Model with interaction
The two-factor model without interaction does not always fit the data that are recorded. The most general model for the effect of the two factors is one that does not place any restrictions on the mean responses — one with a different parameter for each factor combination,
yijk = |
(explained by X & Z) µij |
+ |
(unexplained) εijk |
A more useful way to express this model is in the form,
yijk = µ |
+ |
(explained by X) βi |
+ |
(explained by Z) γj |
+ |
(interaction) δij |
+ |
(unexplained) εijk |
There are redundant parameters here, so we take the approach used in our earlier models without interaction by defining parameters that refer to baseline levels of X and Z to be zero,
The δ-parameters are called the interaction terms and they describe how far the treatment means differ from what would be expected if there was no interaction — if there is no interaction, they will all be zero.
Note that there are (gX - 1)(gZ - 1) of the non-zero interaction parameters.
Music and Alzheimer's disease
The diagram below initially shows a no-interaction model for the Alzheimer's data. Observe that there are gX + gZ - 1 = 4 draggable red arrows to adjust the parameters of the model — these are the degrees of freedom (number of non-zero parameters) for the no-interaction model.
Click Least squares to see the best fit for this model.
Click the checkbox Interaction to add an interaction term to the model and observe that (gX - 1)(gZ - 1) = 2 extra draggable green arrows are added, allowing the model mean to be separately set for each combination of factor levels.
Click Least squares to see the best-fitting model with interaction. Rotate to y-z to help understand this model.
The easy listening music seems to result in particularly high agitation levels for early-stage patients (purple). |