© John Wiley & Sons, Inc.

FIGURE 19-8: Nonlinear model fitted to drug concentration data.

Using equivalent functions to fit the parameters you really want

It’s inconvenient, annoying, and error-prone to have to perform manual calculations on the parameters

you obtain from nonlinear regression output. It’s so much extra work to read the output that contains the

estimates you need, like

and the

rate constant, then manually calculate the parameters you want,

like

and λ. It’s even more work to obtain the SEs. Wouldn’t it be nice if you could get

and λ and

their SEs directly from the nonlinear regression program? Well, in many cases, you can!

Because nonlinear regression involves algebra, some fancy math footwork can help you out. Very

often, you can re-express the formula in an equivalent form that directly involves calculating the

parameters you actually want to know. Here’s how it works for the PK example we use in the

preceding sections.

Algebra tells you that because

, then

. So why not use

instead of

in the formula you’re fitting? If you do, it becomes

. And you

can go even further than that. It turns out that a first-order exponential-decline formula can be written

either as

or as the algebraically equivalent form

.

Applying both of these substitutions, you get the equivalent model:

, which

produces exactly the same fitted curve as the original model. But it has the tremendous advantage of

giving you exactly the PK parameters you want, which are

and λ, rather than

and ke which

require post-processing with additional calculations.