The effects of transformations and preliminary tests for non-linearity in regression

Non-linear relationships between two variables are often detected as a result of a preliminary statistical test for linearity. Common approaches to dealing with non-linearity are to (a) make a linearizing transformation in the independent variable or (b) fit a relationship that is non-linear in the independent variable, such as including a quadratic term. With either approach, the resulting test for association between the two variables can have an inflated type I error. We consider testing the significance of the quadratic term in a quadratic model as a preliminary test for non-linearity. Using simulation experiments and asymptotic arguments, we quantify the type I error inflation and suggest simple modifications of standard practice to protect the size of the type I error. In the case of quadratic regression, the type I error will be increased by roughly 50 per cent. The simple strategy of appropriately correcting the alpha-level is shown to have minimal loss of power if the relationship is truly linear. In the case of a linearizing transformation, the impact on the type I error will depend on the values of the independent variable and on the set of potential linearizing transformations considered. Simulation results suggest that a procedure which adjusts the test statistic according to the results of the preliminary test may offer adequate protection.