*The Business Professor*, updated October 15, 2019, last accessed October 20, 2020, https://thebusinessprofessor.com/lesson/two-way-anova-definition/.

### Two-Way ANOVA Definition

A two-way ANOVA is a one-way ANOVAâ€™s extension, a statistical test used to examine the effect of two different variables on one continuous dependent variable. Besides assessing the main effect of each independent variable, ANOVA also assesses to find out if, between the two variables, there is any kind of interaction.

### A Little More on What is two-way ANOVA

A two-way ANOVA is a test that analyzes the independent variableâ€™s effect on the anticipated outcome, along with its outcomeâ€™s relationship. When it comes to random factors, they are usually considered to have no statistical influence on a set of data. It is contrary to the systematic factors where their statistical are found to be of significance.

A researcher may use a two-way ANOVA when he or she wants to have one measurement variable as well as two nominal variables. Note that the nominal variable (also known as factors), exists in all possible combinations. By using a two-way ANOVA, as a researcher, you can be able to find out whether the outcomesâ€™ variability is as a result of the factors in the analysis or by chance.

Individuals can apply ANOVA in different fields, such as economics, finance, social science, and medicine.

**Two-Way ANOVA Assumptions**

Generally, a two-way ANOVA test is an extension of the one-way-ANOVA test. The two-way ANOVA test has got two independent variables hence the term â€śtwo-way.â€ť Note that there as several assumptions that relate to the two-way analysis of variance. They are as follows:

- The sample population must be approximately normally or normally distributed
- It is mandatory for the samples to be independent
- It is mandatory that the populationsâ€™ variances be equal
- The sample size of the groups must be the same

**A Two-Wat ANOVA Hypotheses**

The two-way ANOVA has several sets to hypotheses. The null hypotheses are as follows:

- The first factorâ€™s population means are equal. It is is the same as the one-way ANOVA when it comes to the row factor)
- The second factorâ€™s population means are equal. It is is the same as the one-way ANOVA when it comes to column factor)
- Between the two factors, no interaction exists. It is the same as administering a test for independence using contingency tables.

**Two-way ANOVA Uses**

When you want to identify which factors are influencing a particular outcome, you will first have to apply the ANOVA test. By performing the test, a tester may be in a position to help you do more analysis on the methodical factors that are contributing to the variability of the data set in a statistical manner.

Also, a two-way ANOVA test will tell you the outcome of the two independent variables on a dependent variable. The results from the ANOVA test can be used in an F-test to determine its significance on the regression formula overall.

It is helpful when you want to test the effects of variables on each other. It is the same as that of multiple two-sample t-tests. Nonetheless, it is suitable for a range of issues because it usually results in fewer type 1 errors.

ANOVA can also be used to group differences where it compares each groupâ€™s means, including spreading out the variance into different sources. It is usually used together with subjects, test groups either between or within groups.

**A Two-Way ANOVA vs. ANOVA**

Analysis of variance exists in two types: one-war and two-way which is also known as unidirectional and bidirectional respectively. The two analyses of variance refer to independent variablesâ€™ number in the analysis of variance test.

**One-Way ANOVA**

It is a type of analysis of variance used to evaluate the sole factorâ€™s impact on a single response variable. It also tests the samples to find out if they are equal. In addition, it can be used to determine whether, between the means of three or more unrelated groups, there are significant differences in the statistics.

**Two-Way ANOVA**

There are two independent variables in this type of analysis of variance. With the two-way variables test, a company can easily compare the productivity of its workers, based on two independent variables such as skills and salary. It can utilize the test to observe how those two factors interact with each other. It also tests the two factorsâ€™ effect simultaneously.

**Key Takeaways**

- A two-way ANOVA is a one-way ANOVAâ€™s extension, a statistical test used to examine the effect of two different variables on one continuous dependent variable.
- The two-way ANOVA test has two independent variables hence the term â€śtwo-way.â€ť
- The two-way ANOVA can be applied in different fields, such as economics, finance, social science, and medicine.

**References for â€śTwo-Way ANOVA****â€ť**

https://statistics.laerd.com/spss-tutorials/two-way-anova-using-spss-statistics.php

https://en.wikipedia.org/wiki/Two-way_analysis_of_variance

https://www.investopedia.com â€ş Business â€ş Business Leaders â€ş Math & Statistics

**Academic research for ****â€śTwo-Way ANOVA****â€ť**

Two-way ANOVA with unequal cell frequencies and unequal variances, Ananda, M. M., & Weerahandi, S. (1997). Two-way ANOVA with unequal cell frequencies and unequal variances. *Statistica Sinica*, 631-646.

Nonparametric competitors to the two-way ANOVA, Toothaker, L. E., & Newman, D. (1994). Nonparametric competitors to the two-way ANOVA. *Journal of Educational Statistics*, *19*(3), 237-273.

Two-way ANOVA models with unbalanced data, Fujikoshi, Y. (1993). Two-way ANOVA models with unbalanced data. *Discrete Mathematics*, *116*(1-3), 315-334.

[HTML] A parametric bootstrap approach for two-way ANOVA in presence of possible interactions with unequal variances, Xu, L. W., Yang, F. Q., & Qin, S. (2013). A parametric bootstrap approach for two-way ANOVA in presence of possible interactions with unequal variances. *Journal of Multivariate Analysis*, *115*, 172-180.

Adjusting for unequal variances when comparing means in one-way and two-way fixed effects ANOVA models, Wilcox, R. R. (1989). Adjusting for unequal variances when comparing means in one-way and two-way fixed effects ANOVA models. *Journal of Educational Statistics*, *14*(3), 269-278.

Testing non-additivity (interaction) in two-way ANOVA tables with no replication, Alin, A., & Kurt, S. (2006). Testing non-additivity (interaction) in two-way ANOVA tables with no replication. *Statistical methods in medical research*, *15*(1), 63-85.

Using two-way ANOVA and hypothesis test in evaluating crumb rubber modification (CRM) agitation effects on rheological properties of bitumen, Aflaki, S., & Memarzadeh, M. (2011). Using two-way ANOVA and hypothesis test in evaluating crumb rubber modification (CRM) agitation effects on rheological properties of bitumen. *Construction and Building Materials*, *25*(4), 2094-2106.

An approximate degrees of freedom test for heteroscedastic two-way ANOVA, Zhang, J. T. (2012). An approximate degrees of freedom test for heteroscedastic two-way ANOVA. *Journal of Statistical Planning and Inference*, *142*(1), 336-346.

Testing for interaction in two-way ANOVA tables with no replication, Tusell, F. (1990). Testing for interaction in two-way ANOVA tables with no replication. *Computational Statistics & Data Analysis*, *10*(1), 29-45.

Performance of two-way ANOVA procedures when cell frequencies and variances are unequal, Bao, P., & Ananda, M. M. (2001). Performance of two-way ANOVA procedures when cell frequencies and variances are unequal. *Communications in Statistics-Simulation and Computation*, *30*(4), 805-829.