# Conduct and Interpret a Factorial ANOVA

What is the Factorial ANOVA?

ANOVA is short for ANalysis Of Variance.  As discussed in the chapter on the one-way ANOVA the main purpose of a one-way ANOVA is to test if two or more groups differ from each other significantly in one or more characteristics.  A factorial ANOVA compares means across two or more independent variables.  Again, a one-way ANOVA has one independent variable that splits the sample into two or more groups, whereas the factorial ANOVA has two or more independent variables that split the sample in four or more groups.  The simplest case of a factorial ANOVA uses two binary variables as independent variables, thus creating four groups within the sample.

For some statisticians, the factorial ANOVA doesn’t only compare differences but also assumes a cause- effect relationship; this infers that one or more independent, controlled variables (the factors) cause the significant difference of one or more characteristics.  The way this works is that the factors sort the data points into one of the groups, causing the difference in the mean value of the groups.

 Independent Variables 1 2+ Dependent Variables 1 One-way ANOVA Factorial ANOVA 2+ Multiple ANOVAs MANOVA

Example: Let us claim that blonde women have on average longer hair than brunette women as well as men of all hair colors.  We find 100 undergraduate students and measure the length of their hair.  A conservative statistician would then state that we measured the hair of 50 female (25 blondes, 25 brunettes) and 25 male students, and we conducted an analysis of variance and found that the average hair of blonde female undergraduate students was significantly longer than the hair of their fellow students.  A more aggressive statistician would claim that gender and hair color have a direct influence on the length of a person’s hair.

Most statisticians fall into the second category.  It is generally assumed that the factorial ANOVA is an ‘analysis of dependencies.’It is referred to as such because it tests to prove an assumed cause-effect relationship between the two or more independent variables and the dependent variables.  In more statistical terms it tests the effect of one or more independent variables on one dependent variable.  It assumes an effect of Y = f(x1, x2, x3, … xn).

The factorial ANOVA is closely related to both the one-way ANOVA (which we already discussed) and the MANOVA (Multivariate Analysis of Variance).  Whereas the factorial ANOVAs can have one or more independent variables, the one-way ANOVA always has only one dependent variable.  On the other hand, the MANOVA can have two or more dependent variables.

The table helps to quickly identify the right Analysis of Variance to choose in different scenarios.  The factorial ANOVA should be used when the research question asks for the influence of two or more independent variables on one dependent variable.

Examples of typical questions that are answered by the ANOVA are as follows:

• Medicine – Does a drug work? Does the average life expectancy differ significantly between the 3 groups x 2 groups that got the drug versus the established product versus the control and for a high dose versus a low dose?
• Sociology – Are rich people living in the country side happier? Do different income classes report a significantly different satisfaction with life also comparing for living in urban versus suburban versus rural areas?
• Management Studies – Which brands from the BCG matrix have a higher customer loyalty? The BCG matrix measures brands in a brand portfolio with their business growth rate (high versus low) and their market share (high versus low).  To which brand are customers more loyal – stars, cash cows, dogs, or question marks?

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The Factorial ANOVA in SPSS

Our research question for the Factorial ANOVA in SPSS is as follows:

Do gender and passing the exam have an influence how well a student scored on the standardized math test?

This question indicates that the dependent variable is the score achieved on the standardized math tests and the two independent variables are gender and the outcome of the final exam (pass or fail).

The factorial ANOVA is part of the SPSS GLM procedures, which are found in the menu Analyze/General Linear Model/Univariate.  In the GLM procedure dialog we specify our full-factorial model.  Dependent variable is Math Test with Independent variables Exam and Gender.

The dialog box Post Hoc tests is used to conduct a separate comparison between factor levels.  This is useful if the factorial ANOVA includes factors that have more than two factor levels.  In our case we included two factors of which each has only two levels.  The factorial ANOVA tests the null hypothesis that all means are the same.  Thus the ANOVA itself does not tell which of the means in our design are different, or if indeed they are different.  In order to do this, post hoc tests would be needed.  If you want to include post hocs a good test to use is the Student-Newman-Keuls test (or short S-N-K).  The SNK pools the groups that do not differ significantly from each other.  Therefore it improves the reliability of the post hoc comparison by increasing the sample size used in the comparison.  Another advantage is that it is simple to interpret.  The Options dialog allows us to add descriptive statistics, the Levene Test and the practical significance (estimated effect size) to the output and also the mean comparisons.

The Contrast dialog in the GLM procedure model us to group multiple groups into one and test the average mean of the two groups against our third group.  Please note that the contrast is not always the mean of the pooled groups! Contrast = (mean first group + mean second group)/2.  It is only equal to the pooled mean if the groups are of equal size.  In our example we do without contrasts.

And finally the dialog Plots… allows us to add profile plots for the main and interaction effects to our factorial ANOVA.

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