Factorial Designs

Chapter 11: Factorial Designs

Study Aids and Important terms and definitions

Many of you have been inquiring about how to better prepare for the quizzes.

One way is to use the online resources that accompany the textbook.

The book website is located at:

http://www.cengage.com/cgi-wadsworth/course_products_wp.pl?fid=M20bI&product_isbn_issn=9781111342258 Links to an external site.

Select the chapter you want to view.

Any item with a symbol of a lock next to it can only be viewed by instructors.

Other, non-locked, items can be viewed by students and for each chapter there is a glossary, flash cards that you can set to view either a word or its definition first, a crossword puzzle, and a practice quiz.

You should be pleased to notice that there are many fewer vocabulary words in chapter 11 than there were in chapter 10! Chapter 11 vocabulary words and terms you should know the definition of include:

factor                                                              

factorial design

single-factor design                                                    

two-factor design                                                                               

levels                                                              

three-factor design                                         

main effect

interaction between factors or interaction

mixed design

combined strategy

higher-order factorial design

Introduction to Factorial Designs

Behavior is complicated.

People are complicated.

Most of our behavior is affected by many things.

In the simple experimental designs described in chapters 8 & 9 a researcher manipualtes one independent variable and measures the change in scores on a dependent variable.

In a factorial design the researcher can maipulate TWO OR MORE independent variables and measure their effects on an idependent variable.

Factorial designs may be experimental, nonexperimental, quasi-experimental or mixed.

We will begin the discussion with consdieration of experimetnal factorial designs.

Experimental Factorial Designs (p. 310)

When two or more independent variables and/or quasi-independent variables are combined in a single study they are called factors

                           An experiment with a single independent variable could be called a single factor experimental design

We will use the example of a study of weight loss to illustrate factorial designs.

A researche r could examine the effects of diet on weight loss - two groups of women, assign one group to eat a typical 2,000 calorie per day diet and a second group to restrict calories to 1,500 per day, and measure weight of particpants at time 1 and then after 8 weeks of dieting.

 

Or we could do a single factor experiment to examine the effect of exercise on weight loss - two groups of women, assign one to conduct their usual exercise and the second to add 30 minutes per day of aerobic activity to their usual exercise, and measure particpants' weights at time 1 and again 8 weeks after the exercise or no exercise condition begins.

 

The beauty of a factorial design is that we could use four groups to conduct a study on the individual and combined effects of calorie restriction AND exercise on weight loss

 

 This is a two-factor design. The two factors are:

1. diet
2. exercise

In factorial designs each factor is noted with a letter, so we could call diet factor A and exercise factor B

Each factor has two levels. Factor A is 1,500 or 2,000 calories and factor B is 0 or 30 minutes of aerobic exercise.

So this experiment would be described as a 2 x 2 (pronounced "two-by-two") factorial design

If we added a third exercise condition, 60 minutes of exercise, then it would be a 2 x 3 factorial design.

If instead of adding a third level we added a third factor with two levels, say taking a new diet pill or not taking a new diet pill, then it would be a 2 x 2 x 2 design.

Make sure you know how to read this sort of notation in terms of how many factos there are and how many levels of each factor there are, and how many treatment condtions there are.

To calculate treatment condtions multiply the number, for example, in a 2 x 2 design there are 4 treatment condtiions becasue 2 times 2 is four.

In a 2 x 3 design there are six treatment condtions because 2 times 3 is 6.

In a 2 x 2 x 2 design there are eight condtions becasue 2 times 2 times 2 equals 8.

So what? What is the big deal about adding conditions and combining variables into a single design?

The main advantage of a factorial design is that you can look at the effect of each variable alone, AND at the combined effects of the variables.

Main effects and interactions (p. 312)

 Considering the example of the weight loss experiment again, if we did two single factor experiments we could determine whether calories consumed affect  weight loss, and whether exercise affects weight loss.

In the 2 x 2 design we can determine if calories consumed affects weight loss, if exercise affects weight loss, AND if the combined effect of caloreis consumed and exercise is different from either intervention alone, or the same as if we just added the two condtions together.

Unlike doing two single factor experiments, in an experiment with two factors we can get three sets of information; the effect of each factor alone, and the combined effect of both factors.

Consider the weight loss example again...

 

Each column of the matrix correponds to a calorie level, either 2,000 or 1,500 and each row corresponds to an exercise level, either no extra exercise, or 30 minutes of exrtra aerobic exercise each day.

The means are noted with the letter M and each mean in this case corresponds to the number of pounds lost after eight weeks in one of the four treatment condtions.

Those who made no changes to their diet or exercise lost an average of 0 pounds.

Those who changed their diet but not their exercise lost an average of 8 pounds.

Those who increased their exercise but did not change thier diet lost an average of 6 pounds.

Those who increased thier exercise AND changed their diet lost an average of 14 pounds.

It looks like those who exercised lost weight, those who cut caloreis lost weight, and those with exercised AND cut calories lost the most weight.

Look at the first row which shows the effects of diet alone with no exercise. Particpants lost 0 pounds with no diet and 8 pound swhen dieting. Since thse number sare different, this means that there is a main effect for diet.

Next, look at the first column, this column shows both exercise condtions when there is no change in diet. The no exercise group lost an average of 0 pounds and the exercise group lost an average of 6 pounds. This means that there is a main effect for exercise. 

Determing if there is an interaction is a little more complicated.

If there is an interaction, that means that the combined effect of diet and exercise is different from what we would predict by their additive effects

• think of that old saying that, "the whole is different from the sum of its parts"
• when this is true, that means that there is an interaction

The next step is to compute row and column means...

The mean of the first colum is the average of 0 and 6, which is 3. The mean of the second column is the average of 8 and 14, which is 11.

Next we calculate the row means.

The mean of the first row is the mean of 0 and 8, which is 4. The mean of the second Row is the average of 6 and 14, which is 10.

If you subtract the column means of 11 and 3 you get 8. THis is the same number as the main effect for weight.

If you subtract the column means of 10 and 4, you get 6. This is the same as the main effect for exercise.

This means that there is no interaction between diet and exercise. Combining the weight loss strategies leads to the same weight loss as if you added the strategies together.

Now look at the matrix below

 

In this example the column means are 3 and 10 and the row means are 4 and 9. If you subtract the row means, you get 7 and not 8. And the subtracted column means are 5, not 6.

In this example, there is an interaction between diet and exercise such that the weight loss is slightly less than what would be predicted by adding together the combined main effects of the two factors.

What are the practical implications of this?

Those who combined two weight loss strategies lost more weight than either strategy alone, but not as much as we would predict by adding the weight loss of the two strategies together.

Now we can work to develop a theory about why this might be the case. Perhaps those who exercised and dieted were tired and did not exercise as vigorously, or their bodies more quickly became efficicient at conserving energy. These are the complex kinds of relationships between variables that factorial designs can help us understand.

On page 317 in your textbook an alternate way of thinking about interactions is considered.

If there is no interaction, then the variables are independent, and if there is an interaction then one of the variables influences the effects of another.

Yet another way of thinking about interactions is to graph the means at each level of each factor. If the lines are parallel, then that is an indication that there is no interaction, and if the lines are not parallel then that is an indication that there is an interaction. In a way, this si the same as the argument of independence. Parallel lines never cross and are independent. Non-parallel lines eventually cross one another suggesting that they ar enot completely independent.

A good rule of thumb

If you need to know something about a second independent variable in order to understand the first, then there is probably an interaction...

We most commonly encounter the idea of interactions when thinking of drugs and drug interactions. The effects of two drugs may be independent, that is taking the drugs together will produce the predicted effects.

If there is an interaction, the combined effect of the drug may be different than wha tyou would expect given knowledge abut the effects of each alone.

For example, many maedications interact with birth control pills such that a niave person taking a new drug and unaware of its interactions with oral contraceptives might have an unwanted pregnancy! The birth control is affected by the additional medication in a way that makes it less effective a preventing pregnancy.

Or combining alcohol with sedatives can make a person more sedated than what we might predict from merely combining the effects of the alcohol plus the effect of the sedative.

It depends...

If you have to answer with "it depends" then there is probably an interaction.

For example, if someone asks you "do you want to come to my party?" and you say, "it dpends, when is it and who is going tobe there?" This suggests that there is some interaction between the variables of other party goers and time of party. Perhaps you are inclined not to go if it is the day before your big exam - a main effect, but if certain people will be there or won't be there (also a main effect) then you might be persuaded to go even though you have an exam.

If you are still confused...

Carefully read the examples in the book.

And there are a couple of nice examples on the web page indicated below.

http://teachpsych.org/otrp/resources/factorial/Factorial3_print.html Links to an external site.

Finally, you can quiz yourself by looking at the sample questions below (you'll have to scroll down to get to them...). They are similar to some of the questions that will be on the quiz for this module.

Interpreting main effects and interaction (p. 319)

 Looking at a matrix of factor means is one way to quickly evaluate outcomes.

However, even if we see that there is an effect a statistical test will need to be done to determine if the main effect or interaction is significant.

One challenge in factorial designs is that interactions can make it hard to interpret main effects if the interaction is large enough to mask the main effect.

That siad the main effects and interactions are always independent of one another.

Independence of main effects and interactions (p. 321)

What does it mean to say that main effects and interactions are completely independent?

It means that any combination of them is possible.

In a 2 X 2 AB design there are several possible outcomes: no main effects or interaction, no main effects and an interaction, main effect for A and no interaction, main effect for B and an interaction, and so on...

You rbook provides several exmaples of the possible outcomes, and the questions below are designed to test your understanding of how to read matrices and determine if there are main effects for interactions.

 Test your understanding of main effects and interactions

The matrix – test yourself and see if you can correctly understand how to read hypothetical data from a factorial design…For the example matrix below:

 

            a.  What mean for the missing cell would result in no main effect for factor A?

            b.  What mean for the missing cell would result in no main effect for factor B?

            c.  What mean for the missing cell would result in no interaction?

           

 

                                                                         Factor B

                                                                B1                      B2

 

 

 

Answers:

a.   A mean of M = 10 would produce no main effect for factor A (both rows would average M = 25).

b.   A mean of M = 30 would produce no main effect for factor B (both columns would average M = 30).

c.   A mean of M = 50 would produce no interaction (within the matrix, there is a 20-point increase in both columns, and a 10 point increase in both rows).

 

More testing yourself:

 

For each of the 8 matrices below, indicate – by answering yes or no - whether there is a main effect for A, a main effect for B, an A x B interaction. The first one is completed already, and answers are below:

 

1. A: No    B:No   A x B: No

                                                                         Factor B

                                                                B1                      B2

 

                                                     

 

 

 

 

 

2. A:     B:     A x B:

                                                                        Factor B

                                                                B1                      B2

 

3. A:      B:       A x B:

 

                                                                         Factor B

                                                                B1                      B2

 

 

4. A:       B:      A x B:

                                                                         Factor B

                                                                B1                      B2

 

 

5. A:     B:     A X B:    

                                                                         Factor B

                                                                B1                      B2

 

 

 

 

6. A:     B:     A x B:

                                                                         Factor B

                                                                B1                      B2

 

 

7. A:     B:     A x B:

                                                                         Factor B

                                                                B1                      B2

 

 

8. A:     B:     A x B:

                                                                         Factor B

                                                                B1                      B2

 

 

 

Answers:

1. No, No, No

2. Yes, No, No

3. No, Yes, No

4. No, No, Yes

5. Yes, Yes, No

6. Yes, No, Yes

7. Yes, Yes, Yes

8. No, Yes, Yes

Common Types of Factorial Designs

Between-subjects and within-subjects designs (p. 323)

The diet example where there are four groups of particpatns in 2 x 2  condtions,  no diet no exercise, diet no exercise,  no diet exercise, and diet exercise is an example of a between subjects factorial design.

One could also do this design as a within-sibjects design where one group of particpants spend eight weeks int he no diet, no exercise condtion, then eight weeks in the diet no exercise condtion, then eight weeks in the exercise no diet condition and finally spend eight weeks in the diet and exercise condtion.

Advantages are the same as for other between-subjects designs.

A disadvantage of a between subjects design is that large numbers of partcipants are needed as the number of factors and levels of factors increases.

A between-subjects factorial design is best when there are many partcipants available, order effects might be a problem, and there is not too much concern about individual differences being a confoudning variable.

A disadvantage of within subject designs is that each particpant must be in each condition so they tend to take a long time and require a lot of particpants. For example, in a 2 x 2 design the particpant woudl be in 4 different condtions, and eight in a 2 x 4 design and so on. Attrition is a problem - many particpants will quit before they complete every condition. Order effects are also problematic as the number of conditions is high in factorial designs.

Within subject designs are best when individual differences are likely to be large and there is little concern about order effects.

Mixed designs: between- and within-subjects

Mixed designs combine two different research designs.

Mixed designs are common when one factor is best suited to a between-subjects design and another factor is best suited to a within-subjects design.

For example, on common type of mixed design is a pretest-posttest control group design.

This design is oftne use din therapy research. We might give subjects a measure of anxiety or depression before and after therapy (the within subject variable) and divide them into two groups that each receive treatment and no treatment or two different types of treatment (between subject variable).

The experimenter would hope that an nteraction would stand out in the post-test scores of one of the experimental groups

Another common mixed design is a pre-post design with multiple treatment condtions, such as no treatment control, therapy, medication, and therapy plus medication, and might determine if ther is an interactions such that therapy or medication affetc post-test scores and possibly an interaction where medication + therapy interacts to produce an outcome different than either treatment alone.

In studies of depression it has been found that at low levels of depression therapy and medication performa bout equally, that therapy produces a lower risk or relapse than medication, and that at higher levels of depression the combination of medication and therapy is superior.

Experimental and nonexperimental or Quasi-experimental factors

Factorial designs can be constructed with experimental variables, for example as in the example at the beginnign of this review where exercise and diet are controlled by the researcher.

They can also be nonexperimental, for example if a researcher examined the impact of gender of student and gender of treacher on grade earned. In this type of design gender is a quasi-experimental variable that isn't controlled by the researcher. The researcher might be interested in seeing if there is an interaction between sex of student and sex of teacher to see if boys or girls do better or worse in school when the teacher is the same or opposite sex.

And combine designs are common - all of the factors do not have to be of the same type.

Combined designs

A combined strategy is used when the factors are a mix of independent and quasi-independent variables.

For example, if instead of looking at the effect of sex of teacher on boys and girls performance (both quasi-indpendent) a researcher looked at the effect of an intervnetion on boys' and girls' academic performance, then sex would be a quasi-independent variable, and the intervention woudl be an independent (experimental) variable.

Usially the first factor is manipualted by the experimenter, such as a treatment, and the second factor is a characteristic of the particpant such as age or gender or ethnicity, or the second factor is time to determine how the effects of a treatment persist or change over time.

Higher order factorial designs

A factorial design is descrbed as a higher order factorial design when there are three or more factors.

For example, we might have an 2 x 2 x 2 or A x B x C design.

As the number of factors increases, so does the number of possible interactions, so these designs are difficult to interpret.

For example, in an A x B x C design, there are three two way itneractions, A x B, B x C, and A x C, as well as as a three-way Ax B x C interaction.

A four factor study would have six two-way, four three-way and one four-way interaction.

These are very complicated to interpret. They are more common outside of psychology

Applications of Factorial Designs (p. 329)

The heading 'applications of factorial designs' is a little misleading, I entered it above because that is how it is listed in your textbook.

It is misleading because all the rest of chapter Chapter 11 is also about applications of factorial designs.

This section covers threel situations where factorial designs are created by adding a second factor in a follow-up study to follow up on the outcome of a study that was originally done with one factor.

If I wrote the book I would have titled the section something like 'special applications of factorial designs" or "using factorial designs in follow-up research"

Expanding and replicating a previous study

After a single research study is completed it is common to do a replication and extension.

The replication is done to see if the same results are obtained again.

THe extension is done to see more detail about the outcome.

For example, imagine a researcher has done an experiment and the results show that a treatment works to decrease depression.

In a follow-up study she might add a factor to tell us more about how the treatment works.

For example, does the treatment work the same for:

• men and women
• people of different ethnicities
• People of different ages
• People with different levels of depression
• If applied as six sessions over six weeks Vs. one six hour workshop
• If delivered in a manual Vs. face-to-face therapy

The second factor in these examples would be sex or ethnicity or age of particpants, or severity of depression, or one of the two therap formats, and of course other factors could also be considered.

Adding a second factor can allow a researcher to have a deeper understanding of how a variable affects behavior since it might affect different types of people differently or affect them differently in different situations.

Reducing variability in between-subjects designs

While this sounds the same as the idea expressed above, the purpose of adding a second factor Vs reducing variance in between subject designs is slightly different.

In the former case, as described above, the purpose of the second factor is to gather more specific information in a follow-up study.

In the case of reducing variability, the researcher may find that results are non-significant in one study,a nd find that when particpants are divided into groups based on a characteristic suh as age or ethnicity the results are more clear.

The procedure is the same as 'adding a secondd factor' and the rationale for doing it is different, in the first case it is to get a better understanding of how the original independent variable operates under different condtions, in the second it  is to reduce variability of particpant groups so that it is more likely that if there is a significant difference - it will be detected.

Evaluating order effects in within-subjects designs

As was discussed in previous chapters, order effects are often regarded as a problem for researchers and they counterbalance designs and create different groups to minimize the impact of the order of treatments.

However, in some cases a researcher may be interested in order effects and treat them as an indpendent variable.

In this case, the researcher would create groups with treatment being delivered in different orders and treat order of treatments as an independent variable and see if treatment order has no effect or the order in which treatments are delivered affects the outcome.

For example, in cancer patients it might matter whether one has chemotherapy or radiation therapy first, and in treatment for depression it might matter whether one takes medication or psychotherapy first.

In sum, factorial designs give the researcher much flexibility in designing a study to answer the specific question of interest.THe results can be more refined and lead to more detailed understanding of psychological phenomena then is possible in a single factor design.