If there was no experimental error we would expect all of the values from one treatment to be identical. Consequently, we should always aim to reduce e ij by careful experimental design. We have made the assumption that e ij are normally distributed, if we suspect that they are non-normal it may be necessary to transform the data prior to the analysis. It is good practice to examine a histogram of the residuals as a subjective test of this assumption. These are discussed in the experimental design notes.
Factorial models have more than one factor.
The simplest factorial model is two-way. The linear model for a two-way analysis of variance is shown below graphically. There are 2 levels of factor A and 3 levels of factor B in this example. Neither factor has any effect. Factor A has no effect, factor B has an effect.