The analysis of variance is in fact a family of statistical methods, rather than a single approach. In all cases, there is one quantitative variable (x) and one or more qualitative variables (factors) which are used to identify the different groups from which the data were collected.

The one factor ANOVA is the most simple case, where there is only one factor dividing the data into groups. In this situation, the test determines whether or not the mean (see average) value of the quantitative variable (x) is the same for each group. As an example, you could use this method to determine whether the mean IQ varies from one state or province to another (the quantitative variable would be IQ, and the factor the region from which the person prevails).

A two factor ANOVA can be of two types:

  • First, the two factor ANOVA without replication, where there is only one value of x for each combination of factors. In this type of ANOVA, the two qualitative variables must be independent, and the ANOVA will let you know whether the mean value of x differs between the groups identified by the first or second factor. However, it cannot determine whether there is an interaction between the two. As an example, you could test to determine whether the mean IQ varies amongst the regions and between men and women (in this case, only 1 man and 1 woman from each region).
  • Second, the two factor ANOVA with replication is similar to the previous case, but there is more than one observation for each combination of factors. This type of ANOVA again determines whether the two factors had an effect on the mean value of x, but also determines whether there exists an interaction between factor 1 and factor 2. In the preceeding example, you could not only determine whether there are IQ differences between the regions and between the sexes, but also whether the IQ differences between the regions depend on the sex of the individual. The example provided above by Cermain is another example of this type of ANOVA.
The two factor ANOVA can be extended to include many factors, but the interpretation of results becomes a little difficult when you have four factors given that the number of interactions grows alarmingly (with three factors, there are four interactions, with four factors there are 11 etc.). When there are more than two factors, the ANOVA is generally termed multiway.

Finally, there is also the nested ANOVA. The nested ANOVA differs from the two factor or multiway ANOVA since the two (or more) factors are not independent. That is, when the experiment is designed, the subjects (on whom x will be measured) are divided into groups according to the first factor, and then are redivided into groups based on the second factor. The difference may seem minor, but the mathematics change in order to account for this modification of experimental design.

The ANOVA can be extended into the multivariate case, where there is more than one quantitative variable (x). When more than one x is measured, a new family of methods becomes useful, called the Multiple ANalysis Of VAriance, or the MANOVA.