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,


Bait Acceptability by Feral Pigs

Feral pigs in Australia are commonly controlled by poisoning with sodium monofluoracetate (1080). Field observations suggest, however, that pigs become bait-shy to 1080 so researchers conducted an experiment using feral pigs that were captured using baited pig traps. Ten male and ten female pigs were selected of approximately the same age and weight.

On Day 1, pigs were offered wheat only and their intake was recorded. On Day 2, five different feeds were offered and the change in intake (Day 2 minus Day 1) was recorded for each pig. The feeds involved combinations of wheat, 1080, dye (for safety reasons) and water (necessary to add the dye and 1080. The changes in intake (kg) are shown in the table below:

  Wheat Wheat
& water
Wheat,
water & dye
Wheat, water
& 1080
Wheat, water,
dye & 1080
Male 0.188
-0.058
0.050
-0.138
0.058
-0.082
-0.712
-1.280
-0.610
-0.830
Female -0.280
-0.062
-0.540
-0.336
-0.260
-0.123
-0.894
-0.672
-0.837
-1.202

The diagram below shows the data. It also displays a plausible model without interaction using a grid of lines that join the 10 treatment means.

Click the y-z rotation button. The two sets of coloured lines represent the males and females. Observe that these two 'profiles' are parallel since the additive nature of the model implies that differences between males and females are the same for each diet.

The profiles are always parallel.

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

Finally, observe that

  • There are similar changes to intake for the three feeds without 1080 and also for the two feeds that include 1080.
  • Feed intake reduces more in Day 2 when 1080 is present than when it is not.