Factorial Designs

 

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

 

A Two by Two Factorial Design

Experimental Conditions

Consider an experiment with a single control and a single experimental group, outlined in the table below.

 

 

CONTROL

 

EXPERIMENTAL

A MALE

3

F MALE

6

B MALE

1

G MALE

7

C FEMALE

5

H FEMALE

5

D FEMALE

2

I FEMALE

4

E FEMALE

4

J FEMALE

8

M

3

M

6

2

2

 

 The means for the control and experimental groups are

 

 

Mean

CONTROL

(3+1+5+2+4)/5 = 3

EXPERIMENTAL

(6+7+5+4+8)/5 = 6

 

The mean difference between the experimental conditions is -3. The question to answer is whether there is a significant difference between the two experimental conditions.

Gender

Taking the gender of subjects into consideration the subjects' scores can be rearranged as

 

 

MALE

FEMALE 

CONTROL

3, 1

5, 2, 4

EXPERIMENTAL

6, 7

5, 4, 8

 

The means for men and women are

 

MALE

FEMALE 

(3+1+6+7)/4= 4.25

(5+2+4+5+4+8)/6 = 4.67

 

The mean difference between men and women is -.42. The question to answer is whether there is a significant difference between males and females.

Combinations

What happened when the two factors, experimental conditions and gender, vary simultaneously?

There are two experimental conditions (control and experimental groups) and the variable gender has two levels (male and female).

 

Condition

Gendert

Combinations

C

M

C M

F

C F

E

M

E M

F

E F

 

Thus, there will be four possible combinations (2*2 = 4). This type of design is also called a two-by-two factorial design.

 

 

MALE

FEMALE 

CONTROL

1, 3

5, 2, 4

EXPERIMENTAL

6, 7

5, 4, 8

 

The mean of each cell of the above table is shown below

 

 

MALE

FEMALE

CONTROL

2.00

3.67

EXPERIMENTAL

6.50

5.67

   

Interactions

The factorial design facilitates the study of interactions. The primary question to answer is whether there is a significant interaction between the experimental conditions and gender.

Do differences in population means between experimental and control groups differ for men and women?  

Compare Cell Means

For our example, the experiment increased the average performance of males from 2 to 6.5 

 

Condition

Gender

Mean

C

M

2
   

E

M

6.50
   

 

and the average performance of females increased from 3.67 to 5.67.

 

Condition

Gender

Mean

C

   

F

3.67

E

   

F

5.67

 

Data Visualization

Select the experimental conditions as the horizontal axis. Observe the effects of the experimental conditions on performance of a task for males and females.

 

 

The effect of treatment for males: connect the following  two means, 2 and 6.5.
The effect of treatment for females: connect the following two means, 3.67 and 5.67.

It seems that the experiment increased the average performance of males slightly more. Notice that the slope of the line representing males is slightly steeper. However, the interaction, suggested by the crossing lines, is only slight and might not be statistically significant. 

Tests of Significance

If the interaction effect is significant, we generally do not interpret main effects (e.g. the experimental effect or the gender effect). If the interaction effect is not significant, next question to answer would be whether there are significant main effects.

Advantages

The advantage of the factorial design is that it permits the study of interactions. Also, addition of new independent variables results in smaller values of the residual variance, making the significance tests more sensitive. For example, we may partition the total variance into four components as shown below.

 

 

Orthogonal Coding of the Factorial Designs

Since the number of observations in our example is not equal, the construction of the orthogonal vectors is more complicated. For the example, the factorial design can be coded on the basis of the following considerations.

Experimental Conditions

Let us construct the first coding vector, X1, to contrast the experimental and control groups. The sums of values of the orthogonal coding vectors have to equal zero; the simplest numbers to fulfill this requirement are 1 and -1. Also, sums of products of orthogonal coding vectors must sum to zero.

Gender

Next, construct the second vector, X2, to contrast the male and female groups. As you have two males and three females in each group you will need two plus 3s and three minus 2s to satisfy the second requirement.

Interaction

Finally, to create the vectors representing the interaction, each vector from the first factor is multiplied by each of the vectors from the second  factor.

Use the product of X1 and X2 as X3 to code the interaction, as

 

 

The table of coefficients of correlation for the above data is

   

 

The table of coefficients of determination is

 

 

Since the coding vectors are orthogonal, the coefficients of determination of predictor variables with the criterion are additive and the summary table for the analysis of variance can be constructed from the right-hand column of the above table as

 

 

Source of

Variance

 

 

Df

Standard

 Variance

 Components

 

 

F

 

 

Probability

X1

1

.529

8.53

.03

X2

1

.0098

0.16

.99

X3

1

.088

1.42

.28

Explained

3

.627

   

Alienation

6

.373

 

 

Total

9

1.00

 

 

 

The F ratio for the experimental effect can be computed as 

The F ratio for the gender effect can be computed as

The F ratio for the interaction effect can be computed as

 

The results indicated a significant main effect for the experimental treatments, F(1,6) = 8.53, p = .03, eta square = .53, a nonsignificant main effect for gender, F(1,6) = .16, p = .99, eta square = .01, and a nonsignificant interaction between treatments and gender, F(1,6) = 1.42, p = .28, eta square = .09.

The standard variance components can be graphed as shown below.

 

 

The above solution may be compared with the traditional solution for the analysis of variance, summarized as

 

Source of

Variance

 

Df

Sum of

Squares

Mean

Square

 

F

 

Probability

X1

1

22.50

22.50

8.53

.03

X2

1

.42

.42

.16

.99

Interaction

1

3.75

3.75

1.42

.28

Explained

3

26.67

8.89

3.37

.10

Residual

6

15.83

2.64

 

 

Total

9

42.50

4.72

 

 

 

Dividing each entry in the sum of squares column by the total sum of squares, one may observe that both solutions are identical. From the obtained results one may conclude that the only statistically significant difference is the difference between the means of the experimental and control groups, irrespective of the gender of subjects. About 53% of the variance in Y is accounted for by the experimental conditions.

 

Interactions

To explain the concept of interactions, let us use another example,

 

Decaffeinated

Coffee

 

Y0

Regular

Coffee

 

Y1

Allen

1

Allen

1+2

Becky

2

Becky

2+2

Cathy

3

Cathy

3+2

M

2

M

4

.67

.67

 

illustrated as

 

In the above example, a group of students is studying for examinations and researcher is recording how many hours past midnight they stay awake. Our research pertains to the question is whether drinking regular instead of decaffeinated coffee increases the number of hours subjects will study past midnight. The subjects, none of them being a regular coffee drinker, were given one cup of decaffeinated coffee at 11:30 P.M. and observed until they fell asleep. The following week, at the same day of the week and the same time, the same subjects were given a cup of regular coffee and, keeping all situational factors the same, observed at what time they retired to bed. The dependent variable was the number of hours past midnight subjects stayed awake.

To illustrate an interaction, as shown below

 

Decaffeinated

Coffee

 

Y0

Regular

Coffee

 

Y1

Allen

1

Allen

1+2

Becky

2

Becky

2+2

Cathy

3

Cathy

3- 2

M

2

M

4

.67

.67

 

 

this hypothetical experiment was modified as for Cathy to have a paradoxical reaction to a stimulant. Instead of helping her to staying awake, drinking a cup of coffee drove her to sleep. In general, the concept of interaction means that some subjects react to the experiment differently than the others. For example, within the area of educational research, the often-heard phrase used to be 'the aptitude-treatment interaction.' In plain language that meant that instruction was to be tailored to needs of different groups of students.

 

Interactions within Factorial Design

Let us return to the experiment discussed at the beginning of this chapter with the obtained means on the dependent variable computed separately for the control and the experimental group in the table below.

 

 

CONTROL (1)

 

EXPERIMENTAL (2)

A MALE

3

F MALE

6

B MALE

1

G MALE

7

C FEMALE

5

H FEMALE

5

D FEMALE

2

I FEMALE

4

E FEMALE

4

J FEMALE

8

M

3

M

6

2

2

 

As shown in the diagram below, the experimental treatment increased the average performance of subjects from 3 to 6.

 

 

When the experiment was conceptualized as a factorial design, the average performance of different subgroups of subjects within the control and experimental conditions is shown in the table below.

 

 

MALE

FEMALE

CONTROL (1)

2.00

3.67

EXPERIMENTAL (2)

6.50

5.67

 

and the computed means are plotted in the diagram that follows.

 

 

The experiment increased the average performance of males from 2 to 6.5. The average performance of females increased from 3.67 to 5.67. The interaction, suggested by the crossing lines in the above diagram, was only slight and did not appear to be statistically significant.