In all the preceding discussion of regression and correlation, both simple and multiple, we have always assumed and tested for linear relationships. This means that in a simple bivariate regression, the regression line is always a straight line; in a multivariate regression, the relationship between the dependent variable and each of the explanatory variables is a straight line. However, we were warned in §11.1.1 that the results obtained by applying the classical linear regression (CLR) model might be seriously distorted if the actual relationship between the variables was not linear.

One illustration of such a non-linear association was given for cost data in figure 11.1. As another, we can take the familiar relationship between age and height. Data for males in Britain from birth to age 34 are plotted as a scatter diagram in figure 12.1. We see a changing pattern with an initial period of rapid increase between birth and 2 years, then steady growth with a slight acceleration from about 12 to 16 years, and after that a period of stability from an age of about 16 or 17. If we extended the chart to later ages we would find a slight decrease in height from about 40 years of age, and this would become more pronounced as osteoporosis takes its toll in old age.