In a two-way anova with interaction, there are two factor effects and an interaction effect.

A key statistical test in research fields including biology, economics and psychology, analysis of variance (ANOVA) is very useful for analyzing datasets. It allows comparisons to be made between three or more groups of data. Here, we summarize the key differences between these two tests, including the assumptions and hypotheses that must be made about each type of test.

There are two types of ANOVA that are commonly used, the one-way ANOVA and the two-way ANOVA. This article will explore this important statistical test and the difference between these two types of ANOVA. 

What is a one-way ANOVA?

A one-way ANOVA is a type of statistical test that compares the variance in the group means within a sample whilst considering only one independent variable or factor. It is a hypothesis-based test, meaning that it aims to evaluate multiple mutually exclusive theories about our data. Before we can generate a hypothesis, we need to have a question about our data that we want an answer to. For example, adventurous researchers studying a population of walruses might ask “Do our walruses weigh more in early or late mating season?” Here, the independent variable or factor (the two terms mean the same thing) is “month of mating season”. In an ANOVA, our independent variables are organised in categorical groups. For example, if the researchers looked at walrus weight in December, January, February and March, there would be four months analyzed, and therefore four groups to the analysis.

A one-way ANOVA compares three or more than three categorical groups to establish whether there is a difference between them. Within each group there should be three or more observations (here, this means walruses), and the means of the samples are compared. 

What are the hypotheses of a one-way ANOVA?

In a one-way ANOVA there are two possible hypotheses.

• The null hypothesis (H0) is that there is no difference between the groups and equality between means (walruses weigh the same in different months).
• The alternative hypothesis (H1) is that there is a difference between the means and groups (walruses have different weights in different months) .

What are the assumptions and limitations of a one-way ANOVA?

• Normality – that each sample is taken from a normally distributed population
• Sample independence – that each sample has been drawn independently of the other samples
• Variance equality – that the variance of data in the different groups should be the same
• Your dependent variable – here, “weight”, should be continuous – that is, measured on a scale which can be subdivided using increments (i.e. grams, milligrams)

What is a two-way ANOVA?

A two-way ANOVA is, like a one-way ANOVA, a hypothesis-based test. However, in the two-way ANOVA each sample is defined in two ways, and resultingly put into two categorical groups. Thinking again of our walruses, researchers might use a two-way ANOVA if their question is: “Are walruses heavier in early or late mating season and does that depend on the sex of the walrus?” In this example, both “month in mating season” and “sex of walrus” are factors – meaning in total, there are two factors.  Once again, each factor’s number of groups must be considered – for “sex” there will only two groups “male” and “female”.

The two-way ANOVA therefore examines the effect of two factors (month and sex) on a dependent variable – in this case weight, and also examines whether the two factors affect each other to influence the continuous variable. 

What are the assumptions and limitations of a two-way ANOVA?

• Your dependent variable – here, “weight”, should be continuous – that is, measured on a scale which can be subdivided using increments (i.e. grams, milligrams)
• Your two independent variables – here, “month” and “sex”, should be in categorical, independent groups.
• Sample independence – that each sample has been drawn independently of the other samples
• Variance Equality – That the variance of data in the different groups should be the same
• Normality – That each sample is taken from a normally distributed population

What are the hypotheses of a two-way ANOVA?

Because the two-way ANOVA consider the effect of two categorical factors, and the effect of the categorical factors on each other, there are three pairs of null or alternative hypotheses for the two-way ANOVA. Here, we present them for our walrus experiment, where month of mating season and sexare the two independent variables.

• H0: The means of all month groups are equal
• H1: The mean of at least one month group is different

• H0: The means of the sex groups are equal
• H1: The means of the sex groups are different

• H0: There is no interaction between the month and gender 
• H1: There is interaction between the month and gender 

Interactions in two-way ANOVA

These effects are not to be ignored. If we put the interactions to one side, with the results mentioned above, an incomplete analysis might conclude that walruses in general lost weight over mating season, which would ignore a reality that the decrease was driven by changes to male walrus weight. Another example could be the efficacy of a candidate drug for a disease; you can see how proper modeling of interaction effects can become critical to many biological research studies. 

The key differences between one-way and two-way ANOVA are summarized clearly below. 

1. A one-way ANOVA is primarily designed to enable the equality testing between three or more means. A two-way ANOVA is designed to assess the interrelationship of two independent variables on a dependent variable. 

2. A one-way ANOVA only involves one factor or independent variable, whereas there are two independent variables in a two-way ANOVA.

3. In a one-way ANOVA, the one factor or independent variable analyzed has three or more categorical groups. A two-way ANOVA instead compares multiple groups of two factors. 

4. One-way ANOVA need to satisfy only two principles of design of experiments, i.e. replication and randomization. As opposed to two-way ANOVA, which meets all three principles of design of experiments which are replication, randomization and local control.

One-Way ANOVATwo-Way ANOVADefinitionA test that allows one to make comparisons between the means of three or more groups of data.A test that allows one to make comparisons between the means of three or more groups of data, where two independent variables are considered. Number of Independent VariablesOne.Two. What is Being Compared?The means of three or more groups of an independent variable on a dependent variable.The effect of multiple groups of two independent variables on a dependent variable and on each other. Number of Groups of Samples Three or more.Each variable should have multiple samples.

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

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 [latex]\text{x}[/latex], [latex]\text{y}[/latex], and [latex]\text{z}[/latex], the ANOVA model includes terms for the main effects ([latex]\text{x}[/latex], [latex]\text{y}[/latex], [latex]\text{z}[/latex]) and terms for interactions ( [latex]\text{xy}[/latex], [latex]\text{xz}[/latex], [latex]\text{yz}[/latex], [latex]\text{xyz}[/latex]). 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 [latex]\text{A}[/latex] and the quantitative variable [latex]\text{X}[/latex] interact (are non-additive) when analyzed with respect to the outcome variable [latex]\text{Y}[/latex].