Factorial Design

Factorial experiments permit researchers to study behavior under conditions in which independent variables, called in this context factors, are varied simultaneously. Thus, researchers can investigate the joint effect of two or more factors on a dependent variable. The factorial design also facilitates the study of interactions, illuminating the effects of different conditions of the experiment on the identifiable subgroups of subjects participating in the experiment.

A Two by Two Factorial Design

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

 

 

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

 

 

The mean of each cell of the above table was is

 

 

The question to answer is whether there are statistically significant differences between the means shown in the above table. This type of design is also called two-by-two factorial design. Here factors really mean predictor variables, i.e., for the example, gender of the subjects and their membership either in the experimental or in the control group. The question to be answered is whether there is a significant difference in mean performance on the criterion (dependent) variable Y. The advantage of the factorial design is that it permits the study of interactions. Also, addition of new predictor variables typically results in smaller values of the residual variance, making the significance tests more sensitive.

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. Let us construct the first coding vector 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. 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. Use the X₁X₂ as X₃ to code the interaction, as

 

 

 

The table of coefficient of correlations for the above data 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

 

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

 

 

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.

Interactions

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

 

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

 

 

 

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.

 

 

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.

 

 

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 is only slight and did not shown to be statistically significant.