Model with curvature and interaction

We have now expanded the linear model,

with quadratic terms to explain curvature and with an interaction term. These can be combined in a model that has both nonlinearity and interaction,

An example illustrates this model.

Energy expenditure of bees

On the previous page, we found that there was strong evidence of interaction between temperature and sucrose on energy expenditure of bees. Much earlier, we also found that there was moderately strong evidence of curvature in the relationship of temperature to energy expenditure, but not of sucrose. The diagram below shows the fit of a model with both interaction and quadratic terms in temperature and sucrose.

In this model, there is still strong evidence of interaction between the effects of temperature and sucrose, and of curvature in the effect of temperature.

Note however that there is now also evidence of nonlinearity in the effect of sucrose — its p-value has reduced from 0.1023 in the earlier model without an interaction term to 0.0243 in this model. There is therefore also moderately strong evidence that the relationship of energy expenditure to sucrose is also nonlinear.

The reason is that the interaction term has explained a considerable amount of the variation in energy expenditure, so the residuals and the estimate of the error standard deviation are much smaller with an interaction term. Since the latter is part of the denominator of all t-ratios, the t-ratios have become bigger and more significant.

The lower the unexplained variation in the response, the greater the chance of detecting explanatory variables (and terms) that have a small effect on the response.

Click the y-x and y-z rotation buttons to help understand the nature of the curvature and interaction in this model.