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.

Soft drink bottling

The diagram below shows models for the fill height deviation data described in the previous page.

Click the checkboxes for Carb and Press to display the least squares fit of the model with main effects for these two variables only. This is a two-factor model without interaction of the kind that was described earlier.

Now click the checkbox Spd to add Line speed to the model. Observe that changing line speed is modelled to have the same effect on mean Fill height deviation for all combinations of Pressure and Carbonation.

Click the y-x button to rotate the diagram. Observe that the effect of changing Carbonation is the same for all combinations of Pressure and Line speed — 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.