CHAPTER 19 Other Useful Kinds of Regression 279

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 inter-

pret 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: c

c x

c x

c x

c x

0

1

1

2

2

3

3

n

n

...

. But in non-

linear regression, the coefficients no longer have to appear paired up to be multi-

plied by predictor variables (like c x

2

2^{). 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 num-

ber 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 regres-

sion 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 the-

oretical 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