Non-linear response

While a linear response is justifiable in many situations, there are also occasions when the response is not expected to be linear. For example, a least squares regression of probability incorrectly implies that predicted probabilities can lie outside the acceptable range of 0 to 1. To deal with such situations, there are two basic approaches. Either you nonlinearly transform the response variable (see normalising transformations, chapter 2) and then do a linear regression using the transformed response, or you non-linearly transform the fitted values, which are a linear combination of the explanatory variables. For example, the widely applied logistic regression uses the logit transformation $y'=\log (y/(1-y))$ (``log odds''). The logarithm transformation is often used when dealing with quantities that are strictly positive such as prices, while the square root transformation is useful for transforming positive and zero count data (e.g. number of storms) prior to linear regression. In a ground-breaking paper, Nelder and Wedderburn (1972) introduced a formal and now widely used procedure for choosing the link function $g(y)$ known as Generalized LInear Modelling GLIM (note ``Generalized'' not ``General'' !).