What is analysis of variance (ANOVA)?

Analysis of variance (ANOVA) is an analysis tool used in statistics that divides the observed aggregate variability in a data set into two parts: systematic factors and random factors. Systematic factors have a statistical influence on the data set, while random factors do not. Analysts use the ANOVA test to determine the influence of the independent variables on the dependent variable in a regression study.

The t- and z-test methods developed in the 20th century were used for statistical analysis until 1918, when Ronald Fisher created the analysis of variance method.

ANOVA is also called Fisher’s analysis of variance and is the extension of t- and z-tests. The term became known in 1925 after it appeared in Fisher’s book Statistical Methods for Research Workers.

It was used in experimental psychology and was later extended to more complex subjects.

The formula of the ANOVA is: $F=\frac{MST}{MSE}$

where: F is ANOVA coefficient, MST = Mean sum of squares due to treatment, MSE = Mean sum of squares due to error.

What does the analysis of variance reveal?

ANOVA testing is the initial step in analyzing the factors that affect a given data set. Once the test is completed, the analyst performs additional tests on the methodical factors that measurably contribute to the inconsistency in the data set. The analyst uses the results of the ANOVA test in an f-test to generate additional data that align with the proposed regression models.

The ANOVA test allows more than two groups to be compared simultaneously to determine if there is a relationship between them. The result of the ANOVA formula, the F-statistic (also called the F-ratio), allows multiple data sets to be analyzed to determine variability between and within samples.

If there is no real difference between the groups tested, null hypothesis, the result of the F statistic of ANOVA will be close to 1. The distribution of all possible values of the F statistic is the F distribution. It is actually a group of distribution functions. It is actually a group of distribution functions, with two characteristic numbers, called degrees of freedom at the numerator and degrees of freedom at the denominator.

Example of the use of ANOVA.

A researcher might, for example, test students at multiple universities to see if students at one university consistently outperform students at other universities. In a commercial application, an R&D researcher might test two different product creation processes to see if one process is better than the other in terms of cost efficiency.

The type of ANOVA test used depends on a number of factors. It is applied when the data are to be experimental. Analysis of variance is used if one does not have access to statistical software and therefore calculates the ANOVA by hand. It is simple to use and is best suited for small samples. In many experimental designs, sample sizes must be the same for various combinations of factor levels.

The ANOVA is useful for testing three or more variables. It is similar to two-sample multiple t-tests. However, it involves less type I error and is appropriate for a variety of problems. The ANOVA groups differences by comparing the means of each group and includes the distribution of variance across different sources. It is used with subjects, test groups, between groups and within groups.

One-way ANOVA and two-way ANOVA.

There are two main types of ANOVA: one-way (or one-way) and two-way. Variants of ANOVA also exist. For example, MANOVA (multivariate ANOVA) differs from ANOVA in that the former analyzes multiple dependent variables simultaneously, while the latter evaluates only one dependent variable at a time. One-way or two-way refers to the number of independent variables in the analysis of variance test. A one-way ANOVA evaluates the impact of a single factor on a single response variable. It determines whether all samples are equal. The one-way ANOVA is used to determine whether there are statistically significant differences between the means of three or more independent (uncorrelated) groups.

The two-way ANOVA is an extension of the one-way ANOVA. With the one-way ANOVA, you have an independent variable influencing a dependent variable. In a two-way ANOVA, there are two independent variables. For example, a two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as wages and skills. It is used to observe the interaction between the two factors and test the effect of two factors simultaneously.

KEY FINDINGS.

• An analysis of variance, or ANOVA, is a statistical method that separates observed data into several components to be used for further testing.
• A one-way ANOVA is used for three or more data sets to obtain information about the relationship between the dependent and independent variables.
• If there is no true variance between groups, the F coefficient of the ANOVA should be close to 1.

Source: www.investopedia.com