Research Methods - Two-Way ANOVA

Research Methods

The Two-Way Analysis of Variance (ANOVA)

Class Interactive Portion

Problem Description 1

As you read the following story, identify the dependent variable and the two independent variables (factors).

The term was over. A dozen students who had all been in the same class were talking about what a bear of a test the comprehensive final exam had been. Competition surfaced.

“Of course, I was ready. Those of us who major in natural science disciplines just study more than you humanities types.”
“Poot, poot, and balderdash,” exclaimed one of the humanities types, who was a sophomore.

A third student said, “Well, there are some of both types in this group; let's just gather some data. How many hours did each of you study for that final?”
“Wait a minute,” exclaimed one of the younger students. “I'm a natural science type, but this is my first term in college. I didn't realize how much time I would need to cover the readings. I don't want my score to pull down my group's tally.”

“Hmmm,” mused the data-oriented student. “Let's see. Look, some of you are in your first term and the rest of us have had some college experience. We'll just take experience into account. Maybe the dozen of us will divide up evenly.”

This story establishes the conditions for a 2 x 2 factorial ANOVA. One of the factors is area of interest; the two levels are natural science and humanities. The other factor is previous experience in college; the two levels are none and some. The dependent variable is hours of study for a final examination.

Problem Description 2

Here's a question for you. Read the question, take 10 seconds, and compose an answer. Who is taller, boys or girls? A halfway good answer is “It depends.” A completely good answer gives the other variable that the answer depends on; for example, “It depends on the age of the boys and girls.”

Factorial ANOVA designs are often used when a researcher thinks that the answer to a question about the effect of variable A is “It depends on variable B.” As you may have already anticipated, the way to express this idea statistically is to talk about significant interactions.

The analysis that follows is for a 2 X 3 factorial design. The factor with two levels is gender: males and females. The second factor is age: 6, 12, and 18 years old. The dependent variable is height in inches.

I've constructed data that mirror fairly closely the actual situation for Americans (Berk, 1998).

For a bit of help on the results, click here.

Problem Description 3

The conditions that make for happy people have been researched by psychologists for a long time. The data that follow are based on a review of the literature by Diener and colleagues (1999). Participants were asked their marital status and how often they engaged in religious behavior. They also indicated how happy they were on a scale of 1 to 10. Analyze the data with a factorial ANOVA and, if appropriate, Tukey_HSD tests.

Problem Description 4

A popularly held belief about university professors is that they don’t work very hard, and that the higher their rank, the less work they do. A statistics student decided to determine whether the belief is true. She took a random sample of 20 university instructors in each of the faculties of business, engineering, arts, and sciences. In each sample of 20, five were instructors, five were assistant professors, five were associate professors, and five were full professors. Each professor was surveyed and asked to report confidentially the number of weekly hours of work. What do the test results tell you?

Problem Description 5

Each year billions of dollars are lost because of worker injuries on the job. Costs can be decreased if injured workers can be rehabilitated quickly. As part of an analysis of the amount of time taken for workers to return to work, a sample of male blue-collar workers aged 35 to 45 who suffered a common wrist fracture was taken. The researchers believed that the mental and physical condition of the individual affects recovery time. Each man was given a questionnaire to complete, which measured whether he tended to be optimistic or pessimistic. Physical condition was also evaluated and categorized as very physically fit, average, or in poor condition. The number of days until the wrist returned to full function was measured for each individual. These data are stored in the file.

--What are the factors in this experiment? What are the levels of each factor?
--Can we conclude that pessimists and optimists differ in their recovery times?
--Can we conclude that physical condition affects recovery times?

Problem Description 6

At PBA, a study was undertaken to investigate whether different training programs and software packages offered by the school were more effective than others. The study recorded the number of words per minute typed by six groups of 40 students who completed the training programs. The training program and software packages assigned to each group are as described below.

                Group 1: hands-on training/MS Word software
                Group 2: computer tutorial/MS Word software
                Group 3: hands-on training/WordPerfect software
                Group 4: computer tutorial/WordPerfect software
                Group 5: hands-on training/AmiPro software
                Group 6: computer tutorial/AmiPro software

The typing speeds for each student were recorded and placed in the data file. Can we conclude that the typing speeds differ among the six groups of students?

Problem Description 7

We dealt with a one-way analysis of variance in which we had only one independent variable. Here, we’ve extended the analysis of variance to the treatment of experimental designs involving two or more independent variables. For purposes of simplicity, we will be concerned primarily with experiments involving two variables, although the extension to more complex designs should be apparent.

Eysenck’s study was pretty complex. He was interested in whether level-of-processing notions could explain differences in recall between older and younger subjects. If older subjects do not process information as deeply, they might be expected to recall fewer items than would younger subjects, especially in those conditions that entail greater processing. This study now has two independent variables, which we shall refer to as factors: Age and Recall Condition. The experiment thus is an instance of what is called a two-way factorial design.

An experimental design in which every level of every factor is paired with every level of every other factor is called a factorial design. In other words, a factorial design is one in which we include all combinations of the levels of the independent variables. In the factorial designs discussed in our class, we will consider only the case in which different subjects serve under each of the treatment combinations. For instance, in our example, one group of younger subjects will serve in the counting condition, a different group of younger subjects will serve in the rhyming condition, and so on. However, for now we will look at a 2 X 2 factorial design. Later, since we have 10 combinations of our two factors (5-Recall Conditions x 2-Ages), we would have 10 different groups of subjects. When the research plan calls for the same subject to be included under more than one treatment combination, we will speak of repeated-measures designs.

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