Overview
Factorial ANOVA can be considered an extension of One Way Analysis Of Variance. Unlike One Way ANOVA, which can only have one variable independent of the test, Factorial ANOVA allows for multiple independent variables or “factors” to be tested.
A Two-Way ANOVA is a type of Factorial ANOVA that has two independent variables, and a Three-Way Anova, on the other hand, is a Factorial ANOVA with three independent variables. In all cases, there is one dependent variable. The number of independent variables is the ‘factor.’ Because the test becomes very complicated and difficult to interpret, we rarely use more than three factors. However, factorial ANOVA is more effective than a series of One-Way ANOVA.
Let’s take an example to illustrate this. The dependent variable is the amount of time a child spends outside. We now have two independent variables. The first is the gender of the children, and the second is the country. Our first independent variable (Gender) has two groups: Male and Female. The second independent variable has 4 levels (levels): the USA, UK, and Australia. This is a 2X4 factorial analysis. In such a case, we will perform a 2 (Gender level) X 4(Country level) Factorial ANOVA.
Hypotheses for Two-Way ANOVA
*These assumptions assume that the population is normally distributed and that the samples are independent.
*The homogeneity or the amount of variance in each group is approximately equal.
*The average sample size for the groups of independent variables is approximately the same. In the above example, 100 male and 100 female kids are included. However, if Germany has only 20 male and 5 female kids, then the cell size will be too small to allow for meaningful ANOVA. (A cell is a point of interaction between the Independent Variables, such as girls from the US, boys from the US, or girls from the UK ….. Each cell in this example should be at most 20.
Interaction and Main Effects
Understanding the meaning of Interaction and Main Effect is crucial to fully comprehending the Factorial ANOVA results. Both the Factorial ANOVA results in the Main effect and interaction are both main effects. The Main effect will look somewhat similar to the One-way ANOVA. In the example above, there will be two Main Effects. One will compare the playing time of males and women, and the other will compare the playing times in different countries. Each factor (Independent variable) is treated separately in this One-Way ANOVA. Interaction is also possible with Factorial ANOVA. This means that the differences between the factors (independent variables) can influence or differ according to each other variable. Unlike Main Effect, here, all factors are considered simultaneously.
Main Effects
As mentioned above, the Factorial ANOVA produces the Main effect for each independent variable. Each Main Effect has its own F value. Understanding the concepts ‘Controlling for’ and ‘Partialling in’ is crucial. Let’s return to our example. We can perform a One-Way ANOVA using one dependent variable, the number of times kids spend outside of their homes, and one independent variable, the gender of the children. The One-Way ANOVA shows that the difference in the mean is significantly different; that male children play more hours than females. We now have a Two-Way Factorial ANOVA. This is where there are two variables, the gender of the children being analyzed and an additional variable: the country they live in.
After running the Factorial ANOVA, we find that the mean for male kids is significantly lower than the mean for female kids. It also shows that children in the USA spend more time outdoors than those from Germany. However, I found a higher proportion of boys in the USA group sample, which could impact my result. The Country Independent variable can show no difference in the means. We need to “control” the Gender Independent variable’s influence to do this. This is how ANOVA breaks down the variance in our dependent variable (the different playing hours) into several components. We can explain part of the variance in our dependent variable by concluding that Male and Female children play outside differently. If someone asks why some children play for 1 hour and others for 3 hours, they will likely find that gender plays a part. There is, however, some variation that countries can explain. To find the country-specific variation, we need to subtract the variance it shares from Gender. Then we will see if the children’s Country can explain the dependent variable. After adjusting for the effect on gender, we will attempt to determine the Main Effect of how much time a child spends outside. Multicollinearity is explained in the blogs that are related to the Model evaluation. If the independent variables are strongly correlated, then the dependent variable will have explained all of the variances.
Interactions
The Factorial ANOVA also includes interaction. The number of interactions increases as a function of the number and type of independent variables. If there are 2 independent variables, there will be 1 interaction and 2 main effects. However, if there are 3 independent variables, there will be 3 main effects, 3 two-way interactions, and 1 three-way interaction. This can get very complicated as more independent variables are added.
Drawing inferences by using Main Effects and Interactions
Let’s say that there is an equal number of children in each group. For example, 120 total kids are 60 male and 60 women, with 30 kids coming from the USA, UK, Germany, and Australia. In each country, there are 15 boys as well as 15 kids. This eliminates the need to control for the effect of any independent variable. The ANOVA test reveals that the mean for male kids is significantly higher than that of female children. The mean of American children is also higher than in any other country. We conclude that Gender and Country are the main effects, but we also consider the interactions. We find that male child play an hour longer than their counterparts in every country except Germany, where they play three hours more than their counterparts. This shows that the hours played by male children depend on their country. It is called a “two-way interaction.” We can draw a more powerful inference by including Interaction.
A new technique can also be introduced for male and female sprinters. After a month of training with the new technique, their performance was measured. It turns out that the mean had increased significantly, which makes it clear that the technique is effective. However, Interaction shows that the mean for the female sprinters was much higher than that of the male sprinters, whose performance improved only marginally. The increase in the mean result of the new technique is because it works for female sprinters.
Analysis of Covariance
The Factorial ANOVA also benefits from the analysis of covariance, which allows us to eliminate the effects of other independent variables partially. We can see that children in the USA spend more time outside than in other countries. However, it can be concluded by analysis that there are other countries that have less favorable weather conditions. This could lead to a decrease in outdoor play hours. The ANCOVA can be used to do this. It can subtract the effects of weather and then determine if the country can account for the variation in outdoor time.
ANCOVA allows us to control the effects of continuous variables, while Factorial ANOVA only allows for categorical independent variables. ANCOVA considers the statistical interaction, allowing us to better understand the relationship between dependent and independent variables.
Multivariate analysis of Variance (MANOVA).
Multivariate ANOVA is possible when more than one dependent variable is being considered. Regular ANOVA can only include one dependent variable. MANOVA can include multiple dependent variables. This gives us greater statistical power and allows us to identify patterns among multiple dependent variables. MANOVA, which can simultaneously assess patterns in multiple dependent variables, has the same benefit as the One Way ANOVA.
ANOVA Factorial is an extension of One-Way ANOVA, where multiple independent variables can be compared with their groups. Another variant of ANOVA is Repeated measures ANOVA.
