CHAPTER 19 Other Useful Kinds of Regression 277

Examining nonlinear trends

The straight line in Figure 19-3 doesn’t account for the fact that the accident rate

remained low for the first few years and then started to climb rapidly after 2016.

Perhaps the true trend isn’t a straight line, where the rate increases by the same

amount each year. It may instead be an exponential increase, where the rate

increases by a certain percentage each year. You can have R fit an exponential

increase by changing the link option from identity to log in the statement that

invokes the Poisson regression:

glm(formula = Accidents ~ Year, family = poisson(link = “log”))

This produces the output shown in Figure 19-4 and graphed in Figure 19-5.

Because of the log link used in this regression run, the coefficients are related to

the logarithm of the event rate. Thus, the relative rate of increase per year is

obtained by taking the antilog of the regression coefficient for Year. This is done

by raising e (the mathematical constant 2.718. . .) to the power of the regression

coefficient for Year: e^{0 10414}

.

, which is about 1.11. So, according to an exponential

increase model, the annual accident rate increases by a factor of 1.11 each year —

meaning there is an 11 percent increase each year. The dashed-line curve in

Figure  19-4 shows this exponential trend, which appears to accommodate the

steeper rate of increase seen after 2016.

Comparing alternative models

The bottom of Figure 19-4 shows the AIC value for the exponential trend model is

78.476, which is about 3.2 units lower than for the linear trend model in

Figure 19-2 (AIC 81.72). Smaller AIC values indicate better fit, so the true trend

is more likely to be exponential rather than linear. But you can’t conclude that the

model with the lower AIC is really better unless the AIC is about six units better.

So in this example, you can’t say for sure whether the trend is linear or exponen-

tial, or potentially another distribution. But the exponential curve does seem to

predict the high accident rates seen in 2020 and 2021 better than the linear trend

model.

FIGURE 19-4:

Output from an

exponential trend

Poisson

regression.