or any polynomial distribution would very happily violate those limits at extreme doses, which is

obviously illogical.

If you have a binary outcome, you need to fit a function that has an S shape. The formula calculating Y

must be an expression involving X that — by design — can never produce a Y value outside of the

range from 0 to 1, no matter how large or small X may become.

Of the many mathematical expressions that produce S-shaped graphs, the logistic function is

ideally suited to this kind of data. In its simplest form, the logistic function is written like this:

, where e is the mathematical constant 2.718, known as a natural logarithm (see

Chapter 2). We will use e to represent this number for the rest of the chapter. Figure 18-2a shows

the shape of the logistic function.

The logistic function shown in Figure 18-2 can be made more versatile for representing observed data

by being generalized. The logistic function is generalized by adding two adjustable parameters named

a and b like this:

.

© John Wiley & Sons, Inc.

FIGURE 18-2: The first graph (a) shows the shape of the logistic function. The second graph (b) shows that when b is 0, the

logistic function becomes a horizontal straight line.

Notice that the

part looks just like the formula for a straight line (see Chapter 16). It’s the rest

of the logistic function that bends the straight line into its characteristic S shape. The middle of the S

(where

) always occurs when

. The steepness of the curve in the middle region is

determined by b, as follows:

If b is positive, the logistic function is an upward-sloping S-shaped curve, like the one shown in

Figure 18-2a.

If b is 0, the logistic function is a horizontal straight line whose Y value is equal to

, as

shown in Figure 18-2b.

If b is negative, the curve is flipped upside down, as shown in Figure 18-3a. Notice that this is a

mirror image of Figure 18-2a.