CHAPTER 19 Other Useful Kinds of Regression 283

closer estimates to the truth, repeating this process until it arrives at the

best-fitting, least-squares solution.

Coming up with starting estimates for nonlinear regression problems can be

tricky. It’s more of an art than a science. If the parameters have physiological

meaning, you may be able to make a guess based on known physiology or past

experience. Other times, your estimates have to be trial and error. To improve

your estimates, you can graph your observed data in Microsoft Excel, and then

superimpose a curve from values calculated from the function for various

parameter guesses that you type in. That way, you can play around with the

parameters until the curve is at least in the ballpark of the observed data.

In this example, C0 (variable C0) is the concentration you expect at the moment

of dosing (at t 0). From Figure 19-6, it looks like the concentration starts out

around 50, so you can use 50 as an initial guess for C0. The ke parameter (variable

ke) affects how quickly the concentration decreases with time. Figure 19-6

indicates that the concentration seems to decrease by half about every few

hours, so λ should be somewhere around 4 hours. Because

0 693

.

/ ke,

a little algebra gives the equation ke = 0.693/X. If you plug in 4 hours for X, you

get ke = 0.693/4 = 0.2, so you may try 0.2 as a starting guess for ke. You tell

R the starting guesses by using the syntax: start=list(C0

50, ke

0.2).

The statement in R for nonlinear regression is nls, which stands for nonlinear

least-squares. The full R statement for executing this nonlinear regression model

and summarizing the output is:

summary nls Conc

C0 exp

ke Time

start

list C0

50

*

*

,

,

ke

0.2

Interpreting the output

As complicated as nonlinear curve-fitting may be, the output is actually quite

simple. It is formatted and interpreted like the output from ordinary linear regres-

sion. Figure 19-7 shows the relevant part of R’s output for this example.

FIGURE 19-7:

Results of

nonlinear

regression in R.