## Two-Way ANOVA

Two-way ANOVA examines the influence of different categorical independent variables on one dependent variable.

### Learning Objectives

Distinguish the two-way ANOVA from the one-way ANOVA and point out the assumptions necessary to perform the test.

### Key Takeaways

#### Key Points

• The two-way ANOVA is used when there is more than one independent variable and multiple observations for each independent variable.
• The two-way ANOVA can not only determine the main effect of contributions of each independent variable but also identifies if there is a significant interaction effect between the independent variables.
• Another term for the two-way ANOVA is a factorial ANOVA, which has fully replicated measures on two or more crossed factors.
• In a factorial design multiple independent effects are tested simultaneously.

#### Key Terms

• two-way ANOVA: an extension of the one-way ANOVA test that examines the influence of different categorical independent variables on one dependent variable
• orthogonal: statistically independent, with reference to variates
• homoscedastic: if all random variables in a sequence or vector have the same finite variance

The two-way analysis of variance (ANOVA) test is an extension of the one-way ANOVA test that examines the influence of different categorical independent variables on one dependent variable. While the one-way ANOVA measures the significant effect of one independent variable (IV), the two-way ANOVA is used when there is more than one IV and multiple observations for each IV. The two-way ANOVA can not only determine the main effect of contributions of each IV but also identifies if there is a significant interaction effect between the IVs.

### Assumptions of the Two-Way ANOVA

As with other parametric tests, we make the following assumptions when using two-way ANOVA:

• The populations from which the samples are obtained must be normally distributed.
• Sampling is done correctly. Observations for within and between groups must be independent.
• The variances among populations must be equal (homoscedastic).
• Data are interval or nominal.

### Factorial Experiments

Another term for the two-way ANOVA is a factorial ANOVA. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. Consequently, factorial designs are heavily used.

We define a factorial design as having fully replicated measures on two or more crossed factors. In a factorial design multiple independent effects are tested simultaneously. Each level of one factor is tested in combination with each level of the other(s), so the design is orthogonal. The analysis of variance aims to investigate both the independent and combined effect of each factor on the response variable. The combined effect is investigated by assessing whether there is a significant interaction between the factors.

The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors $\text{x}$, $\text{y}$, and $\text{z}$, the ANOVA model includes terms for the main effects ($\text{x}$, $\text{y}$, $\text{z}$) and terms for interactions ( $\text{xy}$, $\text{xz}$, $\text{yz}$, $\text{xyz}$). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance.

Fortunately, experience says that high order interactions are rare, and the ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results. Caution is advised when encountering interactions. One should test interaction terms first and expand the analysis beyond ANOVA if interactions are found.

Quantitative Interaction: Caution is advised when encountering interactions in a two-way ANOVA. In this graph, the binary factor $\text{A}$ and the quantitative variable $\text{X}$ interact (are non-additive) when analyzed with respect to the outcome variable $\text{Y}$.