# R Tutorial Series: Two-Way Omnibus ANOVA

January 17, 2011
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As with the one-way case, testing the omnibus hypothesis via two-way ANOVA is simple process in R. This tutorial will explore how R can be used to perform a two-way ANOVA to test the difference between two (or more) group means.

### Tutorial Files

Before we begin, you may want to download the sample data (.csv) used in this tutorial. Be sure to right-click and save the file to your R working directory. This dataset contains a hypothetical sample of 60 participants who are divided by gender (male and female) and treatment group (control and treatment). The values represent a scale that ranges from 1 to 5. For instance, this dataset could be conceptualized as a comparison between two professional training programs, where the control group participated the company’s longstanding program and the treatment group participated in an experimental program. The values could represent the attitudes of employees towards the training programs on a scale from 1 (poor) to 5 (excellent).

### Beginning Steps

To begin, we need to read our dataset into R and store its contents in a variable.

1. > #read the two-way ANOVA dataset into an R variable using the read.csv(file) function
3. > #display the data
4. > dataTwoWay
The first ten rows of our two-way ANOVA dataset.

### Two-Way ANOVA

Now that our data are ready, we can conduct a two-way omnibus ANOVA test using the anova(object) function. Note that the only step necessary to add a second independent variable into our ANOVA model is to incorporate it into our lm(model, dataset) function using the * operator. Whereas our one-way model was lm(Values ~ Group), our two-way model becomes lm(Values ~ Group * Gender). As you can see from the results below, adding a second independent variable in this manner also gives us information about the interaction between our variables.

1. > #use anova(object) to test the omnibus hypothesis in two-way ANOVA
2. > #Are the differences between the group means for treatment and gender statistically significant?
3. > #Is there a statistically significant interaction between treatment and gender?
4. > anova(lm(Values ~ Group * Gender, dataTwoWay))
Our two-way ANOVA table.

The output of our ANOVA test indicates that the difference between our treatment group means is statistically significant (p < .001) and that the difference between genders is not (p = .585). However, in light of the statistically significant interaction between treatment group and gender (p = .032), we would generally elect to forgo these main effects. Subsequently, a series of follow-up procedures could be carried out to examine the simple main effects for treatment group and gender.

### Two-Way Multiple Group ANOVA

Conducting a two-way omnibus ANOVA with multiple groups is identical to the demonstrated two-group test. The only difference is that the values in your dataset would be associated with more than two groups. Subsequently, the omnibus hypothesis would test for mean differences across all of the groups. The anova(object) function and its contained lm(formula, data) function would remain the same.

### Complete Two-Way Omnibus ANOVA Example

To see a complete example of how a two-way omnibus ANOVA can be conducted in R, please download the two-way ANOVA example (.txt) file.

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