Model with only main effects

The simplest model for how three factors affect the mean response simply adds together their separate effects. Using the notation yikjl to denote the l'th replicate for the i'th level of factor B, j'th level of C and k'th level of D, this model can be written in the form:


yijkl  =  µ

 + 
(explained by B)
βi

 + 
(explained by C)
γj

 + 
(explained by D)
δk

 + 
(unexplained)
εijkl

where the red error term is assumed to be normally distributed with mean zero.

As in earlier models, we set the first (baseline) level for each of the sets of parametersi}, j} and k} to be zero:

β1  =  γ1  =  δ1  =  0

The other parameters therefore describe differences from the baseline level.

No interaction

In this simple model, the effect of changing B is assumed to be the same whatever the levels of C and D (and similarly for the other factors). There is said to be no interaction between the effects of the factors.

Least squares estimates

The parameter µ and the sets of parameters i}, j} and k} are usually unknown and must be estimated from the experimental data. As in other models, this is done to minimise the sum of squared residuals — the method of least squares.

Water uptake by toads and frogs

The diagram below shows models for the water uptake data described in the previous page.

Click the checkboxes for Anim and Moist to display the least squares fit of the model with main effects for only Animal type and Pre-experiment moisture level. This is a two-factor model without interaction of the kind that was described earlier.

Now click the checkbox Inject to add Injection to the model. Observe that the hormone injection is modelled to have the same effect on mean Water uptake for all combinations of Animal and Moisture.

Click the y-x button to rotate the diagram. Observe that the difference between water uptake in toads and frogs is the same for all combinations of Moisture and Injection — all four lines are parallel. Similarly, click the y-z rotation button and observe that the lines are parallel since the factors do not interact.