Factorial designs allow researchers to investigate the effects of two or more independent variables, called in this context factors, on one dependent variable. Also, the interaction effect of the factors can be tested.  

Research Problem

A researcher is interested in studying the effect of supervised physical exercise and medication on depression. Thirty patients participated in the study. Use different subjects for all conditions of the experiment. Notice that there are six combinations.

After the treatments were completed, the depression level of each subject was assessed. Higher scores indicate higher levels of depression. 
 

Independent Variables and dependent Variable

1. Independent Variables (Factors)

There are two factors: the Physical Exercise factor with two levels (physical exercise and no exercise) and the Medication factor with three levels (placebo, low dose and high dose). It is a 2-by-3 factorial.

 Since the data are from independent samples, the factors are between-subjects.
 

2. Dependent Variable: Depression scores

	Placebo 1	Low Dose 2	High Dose 3
Physical Exercise 1	4 3 4 3 3	1 1 1 1 2	2 3 2 2 2
No Exercise 2	5 5 5 4 5	3 4 4 4 3	3 2 3 3 2

 

SPSS for Windows

A. Enter Data

a. Create three variables: exercise, drug and score.

b. Assign descriptive labels to values: For the variable Exercise, assign 1 for Physical Exercise and 2 for No Exercise. For the variable Drug, assign 1 for Placebo, 2 for Low Dose, and 3 for High Dose.

In the "Variable View" window, you will find a column named Values. Click on the cell and click on the small grey button to define the values. The result will be like this

If you do not know how to do it, please refer to Lesson 2 (Example 2)

c. Enter data

B. Data Analysis

From the menus choose: Analyze \ General Linear Model \ Univariate.

1. Univariate (one dependent variable) : Select the dependent variable, score.

2. Select the fixed factors: exercise and drug.

2. Click the Options button. Move the two factors (exercise and drug) and the interaction term (exercise*drug) to the Display Means for box. In the Display area, select Descriptive statistics and Homogeneity tests. Click Continue.

3. Click the Plots button.

(1) To observe the effects of the Drug factor on depression at each level of the Exercise factor, move the variable "drug" to Horizontal Axis. Next, move the variable "exercise" to Separate Lines. Click Add.

(2) To observe the effects of the Exercise factor on depression at each level of the Drug factor, move the variable "exercise" to Horizontal Axis. Next, move the variable "drug" to Separate Lines. Click Add. Click Continue and OK.

SPSS Output

The analysis of interaction focuses on the cell means.

A. Find the six cell means for the experiment.

B. Visualization

a. Observe the effects of the Drug factor on depression at each level of the Exercise factor. 
 

Recall that

Physical Exercise: Connect the three means (3.4, 1.2, and 2.2).
No Exercise: Connect the three means (4.8, 3.6, and 2.6).

The two lines are not parallel.

1. For patients who did not receive supervised physical exercise, giving the drug lowered their depression levels. The higher the dose, the lower the depression.

2. However, for patients who received supervised physical exercise, it was not the case. A combination of physical exercise and low doses of the drug is particularly effective in decreasing depression level.
 

Compare the cell means.
 

	Placebo 1	Low Dose 2	High Dose 3
Physical Exercise (1)	3.4	1.2	2.2
No Exercise (2)	4.8	3.6	2.6

Comparing the cell means as (3.4 - 4.8 = -1.4), (1.2 - 3.6 = -2.4), and (2.2 - 2.6 = -0.4) shows that the differences between the cell means are not equal. Note that the effect of the Exercise factor is largest for the low dose group.

This indicates that there may be an interaction between Physical Exercise and Drug.
 

b. Observe the effects of the Exercise factor on depression at each level of the Drug factor. 

 
 

Placebo: Connect the two means (3.4 and 4.8)
Low Dose: Connect the two means (1.2 and 3.6)
High Dose: Connect the two means (2.2 and 2.6)

The Exercise factor has no effect on depression if the patients are given high doses of the drug. Notice that the slope of the line is flatter. However, exercise reduces the depression levels effectively for patients who are given low doses of a drug. Notice that the slope of the line is steeper. 
 

D. Examine the main effects


 

1. The Exercise effect

Is there a significant difference in depression scores between those who received physical exercise and those who did not?

The variation due to the Exercise factor is reflected by the two different row means.

				Row Mean
Physical Exercise (1)	3.4	1.2	2.2	2.27
No Exercise (2)	4.8	3.6	2.6	3.67

2. The Medication effect

Are there significant differences in depression scores among the three groups of patients who take different doses of a drug?

The variation due to the Medication factor is reflected by the three different column means.

	Placebo 1	Low Dose 2	High Dose 3
	3.4	1.2	2.2
	4.8	3.6	2.6
Column Mean	4.1	2.4	2.4

 

E. Tests of significance

The conclusions are that there is a significant interaction between the medication factor and exercise factor (F_(2,24) = 10, p = .001) and that the main effects of both the physical exercise (F_(1,24) = 58.8), p < .001)  and the medication (F_(2,24) = 38.53, p < .001) are significant.  
 

Simple Effects (Simple Main Effects)

Since there was a significant interaction, follow-up tests are needed. A simple effect is the effect of one factor within one level of the other factor.

For example, we may want to know the effect of the exercise factor  for individuals within the placebo condition.

For example, we may want to know the effect of the exercise factor  for individuals within the low dose condition.

For example, we may want to know the effect of the exercise factor  for individuals within the high dose condition.

You will learn how to perform simple effects analysis next.