Factorial Designs

In the experimental designs a single experimental variable with usually only one level was considered. It is possible to test several levels of the experimental variables for example several different ads could be tested, each with a separate experimental group. All but group alternately could be considered as control groups against which to compare the experimental group, or an additional control group not exposed to any advertising could be used to protect against possible negative effects of all ads.

Factorial designs permit the experimenter to test two or more variables at the same time and not only to determine the main effects of each of the variables, but also to measure the interaction effects of the variables. Consider the problem of determining the proper concentration of sugar and flavor in a soft drink. A simple approach would be to make up a batch of the optimum mixture as judged by the producer and to have a sample of consumers taste it and completing products and indicate an order of preferences. The consumers might even be asked to comment on the degree of sugar and flavor in a soft drink. Another approach would be to make up several batches with differing levels of sugar content, but with the flavor held constant. Consumers could then taste a sample of each and indicate a preference. Sugar could then be held constant and flavor varied.

The latter approach might indicate that heavy sugar and heavy flavor were both preferred, but a product with such a mixture might turn out to be unpalatable. When the flavor is strong, sugar may become less desirable. Such considerations make it important to test various levels of sugar content combined with various levels of flavor. Suppose four different degrees of sugar content and four of flavor were selected as possible characteristics of the final product. Sixteen different combinations cam be made from these variations, as shown in the following table below:

Flavor Intensity Sugar Content

1 2 3 4
1 a b c d
2 e f g h
3 I j k l
4 m n o p

Each of the 16 formula variations (a to p) can be given to a sample of consumers and their reactions measured on various bases for example, a preference scale from 1 to 10. The following hypothetical data illustrate results that might be obtained.

Flavor Intensity Sugar Content

1 2 3 4

1 4.9 6.0 5.0 3.6
2 6.1 7.3 5.1 3.8
3 8.1 9.2 8.3 4.6
4 6.2 6.4 6.2 3.2

The second degree of sugar content and third degree of flavor intensity are each preferred over all levels of their own variables, no matter what the level of the other variable. The combination of these two is the preferred product formula; that is, it has the highest preference rating of 9.2. The combination of the fourth level of each of the variables is the least preferred product formula: its preferences rating is 3: 2

In the above example, each of the two variables was tested at four different levels. Actually the number of levels for each variable is determined by the thoroughness with which the experimenter wishes to study the problem, the range over which it is considered useful to study the variable, the degree of change necessary to make a discernible differences to the consumers and the cost.

While two variables were considered in the factorial design above, it is possible to test three or more variables. Assume that color was a third factor that might influence consumers’ preferences of a drink. Four different colors could be tested, but to include each color variation with each possible combination of sugar and flavor would require 64 different cells and would make the experiment an expensive one. To economize in situations of this sort a variation of the factorial design has been developed the Latin Square.