Two-Way Analysis of Variance (ANOVA): What It Is, What It Tells You, vs. One-Way

What Is Two-Way Analysis of Variance (ANOVA)?

A two-way analysis of variance (ANOVA) is a statistical test used to determine the effect of two nominal predictor variables (two independent variables) on a single, continuous outcome variable (dependent variable).

A two-way ANOVA analyzes the main effect of each of the independent variables on the expected outcome, in addition to the variables' relationship to each other—if there is any interaction between the independent variables. Random factors would be considered to have no statistical influence on a data set, while systematic factors would be considered to have statistical significance.

By using analysis of variance (ANOVA), a researcher can determine whether the variability of the outcomes is due to chance or the factors in the analysis. ANOVA is a hypothesis-based test; its goal is to evaluate multiple mutually exclusive theories about a data set. ANOVA is a key statistical test in many research fields and has applications in finance, economics, biology, and psychology.

Understanding Two-Way Analysis of Variance (ANOVA)

A two-way analysis of variance (ANOVA) is a variation of analysis of variance (ANOVA). ANOVA is a method of analysis used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors influence the given data set, while the random factors do not.

ANOVA allows comparisons to be made between three or more groups of data. An ANOVA test is the first step in identifying factors influencing a given outcome—in other words, the effects of variables on one another. It groups differences by comparing the means of each group and includes spreading out the variance across diverse sources. It is employed with subjects, test groups, between groups, and within groups.

Once an analysis of variance test is performed, an analyst may be able to perform further analysis on the systematic factors that are statistically contributing to the data set's variability.

A two-way ANOVA test reveals the results of two independent variables on a dependent variable. ANOVA test results can then be used in an F-test. (An F-test is any statistical test used to compare the variances of two samples or the ratio of variances between multiple samples.)

One-Way ANOVA vs. Two-Way ANOVA

There are two main types of analysis of variance tests: one-way (or unidirectional) and two-way (bidirectional). The designations of "one-way" or "two-way" refers to the number of independent variables in your analysis of variance test. A one-way analysis of variance test evaluates the impact of a single factor on a single response variable. It determines whether the observed differences between the means of independent (unrelated) groups are explainable by chance alone—or if there are any statistically significant differences between groups.

A two-way analysis of variance test is an extension of a one-way analysis of variance test. With a one-way test, you have one independent variable affecting a dependent variable. For example, a one-way analysis of variable allows a company to examine how the level of employee training impacts customer satisfaction ratings. 

With a two-way test, there are two independent variables. For example, a two-way analysis of variance test allows a company to compare worker productivity based on two independent variables, such as department and gender. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time.

A three-way ANOVA, also known as three-factor ANOVA, is a statistical means of determining the effect of three factors on an outcome.

Analysis of Variance (ANOVA) Formula

History of Analysis of Variance (ANOVA)

Analysis of variance was developed by the British statistician Sir R.A. Fisher in the 20th century. Although aspects of this technique were introduced earlier, analysis of variance gained recognition after it was included in Fisher's book, "Statistical Methods for Research Workers," in 1925, one of the most influential books on statistical methods in the 20th century.

While Fisher's initial research using ANOVA was analyzing crop experiments, its use later expanded to many different fields.

​The Bottom Line

Researchers use analysis of variance (ANOVA) to determine whether the variability of the expected outcomes is due to chance or the factors in the analysis. Two-way ANOVA analyzes the main effect of each of the two independent variables on the expected outcome, in addition to the variables' relationship to each other—if there is any interaction between the independent variables. ANOVA is a hypothesis-based test; its goal is to evaluate multiple mutually exclusive theories about a data set.