analysis of variance (ANOVA)

analysis of variance is abbreviated as ANOVA (analysis of variance). There are several kinds of analyses of variance. The simplest kind is a one-way analysis of variance. The term ‘oneway’ means that there is only one factor or independent variable. ‘Two-way’ indicates that there are two factors, ‘three-way’ three factors, and so on. An analysis of variance with two or more factors may be called a factorial analysis of variance. On its own, analysis of variance is often used to refer to an analysis where the scores for a group are unrelated to or come from different cases than those of another group. A repeated-measures analysis of variance is one where the scores of one group are related to or are matched or come from the same cases. The same measure is given to the same or a very similar group of cases on more than one occasion and so is repeated. An analysis of variance where some of the scores are from the same or matched cases and others are from different cases is known as a mixed analysis of variance. Analysis of covariance (ANCOVA) is where one or more variables which are correlated with the dependent variable are removed. Multivariate analysis of variance (MANOVA) and covariance (MANCOVA) is where more than one dependent variable is analysed at the same time. Analysis of variance is not normally used to analyse one factor with only two groups but such an analysis of variance gives the same significance level as an unrelated t test with equal variances or the same number of cases in each group. A repeated-measures analysis of variance with only two groups produces the same significance level as a related t test. The square root of the F ratio is the t ratio.

Analysis of variance has a number of advantages. First, it shows whether the means of three or more groups differ in some way although it does not tell us in which way those means differ. To determine that, it is necessary to compare two means (or combination of means) at a time. Second, it provides a more sensitive test of a factor where there is more than one factor because the error term may be reduced. Third, it indicates whether there is a significant interaction between two or more factors. Fourth, in analysis of covariance it offers a more sensitive test of a factor by reducing the error term. And fifth, in multivariate analysis of variance it enables two or more dependent variables to be examined at the same time when their effects may not be significant when analysed separately.

The essential statistic of analysis of variance is the F ratio, which was named by Snedecor in honour of Sir Ronald Fisher who developed the test. It is the variance or mean square of an effect divided by the variance or mean square of the error or remaining variance:

An effect refers to a factor or an interaction between two or more factors. The larger the F ratio, the more likely it is to be statistically significant. An F ratio will be larger, the bigger are the differences between the means of the groups making up a factor or interaction in relation to the differences within the groups.

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