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Chapter 7: Getting Ahead of the Learning Curve with Nonlinear Regression 125

After you know that a quadratic polynomial seems to be a good fit for the
data, the next challenge is finding the equation for that particular parabola
that fits the data from among all the possible parabolas out there.

Remember from algebra that the general equation of a parabola is
2
y = ax + bx + c. Now you have to find the values of a, b, and c that create the
best-fitting parabola to the data (just like you find the a and the b that create
the best-fitting line to data in a linear regression model). That’s the object of
any regression analysis.

Suppose that you fit a quadratic regression model to the quiz-score data by
using Minitab (see the Minitab output in Figure 7-8 and the instructions for
using Minitab to fit this model in the previous section). On the top line of
the output, you can see that the equation of the best-fitting parabola is quiz
2
score = 9.82 – 6.15 * (study time) + 1.00 * (study time) . (Note that y is quiz
score and x is study time in this example because you’re using study time to
predict quiz score.)

Figure 7-8:
Minitab Polynomial Regression Analysis: Quiz Score versus Study Time
output for
fitting a The regression equation is
Quiz score = 9.823 − 6.149 study time + 1.003 study time**2
parabola
to the S = 1.04825 R-Sq = 91.7% R-Sq(adj) = 90.7%
quiz-score
data.

The scatterplot of the quiz-score data and the parabola that was fit to the
data via the regression model is shown in Figure 7-9. From algebra, you may
remember that a positive coefficient on the quadratic term (here a = 1.00)
means the bowl is right-side up, which you can see is the case here.

Looking at Figure 7-9, it appears that the quadratic model fits this data pretty
well, because the data fall close to the curve that Minitab found. However,
data analysts can’t live by scatterplots alone, so the next section helps you
figure out how to assess the fit of a polynomial model in more detail.

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