Analysis of Variance

Most analyses of relationships between variables involve the use of independent and dependent variables. Analysis of variance (ANOVA) is a collection of methods where the independent variable(s) are categorical (nominal) and the dependent variable is quantitative (metric, interval/ratio) with a numerical scale. The analysis compares the means of the dependent variable for the groups defined by the independent variable(s). A more appropriate name might be analysis of means, but variances are used to determine whether means are different.

The simplest case has one independent variable with two categories and one dependent variable. For example, ANOVA can be used to analyze the relationship between the independent variable gender (categorical) and the dependent variable blood pressure (quantitative). The analysis compares the mean blood pressures for females and males. This comparison can also be made using a t test for the comparison of two means, and such a t test is a special case of ANOVA. Furthermore, the comparison can be made using regression analysis with a dummy variable for gender, and such a regression analysis is also a special case of ANOVA.

ANOVA and regression analysis are special cases of the general linear model, and there are mainly historical reasons why the two methods are seen as distinct. ANOVA grew out of analysis of experimental data in agriculture, where yields were compared for different treatments, such as type of fertilizer. Most of this took place in England under the leadership of the great statistician Ronald Fisher (1850–1921). Much of regression analysis has its foundation in economics with its many quantitative variables. Regression analyses used by economists are often called econometrics. The answer to which method of performing ANOVA is the proper approach can itself be analyzed using regression analysis with the dummy variables, but the construction of such variables can be cumbersome. Most statistical software programs still distinguish between ANOVA and regression, and regression analysis typically provides less detailed output than does ANOVA.

With one independent variable, we perform oneway ANOVA, with two independent variables we perform two-way ANOVA, and so on. Introducing location as a second independent variable with values [Page 34]urban and rural, we can use a two-way ANOVA to study whether there are differences in blood pressure for females and males as well as between an urban and a rural location, using both gender and location as the two variables in one analysis. We could do a one-way analysis for gender and then separately do a one-way analysis for location, but it is more efficient to use both gender and location at the same time in one analysis. With more than one dependent variable, we perform multivariate ANOVA.