Transformation of Covariates
So using squares and cubes or square roots and cubic roots are examples of covariate transformations that can be useful to obtain a linear relationship in specific situations. The most popular transformation of covariates in the medical literature is, however, a logarithmic transformation. Its popularity stems from the fact that for many processes it is rather plausible that with increasing size of x, the effect of x becomes smaller. For example, if we consider the effect of the size of a tumour on the prognosis of a patient, it is plausible that a difference between 2 and 3 centimeter has the same effect as a difference between 3 and 4 centimeters. But it is less plausible to assume that a difference between 10 and 11 centimeter has the same effect. Another example is the intake of certain substances in the food. For example, the recommended daily intake of vitamin B3 as about 18 mg. So it is plausible that in-
creasing the dose from 12 mg to 18 mg has the same beneficial effect as an increase from 18 to 24 mg. However, increasing the dose from 32 to 38 mg has probably a less beneficial effect, as the body got already enough vitamin B3. So we have often to expect some type of saturation in modelling the effect of X on some outcome, in the sense that the effect of X decreases with increasing value of X , and hence we are interested in modelling nonlinear effects of the type shown in Figure 18.1.