Analysis of variance
Introduction
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures used to analyze the differences among group means in a sample. The ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test when more than two groups are involved.
History
The analysis of variance was developed by statistician Ronald A. Fisher. In the year 1918, Fisher was appointed to the Rothamsted Experimental Station, where he would develop the design of experiments and the ANOVA. Fisher's method allowed for the analysis of several independent variables simultaneously, which was a significant advancement in the statistical methodology.
Assumptions
ANOVA assumes that the data is normally distributed. The normality assumption can be evaluated by looking at the distribution of the residuals or by conducting a normality test such as the Shapiro-Wilk or Kolmogorov-Smirnov. Another assumption is the homogeneity of variances, which means that the variance within each group being compared is similar. This can be checked using tests like Levene's or Bartlett's. Finally, ANOVA assumes that the observations are independent of each other.
Types of ANOVA
There are two main types of ANOVA: One-way (or univariate) ANOVA and Two-way ANOVA. One-way ANOVA is used when the experiment has a single factor and the researcher wants to see if there are any significant differences between the means of two or more groups. Two-way ANOVA, on the other hand, is used when there are two factors. It not only shows if there are any significant differences between the means of two or more groups but also if there is any interaction between the factors.
One-Way ANOVA
One-way ANOVA is used to test for differences among at least three groups. It is called one-way because there is only one response variable. It compares the means of the groups you are interested in and determines whether any of those means are statistically significantly different from each other.
Two-Way ANOVA
Two-way ANOVA is a hypothesis test that allows you to compare the mean of two or more groups that are split on two independent variables. The primary purpose of a two-way ANOVA is to understand if there is an interaction between the two independent variables on the dependent variable.
ANOVA Table
The ANOVA table is a way to summarize the overall results of an ANOVA test. The table typically includes the source of variation (between groups, within groups, total), the degrees of freedom associated with each source, the sum of squares, the mean square (which is the sum of squares divided by the corresponding degrees of freedom), the F statistic, and the p-value.
Post Hoc Tests
If the ANOVA test is significant, it means that there are differences somewhere in your groups, but it doesn't tell you where those differences lie. To find out which specific groups differed from each other, post hoc tests are used. Some commonly used post hoc tests include the Tukey's, Bonferroni, and Scheffé's tests.
Applications
ANOVA is widely used in many fields such as psychology, education, sociology, economics, and biology. It is used to compare the means of more than two groups, which can be useful in a variety of experimental designs. For example, in psychology, ANOVA can be used to compare the mean scores of patients in different treatment groups.
Limitations
While ANOVA is a powerful statistical tool, it does have limitations. One limitation is that it cannot be used to test multiple dependent variables. Also, ANOVA can only handle numeric response data; categorical data cannot be used. Furthermore, ANOVA assumes that the data is normally distributed and that variances are equal across groups, which may not always be the case.