CHAPTER 18 A Yes-or-No Proposition: Logistic Regression 261

Calculating effective doses on a logistic curve

One point of special significance on a logistic curve with a numerical predictor is

a median effective dose. This is a dose (X) that produces a 50 percent response,

meaning where Y

0 5. , and is designated ED50. Similarly, the X value that makes

Y

0 8. is called the 80 percent effective dose and is designated ED80, and so on. You

can calculate these dose levels from the a and b parameters of the fitted logistic

model in the preceding section.

Using

your

high-school

algebra,

you

can

solve

the

logistic

formula

Y

e

a bX

1

1

/

_(

) for X as a function of Y. If you don’t remember how to do that,

don’t worry, here’s the answer:

X

Y

Y

a

b

log 1

where log stands for natural logarithm. If you substitute 0.5 for Y in the preceding

equation because you want to calculate the ED50, the answer is a b

/

. Similarly,

substituting 0.8 for Y gives the ED

a

b

80

1 39

as ^.

.

Imagine a logistic regression model based on a study of participants taking a drug

at different doses where the predictor is level of drug dose, and the outcome is

that it produces a therapeutic response. The model has a

3 45

.

and b

0 0204

.

mg/dL.  In this case, the ED80 (or 80 percent effective dose) would be equal to

1 39

3 45

0 0234

.

.

/ .

, which works out to about 207 mg/dL.

FIGURE 18-5:

The logistic curve

that fits the data

from Table 18-1.

© John Wiley & Sons, Inc.