinations of those different factors.

So, to do a caffeine-chocolate factorial experiment, you must randomly assign each participant to some combination of chocolate and caffeine.  For example, you might give one-fourth of the participants plain raisins and decaffeinated cola, one-fourth plain raisins and caffeinated cola, one-fourth chocolate-covered raisins and decaffeinated cola, and one-fourth chocolate-covered raisins and caffeinated cola (see Table 1).

Table 1:  Example of a Factorial Experiment.

Note that participants are randomly assigned to combinations of two different treatments.

 Group 1 gets:  Plain raisins and Caffeine-free cola Group 2 gets:  Plain raisins and Caffeinated cola Group 3 gets: Chocolate-covered raisins and Caffeine-free cola Group 4 gets: Chocolate-covered raisins and Caffeinated cola

## What 3 things can you learn from a factorial experiment?

Such a factorial experiment would tell you about the individual effects of each of your treatment factors as well as special effects of certain combinations of those factors. Specifically,

1. By comparing the caffeine-free cola groups in the left column to the caffeinated cola groups in the right column, you could see whether drinking caffeinated cola put people in a better mood than drinking caffeine-free cola (see Table 1B).

Table 1B:  How a factorial experiment allows you to compare caffeine-free groups to caffeinated groups

 Caffeine-free cola groups Caffeinated cola groups Group 1 gets: Plain raisins and Caffeine-free cola Group 2 gets: Plain raisins and Caffeinated cola Group 3 gets: Chocolate-covered raisins and Caffeine-free cola Group 4 gets: Chocolate-covered raisins and Caffeinated cola

2. By comparing the plain raisins groups in the top row to the chocolate-covered raisins groups in the bottom row, you could see whether eating chocolate-covered raisins puts people in a better mood than eating plain raisins (See Table 1C).

Table 1C:  How a factorial experiment allows you to compare plain raisin groups to chocolate raisin groups

 Plain raisin groups Group 1 gets:  Plain raisins and Caffeine-free cola Group 2 gets:  Plain raisins and Caffeinated cola Chocolate-covered raisin groups Group 3 gets: Chocolate-covered raisins and Caffeine-free cola Group 4 gets: Chocolate-covered raisins and Caffeinated cola

3. By looking at the average effects of each individual factor, you could see whether combinations of your two factors have an effect that is different from what would be expected from knowing only each factor’s average effect. In your chocolate-caffeine factorial experiment, because you would know the individual effects of both caffeine and chocolate and because participants get different combinations of chocolate and caffeine, you could determine whether certain chocolate-caffeine combinations produce effects that are different from the sum of the average of caffeine’s and chocolate’s effects. For instance, it might be that eating plain raisins usually makes people feel better and drinking caffeine-free cola usually makes people feel better, but that combining plain raisins with caffeine-free cola makes people feel much worse by producing a raisin-cola effect in people’s stomachs that is very similar to a  Mentos-cola effect.

## Why you can’t learn about each of those 3 separate effects with one simple two-group experiment: With two groups you can twist two factors together, but you can’t disentangle their effects.

To appreciate how wonderful the factorial experiment is, let’s see what would happen if you tried to separate out the individual effects of both caffeine and chocolate—as well as any effects due to certain caffeine-chocolate combinations being unexpectedly good or unexpectedly bad—in a two-group experiment. Specifically, imagine you fused the caffeine and chocolate factors  together in a  two-group experiment by comparing a group who ate plain raisins and drank caffeine-free colas to a group who ate chocolate-covered raisins and drank caffeinated colas (see Table 2A).

Table 2A: Two-Group Experiment Comparing Plain Raisin/Caffeine-Free Cola Group to Chocolate-Covered Raisins/Caffeinated Cola Group.

 Group 1: Plain Raisins and Caffeine-Free Cola Group 2: Chocolate-Covered Raisins and Caffeinated Cola

With such an experiment, could you separate out the individual effect of each factor? No—you have tied the variables together in a way that doesn’t allow you to untie them (in technical terms, you have confounded the two variables).  To understand why such an experiment would not allow you to isolate each factor’s individual effect, imagine that the experiment obtained the results summarized in Table 2B.

Table 2B: Comparing Average Mood of Plain Raisin/Caffeine-Free Cola Group to Chocolate-Covered Raisins/Caffeinated Cola Group.

 Group 1: Plain Raisins and Caffeine-Free Cola Average Mood: 3 Group 2: Chocolate-Covered Raisins and Caffeinated Cola Average Mood: 6

Average Mood on a 0 (Bad) to 10 (Great) Scale.

Assuming the difference between the groups was statistically significant, you could conclude that Group 2 (the group eating chocolate-covered raisins and drinking caffeinated cola) was in a better mood than Group  1 (the group eating plain raisins and drinking caffeine-free cola). But you could not say how much—if any—of the difference in mood was due to eating chocolate-covered raisins because the groups also differ in what type of cola they drank.

In a way, trying to isolate the cause of Group 2’s better mood is like trying to determine what made your stomach queasy right after eating an unfamiliar food and drinking an unfamiliar beverage. You would wonder: “If it’s not just a coincidence that I feel queasy after what I ate and drank, am I feeling that way because of what I ate? Or was it what I drank? Or was it that the combination of what I ate and drank didn’t go well together?  Or did what I ate upset my stomach, and what I drank made it even worse?  Or….?”

Applying the same logic just used in the upset stomach example to the results described in Table 2, Group 2 might be in a better mood that Group 1 is due because

1. Eating chocolate-covered raisins putting people in a better mood than eating plain raisins, and/or

2. Drinking caffeinated cola putting people in a better mood than drinking caffeine-free cola, and/or

3. Combining chocolate-covered raisins with drinking caffeinated cola  producing an effect that is better than just the sum of their separate effects (like the combination of peanut butter and jelly  complementing each other)  or the eating of plain raisins while drinking caffeine-free cola producing a combined effect that is worse than their individual effects (like the combination of mustard and oatmeal tasting worse than either alone).

To reiterate, if you found that a difference in mood between the caffeine-chocolate group and the no-chocolate, no-caffeine group, you wouldn’t know which difference(s) in how the groups were treated was/were responsible.  Even if you knew that there was only one reason for the difference in mood, you wouldn’t know which of following reasons was the right one:

1.  chocolate-covered raisins boosted mood more than plain raisins, or

2.  caffeinated colas boosted mood more than decaffeinated colas, or

3.  even though neither chocolate nor caffeine affect mood by themselves, combining chocolate-covered raisins with caffeinated colas puts people in a better mood than combining plain raisins with decaffeinated cola.

But you wouldn’t know that there was only one reason for the difference between the groups! For example, it could be that

eating chocolate-covered raisins puts people in a better mood than eating plain raisins, and

drinking caffeinated cola puts people in a better mood than drinking decaffeinated cola , and

caffeinated cola goes especially well with chocolate-covered raisins.

## A Simple, Two-Group Experiment Would Allow You to Vary One Factor (Caffeine) While Giving All Participants Plain Raisins

You have seen that a two-group experiment would not allow you to separate out the effects of chocolate from either the effects of caffeine or the unique effects of certain chocolate-caffeine combinations.   But could you isolate the individual effects of both chocolate and type of cola by doing a simple, two-group experiment like the one outlined in Table 3?

Table 3: Simple experiment comparing caffeine-free cola to caffeinated cola for people eating plain raisins.

 Group A: Plain raisins and Caffeine-free cola Group B: Plain raisins and Caffeinated cola

Average Mood on a 0 (Bad) to 10 (Great) Scale.

No, such an experiment obviously could not tell you anything about the difference that different types of raisins make because participants get only one type of raisin. However, because the only difference between the groups is caffeine, such a simple experiment would allow you to determine the difference that caffeine makes for people who are eating plain raisins.

## A Second Simple, Two-Group Experiment Would Allow You to Vary One Factor (Caffeine) While Giving All Participants Chocolate-Covered Raisins

Similarly, the simple experiment outlined in Table 4 would not tell you anything about the difference that different types of raisins make because participants get only one type of raisin. However, that experiment would allow you to isolate the effects of caffeine for people who are eating chocolate-covered raisins.

Table 4: Simple experiment comparing caffeine-free cola to caffeinated cola for people eating chocolate-covered raisins.

 Group C: Chocolate-covered raisins and Caffeine free cola Group D : Chocolate-covered raisins and Caffeinated cola

Average Mood on a 0 (Bad) to 10 (Great) Scale.

## Looking at Differences Between Groups That Differ in Whether They Drank Caffeinated Colas Allows You to See the Difference That Caffeinated Colas Make

So, if you did both of these simple experiments, the first simple experiment would tell you the effect of caffeinated cola for people who ate plain raisins, and the second simple experiment would tell you the effect of caffeinated cola for people who ate chocolate-covered raisins. Specifically, for each experiment, you could see what difference caffeinated cola made by taking the difference between the average mood of the caffeinated cola group and the average mood of the caffeine-free cola group. If we do that subtraction using the data in Table 5, the effect of caffeine for the plain raisin groups seems to be 2. Similarly, doing that subtraction using that data in Table 6, suggests that the effect of caffeine for the chocolate-covered raisin groups also seems to be 2.

Table 5:

 Simple Experiment 1 Group A: Plain  raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Difference between caffeine-free and caffeinated cola groups   Caffeine Effect = 2 (8 – 6) for Simple Experiment 1

Table 6:

 Simple Experiment 2 Group C: Chocolate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Difference between caffeine-free and caffeinated cola groups   Caffeine Effect = 2 (9 – 7) for Simple Experiment 2

## How Combining 2 Simple Experiments Gives You (At Least) 2 Simple Main Effects

What happens if you combine Simple Experiment 1 and Simple Experiment 2 into one experiment, as we have done in Table 7?

Table 7:  How Combining Two Simple Experiments That Manipulated Caffeine Allows You To See Two Caffeine Simple Main Effects

 Caffeine Simple Main Effects Simple Experiment 1 (both groups get plain raisins, but one group gets more caffeine) Group A: Plain raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Caffeine Simple Main Effect = 2 (8 – 6) for Simple Experiment 1 Simple Experiment 2 (both groups get chocolate-covered raisins, but one group gets more caffeine) Group C: Chocolate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Caffeine Simple Main Effect = 2 (9 – 7) for Simple Experiment 2

Now, your table has two rows: one for each simple experiment. You can find the difference caffeine makes for the plain raisin groups (by taking the difference between the plain raisin, caffeine-free cola group’s average and the plain raisin, caffeinated group’s average)--just as you would do if you had a simple experiment that varied caffeine while having all participants eat plain raisins.  Similarly, you could find  the difference caffeine makes for the chocolate-covered raisin groups (by taking the difference between the chocolate-covered raisin, caffeine-free cola group’s average and the chocolate-covered raisin, caffeinated group’s average) )--just as you would do if you had a simple experiment that varied caffeine while having all participants eat chocolate-covered raisins. These caffeine effects, which could have been discovered by doing two separate simple experiments, are called simple main effects.

To illustrate that calculating simple main effects is simply a matter of subtraction, look at the last column of Table 7: the column labeled “Caffeine Simple Main Effects.” How did we estimate those caffeine simple main effects? Subtraction--Subtracting the average for the plain raisins/caffeine free cola group (6) from the plain raisins/ caffeinated cola group (8) suggests that, for the plain raisin groups, the simple main effect of caffeine is 2.  Similarly, subtracting the average for the chocolate-covered raisin/caffeine free cola group (7 ) from the chocolate-covered raisin/ caffeinated cola group (9 ) suggests that, for the chocolate-covered raisin groups, the simple main effect of caffeine is 2.

So far, combining two simple experiments seems to have accomplished nothing other than give you a new term: simple main effects. That is, instead of talking about caffeine effects in each simple experiment, we are talking about the simple main effects of caffeine in our combined experiment. How can you learn more from combining the two experiments than you could have learned from doing two separate experiments?

## How Combining 2 Simple Experiments Gives You 4 Simple Main Effects

Start by looking at both the rows and the columns. As you can see from Table 8, it could be argued that not only does each row represent a simple experiment—but that each data column also represents a simple experiment. So, in a way, our factorial experiment contains 4 simple experiments:

Table 8:  A factorial experiment can be seen as containing 4 simple experiments

 Simple Experiment 3 (Caffeine-free cola groups getting different kinds of raisins) Simple Experiment 4 (Caffeinated cola groups getting different kinds of raisins) Simple Experiment 1 (plain raisin groups getting different kinds of cola) Group A: Plain raisins and Caffeine-free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Simple Experiment 2 (chocolate- covered raisin groups getting different kinds of cola) Group C: Chocolate-covered raisins and Caffeine-free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Simple Main Effect of raisins for caffeine-free groups = 1 (7 – 6) Simple Main Effect of raisins for caffeinated groups = 1  (9 – 8)

To find the simple main effect of raisins for the caffeine-free cola groups, look at the Simple Experiment 3 column. Specifically, subtract the plain raisin/ caffeine-free cola group mean (which is 6) from the chocolate-covered raisins/ caffeine-free cola group mean (which is 7). From that subtraction, you can conclude that the best estimate is that eating chocolate-covered raisins while drinking caffeine-free cola boosts mood by about 1 point (because 7 6 = 1).

Similarly, to find the simple main effect of raisins for the caffeinated cola groups, look at the Simple Experiment 4 column. Specifically, subtract the plain raisin/ caffeinated cola group mean (which is 8) from the chocolate-covered raisins/ caffeinated cola group mean (which is 9). From that subtraction, you can estimate that eating chocolate-covered raisins while drinking caffeinated cola boosts mood by about 1 point (because 9 8 = 1).

## Comparing Effects From the Same Experiment Is Better Than Comparing Effects From Different Experiments

At this point, it may seem that combining your two simple experiments into one is no better than having performed four separate simple experiments. But that is not true: If you had done four separate experiments, it would be hard to compare those experiments because the participants in the different experiments may have differed from each other in some systematic way. That is, if you had done different experiments, comparing those different experiments would be like comparing apples that get one treatment with oranges that get another treatment.   But by doing one experiment, you are, in a sense, comparing apples that randomly got one treatment combination with apples that randomly got another treatment combination. As a result, you do not have to worry that your first experiment consisted of participants who were systematically different (e.g., more eager to be in an experiment) from the participants who were in your last experiment. So, by doing one experiment in which one group of participants were randomly assigned to four different groups, your “four” experiments are comparable.

## Two Ways to Compare Your Two Pairs of Simple Main Effects

“Experiments” 1 and 2 (the two rows representing two experiments on the effects of caffeine) are comparable-- as are “Experiments” 3 and 4 (the two data columns representing two experiments on the effect of chocolate). You can take advantage of having two pairs of simple main effects--a caffeine pair and a chocolate pair—in two ways.

### #1 Averaging Each Factor’s Simple Main Effects to Estimate Each Factor’s Overall Main Effect

You can average each pair of simple main effects to find the factor’s average effect. For example, you could average the two caffeine simple main effects to get the effect of caffeine averaged across both the plain raisin and chocolate-covered raisin groups. This average, overall effect of a factor is called the overall main effect—often referred to as just a main effect.  So, if you got the results in Table 9, you would estimate that the overall main effect of caffeine is 2. That is, on average, across participants who ate plain raisins and who ate chocolate-covered raisins, caffeine boosted mood by about 2 points.

Table 9: Caffeine’s Two Simple Main Effects and Caffeine’s Overall Main Effect

 Simple Experiment 1 Group A: Plain raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Caffeine Effects     Caffeine Simple Main Effect = 2 (8 – 6) for Simple Experiment 1 Simple Experiment 2 Group C: Chocolate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Caffeine Simple Main Effect = 2 (9 – 7) for Simple Experiment 2 Average caffeine overall main effect = 2  because (2 + 2)/2 = 2

Similarly, you can average the pair of chocolate simple main effects to get an estimate the average overall main effect of chocolate. As you can see from the last row of Table 10, the effect of chocolate-covered raisins averaged over caffeinated and decaffeinated colas—the overall main effect of chocolate (often called the chocolate main effect)—seems to be about 1 point.

Table 10:  Raisin’s Two Simple Main Effects and Raisin’s Overall Main Effect

 Simple Experiment 3 Simple Experiment 4 Group A: Plain raisins and Caffeine-free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Group C: Chocolate-covered raisins and Caffeine-free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Chocolate Simple Main Effect for Caffeine-Free Groups= 1 (7 – 6) Chocolate Simple Main Effect for Caffeinated Groups = 1 (9 – 8) Average overall raisin main effect = 1 because (1 + 1)/ 2 = 1

### #2 Subtracting A Factor’s Simple Main Effects From Each Other to Estimate the Interaction Effect

In addition to averaging a factor’s simple main effects to find its average effect, you can  subtract a factor’s simple main effects to see whether the factor’s effect differs depending on other factor(s) you manipulated. For example, you can see whether caffeine’s effect differs depending on what type of raisin people eat by looking at the difference between caffeine’s pair of simple main effects.

## What are interactions?

If caffeine’s simple main effect in the plain raisin groups is significantly different from its simple main effect in the chocolate covered raisin groups, there is an interaction: the effect of one factor differs depending on another factor. In other words, you have an interaction when the effect of one factor is strengthened, weakened, or reversed depending on a second factor. But, when you look at Table 11, does it seem that caffeine’s effect differs depending on what type of raisin people eat?

Table 11:  Looking for the chocolate-caffeine interaction by looking at the difference between the caffeine simple main effects.

 Simple Experiment 1 Group A: Plain raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Caffeine Effect for Simple Experiment 1 (the simple main effect of caffeine for plain raisin groups) =  2 (8 – 6) Simple Experiment 2 Group C: Chocolate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Caffeine Effect for Simple Experiment 2 (the simple main effect of caffeine for plain raisin groups) = 2 (9 – 7) Interaction Effect = Difference between caffeine simple main effects = 0 (2 - 2  = 0)

No, as you can see from Table 11, there is no difference between the two caffeine simple main effects. Specifically, the caffeine simple main effect in Simple Experiment 1, in which participants ate plain raisins, is 2, and the caffeine simple main effect in Simple Experiment 2, in which participants ate chocolate-covered raisins, is also 2. So, there is no hint of an interaction because the caffeine simple main effects are the same as each other.

Because both caffeine simple main effects are both 2, the average of those caffeine simple main effects—the overall caffeine main effect-- will also be 2. This fact illustrates a general rule:  If a factor’s simple main effects are the same as each other, those simple main effects will also be the same as that factor’s overall main effect. Put another way, if there is no difference between a factor’s simple main effects, there will also be no difference between those simple main effects and their overall main effect. If you had the Results from Table 11,  because the caffeine simple main effects are the same as the caffeine overall main effect, you wouldn’t need to talk about each caffeine simple main effect separately: You could just talk about caffeine’s overall main effect.

## Drawing Conclusions From Overall Main Effects When There Is No Interaction

In the example we have been discussing, caffeine’s effect does not differ depending on what type of raisin people ate. That is, caffeine’s effect is not weakened, strengthened, or reversed by what type of raisin people ate. When the effect of one factor does not depend on the level of a second factor, there is no interaction between those factors. In such cases, we can talk about the effect of our first factor without saying “But the effects of that factor are different depending on the second factor.”

So, when there is no interaction involving a factor, that factor’s

1.  simple main effects are not different from each other (for example, in the study we’ve been discussing, the caffeine simple main effects are both 2),

2. simple main effects are not different from that factor’s overall main effect (for example,  in the study we’ve been discussing, the caffeine simple main effects and caffeine’s overall main effect are both 2), and

3. overall main effect does not need to be qualified by saying that its effect varies depending on the other factor(s) in our study. For example, we can simply say caffeine’s effect is to increase mood by 2—we don’t have to say that its effect averages out to 2 but was not 2 when participants were eating chocolate-covered raisins.

Thus far, we have shown you how to look for the caffeine by chocolate interaction (or, if you prefer, the chocolate by caffeine interaction) by comparing caffeine’s simple main effects. You could, however, also find the caffeine by chocolate interaction by comparing chocolate’s simple main effects (see Table 12).

Table 12:  Looking for the chocolate-caffeine interaction by looking at the difference between the chocolate simple main effects

 Simple Experiment 3 Simple Experiment 4 Group A: Plain raisins and Caffeine-free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Group C: Chocolate-covered raisins and Caffeine-free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Simple main effect of raisins in the caffeine-free cola groups =   1 (7 – 6) Simple main effect of raisins in the caffeinated cola groups = 1 (9 – 8) Interaction Effect = Difference between raisin effects in Experiment 3 and raisin effect in Experiment 4 = 0 (1-1 is 0)

As you can see from looking at the columns in Table 12, there is no difference between chocolate’s simple main effects: Both are 1. Because the chocolate simple main effects are the same regardless of whether people drink a caffeinated beverage, there is no chocolate-caffeine interaction.

To sum up, just as you can talk about the combination of peanut butter and chocolate by saying that (1) adding salty peanut butter brings out the sweetness in the  chocolate  or (2) adding  chocolate brings out the saltiness of the peanut butter, you have two equally valid ways to talk about a caffeine-chocolate interaction—or, in this case, the lack of a caffeine-chocolate interaction. For instance, you can say that

(1) the effect of caffeine does not depend on what type of raisin participants ate: The effect of caffeine is 2, regardless of what raisins people ate or

(2) the effect of chocolate does not depend on whether participants drank a caffeinated beverage: The effect of chocolate is 1, regardless of what cola people drank.

Both ways are correct.

## Two Main Effects Don’t Make an Interaction: Avoiding Two Common Mistakes

The wrong way to talk about the data in Table 12 is to say that there is an interaction. Clearly, there is not: The effect of chocolate does not depend on the effect of cola. So, why do many students make that mistake? They buy into two misconceptions that you should avoid:

1.      The misconception that if all main effects are not the same or if all simple main effects are not the same, there is an interaction. The truth is that you have an interaction only if a factor’s  simple main effects are different from each other. So, for determining whether there is an interaction involving caffeine, you compare only the caffeine simple main effects with each other. For determining whether there is an interaction, it doesn’t matter whether any of the caffeine main effects differ from any of the raisin effects.  Thus, in our caffeine-chocolate experiment, for determining whether there is an interaction, it is irrelevant that

1. The caffeine overall main effect is different from the chocolate overall main effect, and that

2. The caffeine simple main effects differ from the chocolate simple main effects.

2.      The misconception that if one of the groups has a higher average score than the others, there is an interaction. The truth is that you have an interaction only if the combined effect of two factors is different from the sum of their individual effects. In the example in Table 12, the group getting caffeine and chocolate scores 3 points higher than the group getting no caffeine and no chocolate. This 3-point difference gives us no evidence that there is anything special about the combination of chocolate and caffeine because 3 points is just the sum of caffeine's 2 point effect and chocolate’s 1-point effect. That is, on average, caffeine groups score 2 points higher than no caffeine groups; on average, chocolate groups score 1 point higher than no-chocolate groups.  So, just from knowing the average, individual effect of caffeine and chocolate, we would expect the caffeine-chocolate group to score 3 (2 + 1) points higher than no caffeine-no chocolate group. Since the combination of chocolate and caffeine doesn’t have an effect that is different from the sum of their individual effects, there is no interaction.

·        Tip: To avoid being fooled about whether you have an interaction, simply compare a factor’s simple main effects: If that factor’s simple main effects do not differ, you do not have an interaction.

Table 13:  An Experiment Resulting in Two Main Effects But No Interactions

 Simple Experiment 3 Simple Experiment 4 Caffeine effect Simple Experiment 1 Group A: Plain raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Simple Main Effect for plain raisin groups =     2 (8 – 6) Simple Experiment 2 Group C: Chocolate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 9 Simple Main Effect for chocolate-covered raisin groups =     2 (9 – 7) Average Caffeine Effect = 2 (2 + 2)/2 Raisin Effects Simple Main Effect for caffeine-free groups =  1 (7 – 6) Simple Main Effect for caffeinated groups =                1 (9 – 8) Average raisin effect = 1    ( 1 + 1)/2

### Main Effects Are Easy to Interpret When There Are No Interactions

In the example we have been discussing (see Table 13), there was no interaction between caffeine and type of raisin. How do we interpret results when there is no interaction?

The answer is simple: We would just talk about the overall main effects. In this case, we would say that, because the effect of caffeine did not depend on what type of raisin participants ate, caffeine’s (2 point) effect seems to generalize across both plain raisin and chocolate-covered raisin conditions. We would also say that raisin’s (1 point) effect seems to generalize across both caffeine-free and caffeinated colas.

## Interpreting Main Effects When There Is an Interaction: Interactions Limit the Generalizations We Can Make About Our Overall Main Effects

Because we are giving participants different combinations of chocolate and caffeine, we can detect whether a particular combination of caffeine and chocolate produced an effect that was different from just the sum of their individual effects. If combining the effects of two treatments produces effects different from the sum of their individual effects, there is an interaction.

You are probably most familiar with the term interaction from drug interactions. Two drugs that, separately, may both be helpful might, when taken together, be harmful. Indeed, you probably have an older relative who has had problems due to having been given drugs that interact.  (To see whether two drugs interact, you can use this drug interaction checker.)

As you may know from drug interactions, when you have an interaction, the effect of one factor depends on some other factor.  Below (see Table 14), you can see an example of an interaction between text color and background color on a message’s readability.  As you can see, against a white background, black text is easier to read than white text, but black text is harder to read than white text against a black background. So, is black text easier to read than white text? Because the answer depends on the background color, making a general statement about the superiority of black text is unwise.

Table 14: An interaction between the effect of text color and background color on readability

 Text color Black White Background Color Black 1 8 White 8 1

Scores are based on how easy the text is to read rated on a 0 (not readable) to 10 (very readable) scale.

So, if we have an interaction in a factorial experiment, we shouldn’t make general statements based solely on our main effects. Instead, we should qualify our main effects. For another example of how interactions force us to qualify conclusions suggested by our main effects, consider how we would need  to qualify our conclusions about chocolate’s overall main effect if we got results like the ones in Table 15.

Table 15:  Detecting an Interaction by looking at the difference between the chocolate simple main effects.

 Simple Experiment 3 Simple Experiment 4 Group A: Plain raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Group C: Chocolate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 7 Effect = +1 (7 – 6) Effect = -1 (7 – 8) Average chocolate effect = 0 (1 + -1] /2) Difference between simple effects = 2 (1- [-1])

As you can see, the simple main effect of chocolate-covered raisins for the caffeine-free cola groups (+1) differs from simple main effect of chocolate-covered raisins for the caffeinated cola groups (-1)—and both chocolate simple main effects differ from chocolate’s overall main effect(0). So, it would be misleading to refer only to chocolate’s overall main effect by saying that chocolate has no effect. Instead, we should state that, although the overall effect of chocolate averaged out to zero, there was an interaction such that chocolate’s effect depended on which type of cola participants drank.

To look at the caffeine-chocolate interaction focusing on caffeine’s effects, see Table 16.

Table 16: Detecting an Interaction by looking at the difference between the caffeine simple main effects.

 Simple Experiment 1 Group A: Plain raisins and Caffeine free cola Average = 6 Group B: Plain raisins and Caffeinated cola Average = 8 Caffeine effect = 2 (8 – 6) Interaction Effect Simple Experiment 2 Group C: Chocol ate-covered raisins and Caffeine free cola Average = 7 Group D : Chocolate-covered raisins and Caffeinated cola Average = 7 Caffeine effect = 0 (7 – 7) Average caffeine effect =1 (2+0)/2 Difference between caffeine simple main effects = 2  (2-0 = 2)

From Table 16, you can see that there seems to be an interaction because there is a difference between caffeine’s effect in the plain raisin groups and caffeine’s effect in the chocolate-covered raisin groups. Specifically, caffeine seems to boost mood by 2 points in the plain raisin groups, but it seems to have zero effect in the chocolate-covered raisins groups. Because the overall, average effect of caffeine (1) is different from both the effect of caffeine in the plain raisin groups (+2)  and the chocolate-covered raisin groups (0), making a general statement about the effect of caffeine based on its average effect would be a mistake. In other words, because there is an interaction involving caffeine and raisins, we should not talk about the overall main effect of caffeine without also pointing out that this caffeine overall main effect is qualified by a caffeine-raisin interaction.

## Reviewing What the 4-Group Factorial Experiment Gives You: 4 Simple Main Effects, 2 Overall Main Effects, and One Interaction

Thus far, you have been exposed to some essential and complex concepts, so let’s recap.

The factorial experiment essentially combines 4 simple experiments into 1. So, if you did a caffeine-chocolate factorial experiment, you could have learned about

1. the simple main effect of type of cola for groups getting plain raisins (as you would have if you had just done Simple Experiment 1),

2. the simple main effect of type of cola for groups getting chocolate-covered raisins (as you would have if you had just done Simple Experiment 2),

3. the simple main effect of type of raisins for groups getting caffeine-free cola (as you would have if you had just done Simple Experiment 3), and

4. the simple main effect of type of raisins for groups getting caffeinated cola (as you would have if you had just done Simple Experiment 4).

In addition, after pairing up the cola simple main effects and then pairing up the chocolate simple main effects, you would have been able to

1.  Average the two simple cola main effects to find the overall main effect of type of cola: the difference cola makes averaged across groups eating plain raisins and groups eating chocolate-covered raisins.  (Note that you find the difference type of cola makes by looking at differences within the raisin rows because, within the raisin rows, type of cola differs but type of raisin is kept constant.)

2.  Average the two simple raisin main effects to find the overall main effect of type of raisin: the difference type of raisin makes averaged across groups drinking caffeine-free and caffeinated colas.  (Note that you find the find the raisin simple main effects by looking at differences within the cola columns. Because, within the cola columns, type of raisin differs but type of cola is kept constant.)

3. Subtract the two simple cola main effects to see whether there was an interaction effect: the effect of cola being different for plain raisin groups than it was for chocolate-covered raisin groups. The interaction effect allows you to know whether the average effect of cola applies equally to both participants regardless of what type of raisin they ate.  Instead of determining whether there was an interaction by seeing whether the two cola simple main effects differ from each other, you could determine whether there was an interaction by seeing whether the two raisin simple main effects differ from each other.

## Is There an Interaction?: A Way to Double-Check

You could be doubly sure about whether there is an interaction by seeing whether the pair of cola simple main effects differ and then seeing whether the pair of raisin simple main effects differ. If the two caffeine simple main effects are the same, the raisin simple main effects will both be the same. In that case, each pair would confirm that there is no interaction. Conversely, if the caffeine simple main effects differ from each other, the chocolate simple main effects will differ from each other. In that case, each pair would confirm that there is an interaction.

You now probably know a lot about the 2 X 2 factorial experiment. You know that they can determine whether there is an effect for the first factor, the second factor, and whether there is an interaction effect.
You may have also figured out that these three effects are independent. For example, you can have main effects without interactions, and you can have an interaction without any main effects.
So, you may feel ready to interpret the results of factorial experiments.
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• If you feel you are ready to test your ability to decipher the results of a factorial experiment, you can test your skills here:

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