Analysis of Variance (ANOVA)

Explanation

A study of the impact of gender and race on income could employ analysis of variance to see whether mean incomes in the various gender/racial groups are different. It would perhaps be better to use a name such as analysis of means for these methods, but variances are actually used to determine whether group means are different—thus the name.

Historical Account

Analysis of variance has its origins in the study of data from experiments. Many of the earliest applications were from agriculture in the study of how yield on a piece of land is affected by factors such as type of fertilizer and type of seed. The British statistician Sir Ronald A. Fisher did much of the early work developing analysis of variance.

Experiments can be set up in many different ways, and there are corresponding methods of analysis of variance for the analysis of the data from these designs. Thus, the name analysis of variance covers a large number of methods, and the ways in which an experiment is designed will determine which analyses are used. The ties between the design of experiments and analysis of variance are very strong.

There are also strong ties between analysis of variance and regression analysis. Historically, the two sets of statistical methods grew up separately, and there was little contact between the people who worked in the two areas. The old, elaborate methods of computations used in analysis of variance seemed very different from computations used in regression analysis. But now that statistical software has taken over the computations, it is not very difficult to demonstrate how the two sets of methods have many things in common. Most analyses of variance can be represented as regression analysis with dummy variables. Indeed, analysis of variance and regression analysis are special cases of the so-called general linear model. This became clearer when people started to use analysis of variance on data obtained not only from experiments but also from observational data.

Applications and Examples

One-Way Analysis of Variance

Many of the central ideas in analysis of variance can be illustrated using the simplest case in which there are data on a dependent variable Y from two groups. Thus, we have a nominal variable with two values as the independent variable. This becomes a one-way analysis of variance because there are data only on one independent variable. The data could have been obtained from a psychological experiment whereby one group consisted of the control group and the other group consisted of elements in an experimental group that received a certain stimulus. Then, is there any effect of the stimulus? Also, the data can come from an observational study in which there are incomes from people in different ethnic groups, and we want to see if the income means are different in the underlying ethnic populations.

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