Advanced Statistics with Applications in R

Chapter 9Nonlinear regression

Nonlinear regression is a powerful statistical tool rarely covered in traditional statistics textbooks. Nonlinear regression is not only a practically important technique, but also an important example of a real‐life statistical model where classical theory of unbiased estimation and sufficient statistics do not work. Undoubtedly, the linear model is the champion among statistical techniques when it comes to modeling relationships between variables. However, sometimes the association is not linear such as when the response has a sigmoid shape – then nonlinear regression must be applied. Unlike linear regression, nonlinear regression is a complex statistical model where small sample properties are difficult to study – here, we rely on asymptotic properties. The major method of estimation is the nonlinear least squares, which, unlike linear least squares, requires iterations. Various numerical issues arise in the nonlinear regression model: (1) finding satisfactory starting value, (2) existence of the solution of the nonlinear least squares, (3) multiple local minima for the residual sum of squares. This chapter covers major concepts of nonlinear regression, illustrated with various examples, and its implementation in R. The chapter ends with the design of experiments emerging in engineering or chemical sciences where the values of the independent variable may be chosen by the experimentalist.

9.1 Definition and motivating examples

Nonlinear regression ...