Anything Goes with Nonlinear Regression

Here, we finally present the potentially most challenging type of least-squares regression, and that’s

general nonlinear least-squares regression, or nonlinear curve-fitting. In the following sections, we

explain how nonlinear regression is different from other kinds of regression. We also describe how to

run and interpret a nonlinear regression using an example from drug research, and we show you some

tips involving equivalent functions.

Distinguishing nonlinear regression from other kinds

In the kinds of regression we describe earlier in this chapter and in Chapters 16, 17, and 18, the

predictor variables and regression coefficients always appear in the model as a linear combination:

. But in nonlinear regression, the coefficients no longer have

to appear paired up to be multiplied by predictor variables (like

). In nonlinear regression,

coefficients have a more independent existence, and can appear on their own anywhere in the formula.

Actually, the term coefficient implies a number that’s multiplied by a variable’s value. This means that

technically, you can’t have a coefficient that isn’t multiplied by a variable, so when this happens in

nonlinear regression, they’re referred to instead as parameters.

The formula for a nonlinear regression model may be any algebraic expression. It can involve

sums, differences, products, ratios, powers, and roots. These can be combined together in a

formula with logarithmic, exponential, trigonometric, and other advanced mathematical functions

(see Chapter 2 for an introduction to these items). The formula can contain any number of

predictor variables, and any number of parameters. In fact, nonlinear regression formulas often

contain many more parameters than predictor variables.

Unlike other types of regression covered in this chapter and book, where a regression

command and code are used to generate output, developing a full-blown nonlinear regression

model is more of a do-it-yourself proposition. First, you have to decide what function you want to

fit to your data, making this choice from the infinite number of possible functions you could

select. Sometimes the general form of the function is determined or suggested by a scientific

theory. Using a theory to guide your development of a nonlinear function means relying on a

theoretical or mechanistic function, which is more common in the physical sciences than life

sciences. If you choose your nonlinear function based on a function with a generally similar

shape, you are using an empirical function. After choosing the function, you have to provide

starting estimates for the value of each of the parameters appearing in the function. After that, you

can execute the regression. The software tries to refine your estimates using an iterative process

that may or may not converge to an answer, depending on the complexity of the function you’re

fitting and how close your initial estimates are to the truth. And in addition to attending to these

unique issues, analysts running a nonlinear regression face all the other complications of

multivariate regression, such as collinearity, as described in Chapter 17.