Page 139 - Statistics II for Dummies

Chapter 7: Getting Ahead of the Learning Curve with Nonlinear Regression 123

4. If you see curvature that’s more complex than a parabola, try to fit a
3
2
third-degree polynomial: y = b + b x + b x + b x .
0 1 2 3
If the data fit the model well, stop here and refer to the later section
“Assessing the fit of a polynomial model.” If the model still doesn’t fit
well, move to step five.
5. Continue trying to fit higher-order polynomials until you find one that
fits or until the order of the polynomial (largest exponent) simply gets
too large to find a reliable pattern.
How large is too large? Typically, if you can’t fit the data by the time the
degree of the polynomial reaches three, then perhaps a different type of
model would work better. Or you may determine that you observe too
much scatter and haphazard behavior in the data to try to fit any model.

Minitab can do each of these steps for you up to degree two (that’s step
three); from there, you need a more sophisticated statistical software pro-
gram, such as SAS or SPSS. However, most of the models you need to fit go up
to the second-degree polynomials.

Using a second-degree polynomial

to pass the quiz

The first step in fitting a polynomial model is to graph the data in a scatter-
plot and see whether the data fall into a particular pattern. Many different
types of polynomials exist to fit data that have a curved type of pattern. One
of the most common patterns found in curved data is the quadratic pattern,
or second-degree polynomial, which goes up and comes back down, or goes
down and comes back up, as the x values move from left to right (see Figure
7-3). The second-degree polynomial is the simplest and most commonly used
polynomial beyond the straight line, so it deserves special consideration.
(After you master the basic ideas based on second-degree polynomials, you
can apply them to polynomials with higher powers.)

Suppose 20 students take a statistics quiz. You record the quiz scores (which
have a maximum score of ten) and the number of hours students reported
studying for the quiz. You can see the results in Figure 7-6.
Looking at Figure 7-6, it appears that three camps of students are in this
class. Camp 1, on the left end of the x-axis, understands the stuff (as reflected
in their higher scores) but didn’t have to study hardly at all (see that their
study time on the x-axis is low). Camp 3 also did very well on the quiz (as
indicated by their high quiz scores) but had to study a great deal to get those
grades (as seen on the far-right end of the x-axis). The students in the middle,
Camp 2, didn’t seem to fare well.

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