254 PART 5 Looking for Relationships with Correlation and Regression

»

» ^{If b is 0, the logistic function is a horizontal straight line whose Y value is equal}

to 1

1

/

e^a , 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.»

» ^{If b is a very large number, either positive or negative, the logistic curve}

becomes so steep that it looks like what mathematicians call a step function, as

shown in Figure 18-3b.

Because the logistic curve approaches the limits 0.0 and 1.0 for extreme values of

the predictor(s), you should not use logistic regression in situations where the

fraction of individuals positive for the outcome does not approach these two lim-

its. Logistic regression is appropriate for the radiation example because none of

the individuals died at a radiation exposure of zero REMs, and all of the individu-

als died at doses of 686 REMs and higher. If we imagine a study of patients with a

disease where the outcome is a cure, if taking a drug in very high doses would not

always cause a 100 percent cure, and the disease could resolve on its own without

any drug, the data would not be appropriate. This is because some patients with

high doses would still have an outcome value of 0, and some patients at zero dose

would have an outcome value of 1.

Logistic regression fits the logistic model to your data by finding the values of a

and b that make the logistic curve come as close as possible to all your plotted

points. With this fitted model, you can then predict the probability of the outcome.

See the later section “Predicting probabilities with the fitted logistic formula” for

more details.

FIGURE 18-3:

The first graph (a)

shows that when

b is negative,

the logistic

function slopes

­downward. The

second graph (b)

shows that when

b is very large,

the logistic

function becomes

a “step function.”

© John Wiley & Sons, Inc.