data analysis

If you recall from Chapter 13, the ANOVA is used when there are more than two groups or levels (but one independent or treatment variable) being compared on a single dependent (measured) variable. When there is more than one independent or treatment variable and two or more groups, we use the Factorial Analysis of Variance, or a Two-Way Analysis of Variance (Two-Way ANOVA). The test statistic generated will still be the F-Ratio. These are still tests of F. Yes, it is getting deep now!

This chapter provides discussion on the use of this type of ANOVA. Two terms that are important when we make these kinds of comparisons are: Main Effects and Interaction Effects.

The chapter also provides a flowchart (decision tree) on page 263 that indicates why we would use this statistical calculation. Once again, a statistical tool is selected based on the nature of the research questions or hypotheses, the number of groups involved in the study, and nature of the data; and, in this case how many additional factors are being introduced in the study (an additional treatment, independent variable or factor related to the grouping).

Example: The text uses the example of two types of exercise programs (independent variable) on the weight loss (dependent variable) on two groups divided by gender (second independent variable). See the diagram on page 264 that reflects the research design.

The text states three research questions – that should be viewed as Null Hypotheses. Remember, we do not present research questions that result in yes/no answers.

Question 1. Is there a difference between the effects on weight loss between two levels of exercise programs: high impact vs. low impact? The Null would state what?.......That there would be no difference between the effects…..

Question 2. Is there a difference between the effects on weight loss between two categories of gender: male and female? Once again, a Null hypothesis could be inserted here. There will be no difference between the effects on weight loss…

Question 3. Is the effect of being in the high-or-low-impact program different for males or females? A Null could also be used here. Because this research question is searching for any interaction between the two factors, it would be necessary to actually state two separate Null Hypotheses. How would they be written?

Questions 1 and 2 deal with the presence of MAIN EFFECTS; whereas, Question 3 deals with the INTERACTION between two factors.

The purpose of an ANOVA is to identify differences between two or more groups cast against a dependent variable (a measured variable). This is called a MAIN EFFECT. See page 265 for Main Effects in Factorial ANOVA.

Note there is a difference that indicates men and women are affected differently across treatments – this indicates the presence of an INTERACTION EFFECT.

Based on the table on page 267, the main effect for treatment OR gender is p. = .127 and p. 176 respectively. However, the probability for the treatment by gender interaction is p.004. In other words, it did not matter whether you are in the high or low-impact treatment group, or whether you are male or female. But it does matter a great deal if you in both conditions simultaneously – such that the treatment does have a different impact on the weight loss of males.

In answer to the three research questions:

1. There is no main effect for type of exercise (high vs. low impact)

2. There is not main effect for gender

3. There is a clear interaction between treatment and gender, which means females lose more weight than males under the high-impact treatment, and males lose more weight than females under the low-impact treatment.

In these types of studies, using this particular statistic, the most important, or interesting, finding will often by the interactions.

To develop this study, or one like it, we follow 10 steps, rather than 8 (as we reviewed previously).

1. State the Null Hypotheses (in this case 3)

First for treatment (high vs. low impact)

Ho: μ high = μ low

And for Gender

Ho : μ male = μ female

And now for the interaction between treatment and gender

Ho : μ high male = μ high female = μ low male = μ low female

The research hypotheses would be presented as:

For the treatment:

H1: x̄ high ≠ x̄ low

And for gender:

H1: x̄ male ≠ x̄ female

And for the interaction between treatment and gender:

H1: x̄ hig male ≠ x̄ high female ≠ x̄ low male ≠ x̄ low female

(FYI: creating these hypotheses is a pain in the….)

2. Set the alpha level (level of risk – level of probability, level of significance) for a Type I error associated with a Null Hypothesis (p<.05)

3. Set the appropriate test statistic. Using the flowchart, we know that the Factorial ANOVA – or Two-Way ANOVA would be the best choice.

4. Compute the test statistic value (called the obtained value)

(Note the directions provided on pp. 270-271). We will be running the data using SPSS. This statistic has a few more steps in it for SPSS.

5. The additional five steps involve the entering the data in SPSS.

Key terms:

Factorial analysis of variance (Two-Way ANOVA). This ANOVA is used when there is more than one independent variable/treatment or factor.

Main Effect: An analysis of variance, which identifies a significant effect of a factor or independent variable on the dependent variable (outcome or measured variable).

Interaction Effect: The varying effect of one independent variable on the dependent variable depending on the level of a second independent variable.

Source table: a listing of sources of variance in an analysis of variance summary table. This table will be produced in SPSS output and will show the obtained F-ratio value and level of significance for each factor and the interaction.

Don’t panic over these ANOVAs. Just have an idea of when to use them and what they will tell you. It is the application of statistics for our research questions and hypotheses that really count. Formulas are not important enough to memorize – you will not have to calculate from a formula – a computer (SPSS) calculates, we think - make interpretations and draw conclusions based on what the computer tells us (our findings, from which we draw conclusions – the truth of our hypothesis).