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 = f (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 = f (xi) + g (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,
Drilling thrust
An experiment was conducted to determine how drilling speeds and feeds affected driling thrust forces in a particular material, B10. Three feed rates and five drilling speeds were used in the experiment and the order of runs was completely randomised.
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 different drilling speed. Observe that all of these 'profiles' are parallel since the additive nature of the model implies that the effect of changing the feed rate is the same whatever the drillng speed.
The profiles are always parallel.
Click the y-z rotation button and observe that the three 'profiles' are again parallel.