CHAPTER 18 A Yes-or-No Proposition: Logistic Regression 253

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: Y

e

X

1

1

/

, where e is the mathematical con-

stant 2.718, known as a natural logarithm (see Chapter 2). We will use e to repre-

sent 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 repre-

senting observed data by being generalized. The logistic function is generalized by

adding two adjustable parameters named a and b like this: Y

e

a bX

1

1

/

(

) .

Notice that the a

bX 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 Y

0 5. ) always occurs when

X

b

a

/

. 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.

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.

© John Wiley & Sons, Inc.