Chapter 11, Problem 1E

In addition to the key words, you should also be able to define the following terms:

Two-factor design

Single-factor design

Levels

Three-factor design

Higher-order factorial design

To determine

The meaning of the terms used in factorial designs:

Answer to Problem 1E

Solution:

Factorial design: It is research design, which involves experimenting with different independent factors. There can be two or more factors in a factorial design. There are designs with one factor as well. Each factor may have discrete values. Each factor is tested at different values, thus providing multiple combinations of different factors. This allows the researcher to study the factor at all possible values and it also lets the researcher know about the interaction effect of the factors on the response variable.

Explanation of Solution

Factorial designs: These are the experimental designs, that allow to experiment with multiple independent variables, and also study the interaction effect of these independent variables when they influence the response variable simultaneously.

There are various terms used in the Factorial Designs. They are described as follows:
Level: Each factor has discrete possible values, or it can be subdivided into different categories. These discrete values or categories are called level of the factor. For example if the factor is gender: it has two levels: male and female. If the factor is income level ; it can be categorized as ; high, medium and low. Thus it has three levels. Like this every factor can be categorized in different levels. These levels in a factorial design are used to define the design. For example if it is said that the design is a 2X3 factorial design, this implies the first factor has 2 levels and the second factor has 3 levels. Similarly if the design is: 4X3X5 this implies that there are three factors in this design. First factor has 4 levels, second factor has 3 levels and the third factor has 5 levels.

Two factor design: The factorial design two factors is called a two factorial design. These two factors may have any number of discrete values to be studied. For example we can study average hours of sleep needed by males and females of different age group. In this study 'hour of sleep' is the response variable. 'gender' is one independent variable. It can be divided into two groups: males and females. Thus it has two levels. The second independent variable in this study is age. We are free to define the levels for it. Let's us say people are divided based on age in three groups: teenagers, Adults and seniors. Thus the second variable has 2 levels.

Single factor design: In this factorial design there is just one factor under which the response variable is studied. This one independent factor can be divided into any number of levels. Since there is only one factor, there cannot be any interaction effect to be studied in this specific factorial design. For example: Suppose the research is conducted to observe the learning power under different teaching techniques. Here the response variable is learning power. The independent factor is teaching techniques. Now the different types of teaching techniques can be: Audio, Video, Mentor. Thus there are three levels of the teaching techniques.

Three factor design: It is type of factorial design which has three independent variables to be studied together. Their main effects and the interaction effects can be studied together. We can consider the example where we study the impact of : IQ level, hours of study, Teaching techniques on the academic performance of students. Thus we got three independent variables here. Each variable can have different levels. IQ level can be divided as: High IQ level, average IQ level and Low IQ level. Hours of study can be categorized as : less than 2 hours a day, 2 to 4 hours a day, 4 to 6 hours and more than 6 hours. Teaching techniques can be divided into 3 levels. Thus the overall structure of the study can be described as follows:

1. Factor I: 3 levels
2. Factor II: 4 levels
3. Factor III: 3 levels

Thus this design can be considered to be a complex 3 factorial design.

Higher order factorial design: this design involves more than 3 factors. These designs may involve 4 or five or more factors and help is studying complex functioning of the entire model, observing how all the factors individually and together influence the response variable. These are one of the most complex research designs.

For example if the aim is to study the average spending on shopping in a country, the following independent factors can be involved to thoroughly study the same:

1. Age
2. Income level
3. Education level
4. City
5. Gender
6. Occupation type

Thus we can see that this would be a detailed design with 6 different factors involved in the study. Such a design is called higher order factorial design.

Conclusion:

Thus we get to know that in a factorial design, the researcher is free to include as many factors in the design to make it more realistic and study the main effect of all the factors and simultaneously study the interaction effects as well. Each factor may have different levels. These levels are used to define the factorial design.