# Three-Way ANOVA: Definition

## What is a Three-Way ANOVA?

The terms “three-way”, “two-way” or “one-way” in ANOVA refer to how many factors are in your test. A three-way ANOVA (also called a three-factor ANOVA) has three factors (independent variables) and one dependent variable. For example, time spent studying, prior knowledge, and hours of sleep are factors that affect how well you do on a test. For comparison, a two-way ANOVA has two factors (e.g. time spent studying and prior knowledge) and one dependent variable. Four-way ANOVA and above are rarely used because the results of the test are complex and difficult to interpret.

## More on Factors and Levels in a Three-Way ANOVA

Let’s say you wanted to find out if there is an interaction between income, age, and gender for how much anxiety job applicants experience at job interviews. The amount of anxiety is the outcome, or the variable that can be measured. The categorical variables Gender, Age, and Income are the three factors.

Factors can also be split into levels. In the above example, income could be split into three levels: low, middle and high income. Age could be split into multiple levels (e.g. under 30, 30-50, over 50). Gender could be split into three levels: male, female, and transgender. If you’re working with treatment groups, you’ll want to include all possible combinations of all factors. In this example there would be 3 x 3 x 3 = 27 treatment groups.

## Calculations — Why to Use Software

Three-way ANOVA uses the same basic procedure from two-way ANOVA; the same formulas to calculate the sum of squares (SS) are used, only in greater amounts. This sounds simple in theory, but in practice you’ll want to use software for a three-way ANOVA— especially if your design is unbalanced. Reasons to use software in general include:

• Multiple calculations are required for multiple process steps, including eight different sums of squares.
• Finding SS for two-way interactions in a three-way design can be “tricky” (Cohen, 2008),
• Multiple F ratios* have to be calculated; Each ratio then has to be hypothesis tested for significance.

* F value = variance of the group means (Mean Square Between) / mean of the within group variances (Mean Squared Error).

A two-way uses four SS while the three-way uses eight: SSA, SSB, SSC, SSAB, SSAC, SSBC, SSABC and SSW (Cohen, 2008). A,B, and C represent the three independent variables. SSW (sum of squares within) is calculated, either by:

• Multiply dfW (degrees of freedom within*) by the mean of the cell variances, or
• Subtract SSABC from SStotal.

This is only the start of the calculations. Several other SS components have to be calculated:

• SSA × B = SSAB − SSA − SSB
• SSA × C = SSAC − SSA − SSC
• SSB × C = SSBC − SSB − SSC
• SSA × B × C = SSABC − SSA × B − SSB × C − SSA × C − SSA − SSB − SSC

*Degrees of freedom within (dfw) is equal to N-K. N is the total number of observations in all groups and K is the numbers of trials or tests in all groups.

## References

Cohen, B. (2008). Explaining Psychological Statistics. John Wiley & Sons.
Thorne, M. & Giesen, M. Statistics for the Behavioral Sciences, 4/e.
William, R. (2004). One way analysis of variance. Retrieved March 9, 2018 from: https://www3.nd.edu/~rwilliam/stats1/x52.pdf

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