# Analysis of Variance (ANOVA)

### Analysis of Variance (ANOVA)

ANOVA is a statistical analysis tool that splits an observed aggregate variability found inside a data set into two parts:

• Systematic factors – These factors have statistical influence on the data set.
• Random factors – Random factors do not have a statistical influence on the given data set.

Analysis of variance tests are used by analysts in regression study to determine the influence that an independent variable has on the dependent variable. ANOVA was created by Ronald Fisher in 1918. However, it was not until 1925, after appearing in Fisher’s book “Statistical Methods for Research Workers” that it was employed in experimental psychology. Later on, analysis of variance was expanded to more complex subjects.

The formula for Analysis of Variance is:

F = MST/MSE

In the formula:

• F is the ANOVA coefficient
• MSE – Is the mean sum of squares due to error
• MST – is the mean sum of squares due to treatment

The analysis of variance is the initial stage in analyzing factors that affect a data set. After ANOVA, the analyst performs other tests on the methodical factors that contribute to the inconsistency in a data set. The test results from ANOVA are utilized in an f –test to generate additional data that aligns with the proposed regression models.

Analysis of variance test supports the comparison of more than two groups at the same time. This is used to determine whether a relationship between them exists. The F statistic is the result of the ANOVA formula result. It is also called the F-ratio and allows for multiple groups data analysis to find the variability between samples and within samples.

The null hypothesis is the difference that exists between the tested groups. If there is no real difference, the F-ratio statistic will be close to 1. Sampling fluctuations also likely to follow Fisher’s F distribution.

How to use Analysis of Variance

Several factors determine the type of ANOVA test to be used. ANOVA should be used when data needs to be experimental. Also, it is employed in situations where there is no access to statistical software resulting in computing ANOVA by hand.

ANOVA is quite simple to use and perfectly suited for small samples. Since there are several experimental designs, the sample sizes must be the same for the various factor level combinations. You can use ANOVA to test three or more variables. It works the same as the two-sample t-test. However, it is suitable for a range of issues and results in fewer type I errors. In grouping differences, Analysis of Variance compares the means of each group and includes spreading out the variance into diverse sources. It is used with test groups, subjects, within groups, and between groups.

One-way and two-way ANOVA

The terms one-way and two-way refers to the number of independent variables in the ANOVA test.

• One-way ANOVA – it has one independent variable with two levels
• Two-way ANOVA – It has two independent variables but can have multiple levels

So you are probably asking yourself “what are levels or Groups”?

Levels are the different classes within the same independent variable. For example, a variable called brand of cereal might have cornflakes, raisin brans, and lucky charms. These constitute a total of three levels.

Suppose you want to study if a support group for alcoholics and individual counseling are effective ways of lowering alcohol consumption. You can decide to split the study participants into three levels or groups:

• Those that need medication only
• Those that require medication and counseling
• And those that need counseling only

In this study, you can take your dependent variable to be the number of alcoholic beverages consumed by the participants in a single day. You will have to use a nested ANOVA for the analysis if your groups or levels have a hierarchical structure.

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Replication in ANOVA

Replication means duplicating your tests with multiple groups. The two-way ANOVA with replication, you have two groups, and individuals within that group are doing more than one thing. For example, two teams of football players, from two different leagues, participating in two competitions. We wouldn’t need replication if we only had one team participating in the competition

Types of Analysis of Variance Tests

The main types of ANOVA tests are one-way and two-way tests. The two-way tests can be with or without replication:

• One-way ANOVA (between groups) – used to test if two groups have a difference between them
• Two-way ANOVA (without replication) – Used if we have one group but is double testing it
• Two-way ANOVA (with replication) – used with two groups and the members of those groups are doing more than one thing.

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