The Factorial ANOVA (with independent factors) is kind of like the One-Way ANOVA, except now you’re dealing with more than one independent variable.

Here's an example of a Factorial ANOVA question:

Researchers want to test a new anti-anxiety medication. They measure the anxiety of 36 participants on different dosages of the medication: 0mg, 50mg, and 100mg. Participants are also divided based on what school they are attending, which researchers hypothesize will also affect anxiety levels. Anxiety is rated on a scale of 1-10, with 10 being “high anxiety” and 1 being “low anxiety”. Use alpha = 0.05 to conduct your analysis.

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Let's try a full example:

Steps for Factorial ANOVA, Two Independent Factors |
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1. Define Null and Alternative Hypotheses 2. State Alpha 3. Calculate Degrees of Freedom 4. State Decision Rule 5. Calculate Test Statistic 6. State Results 7. State Conclusion |

1. Define Null and Alternative Hypotheses

Here, we have three. One for each main effect, and one for the interaction.

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2. State Alpha

alpha = 0.05

3. Calculate Degrees of Freedom

Degrees of freedom are calculated as follows. "a" is the number of a groups you have, "b" is the number of b groups you have, and "N" is your total number of scores.

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4. State Decision Rule

We have three hypotheses, so we have three decision rules. Critical values are found using the effect and error degrees of freedom for our three effects:

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We now head to the F-table and look up our critical values using alpha = 0.05. In the table, we find the critical values shown below:

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These critical values bring us to our three decision rules:

[School] If F is greater than 4.17, reject the null hypothesis.

[Dosage] If F is greater than 3.32, reject the null hypothesis.

[Interaction] If F is greater than 3.32, reject the null hypothesis.

5. Calculate Test Statistic

First, we'll put the degrees of freedom that we've already calculated into our source table:

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Next, we need to find the five SS values we are missing:

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These SS values are then placed into our source table. We find the last missing SS value by subtracting everything we've found so far from the total.

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Each MS value is found by dividing each SS by their respective degrees of freedom:

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Finally, our three F values are found:

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6. State Results

[School] If F is greater than 4.17, reject the null hypothesis.

Our F = 4.166. Retain the null hypothesis.

[Dosage] If F is greater than 3.32, reject the null hypothesis.

Our F = 162.20. Reject the null hypothesis.

[Interaction] If F is greater than 3.32, reject the null hypothesis.

Our F = 15.89. Reject the null hypothesis.

7. State Conclusion

High school students and college students did not have significantly different anxiety levels, F(1, 30) = 4.166, p > 0.05. There was a significant difference between the three different levels of dosage, F(2, 30) = 162.20, p < 0.05. An interaction effect was also present, F(2, 30) = 15.59, p < 0.05.