Importance of models without interaction

If the explanatory variables in a general linear model (GLM) do not interact in their effect on the response, the model can be written in the form,

The effect of changing any explanatory variable is then the same whatever the values of the other variables.

If there is no interaction, we can separately describe the effects of each explanatory variable on the response.

This greatly simplifies the description of the relationships and makes the model much easier to understand.

Models for 2 numerical explanatory variables, X and Z

The simplest type of general linear model (GLM) for two numerical explanatory variables, X and Z, is:

In this model, the effect on the response of changing X by 1 is β1 whatever the value of Z. In a similar way, the effect on the response of changing Z by 1 is β2 whatever the value of X. We therefore say that there is no interaction between the effects of X and Z.

In some applications, the explanatory variables do interact in their effect on the response. For example, increasing the amount of fertiliser may increase crop yield if the temperature is high, but have no effect at low temperatures. A simple model that allows for interaction is:

This model was described more fully in an earlier section.

An interaction between 2 variables is often modelled with an extra explanatory variable that is their product.

Graphical display of models

Both the models with and without interaction can be represented as surfaces on a 3-dimensional scatterplot of the data. For the model without interaction, the surface is a plane, but the model with interaction allows some curvature.

If there is no interaction, slices through this surface at different values of Z are parallel.

If there is interaction, these slices are not parallel.

Energy expenditure of bees

In an experiment, an entomologist recorded energy expenditure (joules/sec) for bees drinking water with different sucrose concentrations (%) and at different temperatures. The following 3-dimensional scatterplot shows the data.

A surface represents the model in the scatterplot. Click Least squares to see the least squares estimates for the model parameters. Rotate the diagram by dragging with the mouse or using the rotation buttons on the right to get a feel for where the data lie in relation to the surface.

Estimated effect of increasing temperature by 1 degree:


Finally, select Grid from the Display type menu, then click the y-x rotation button. The grid lines correspond to slices through the surface at different z-values. Observe that they are not parallel, corresponding to an interaction between the effects of temperature and sucrose.

Increasing temperature at low sucrose levels seems to increase energy consumption by less than at high sucrose levels.


Inference

The existence of interaction between two numerical explanatory variables can be tested using a t-test for whether the coefficient of the interaction term in the GLM is zero.

Energy expenditure of bees

We will now test whether the interaction between sucrose and temperature that is apparent in the least squares estimates could have arisen by chance.

From the p-value associated with the interaction term, 0.0013, we conclude that there is extremely strong evidence of an interaction between the effects of temperature and sucrose.

Note that we should not attempt to interpret the p-values for the main effects of temperature or sucrose if we believe that there is an interaction.

Since we have concluded that the effect of temperature is different at different sucrose levels, it makes no sense to test whether temperature has any effect on the response.