Modelling how the mean response depends on the factor levels

The most general model for how the mean response for the i'th experimental unit might depend on the levels of the two factors, X and Y, can be written in the form

µi   =   (xi, zi)

In practice, we usually simplify this with the assumption that the effects of the two factors are additive. This implies a model of the form,

µi   =   (xi)   +   (zi)

and is called a model without interaction. This type of model is not appropriate for all types of experiment and we will examine situations where the assumption of additive effects does not hold in a later section about interaction. However if the model without interaction holds,

The effect of changing the level of one factor is the same, regardless of the level of the other factor.

For example,


Cholesterol determination

In a hospital laboratory, three technicians are making serum cholesterol determinations in milligrams per centimeter. To test the similarity of these techicians, sera from five normal subjects are split into 1/6 aliquots to be tested twice by each technician.

The three technicians can be treated as one factor (a = 3 levels) and the five subjects are the other factor in the experiment (b = 5) and there are r = 2 replicates since each techician makes two cholesterol determinations from each subject.

The diagram below shows the data. It also displays a plausible model without interaction using a grid of lines that join the 15 treatment means.

Click the y-x rotation button. The five sets of coloured lines represent the five subjects. Observe that all of these 'profiles' are parallel since the additive nature of the model implies that differences between the observers are the same for each subject.

The profiles are always parallel.

Click the y-z rotation button and observe that the three 'profiles' are again parallel.