Page 155 - Intermediate Statistics for Dummies

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Part II: Making Predictions by Using Regression
Searching for the best polynomial model
When fitting a polynomial regression model to your data, the most important
idea is to always start with the simplest model possible and work your way
up as you need to. Don’t plunge in with a high-order polynomial regression
model right off the bat. Here are a couple reasons why:
High-order polynomials are hard to interpret, and their models are
complex. For example, with a straight line you can interpret the values
of the y-intercept and slope easily, but interpreting a tenth-degree poly-
nomial is hard (putting it mildly).
High-order polynomials also tend to cause overfitting. If you’re fitting
the model as close as you can to every single point in a data set, your
model may not hold for a new data set; your estimates for y could be
way off.
To fit a polynomial to a dataset in Minitab, go to Stat>Regression>Fitted Line
Plot> and click on the type of regression model you want: linear, quadratic,
or cubic. (It doesn’t go beyond a second-degree polynomial; however, these
options should cover 90 percent of the cases.) Click on the y variable from
the left-hand box and click Select; this variable will appear in the Response
(y) box. Click on the x variable from the left-hand box and click Select; it will
appear in the Predictor (x) box. Click OK.
Following are the steps for fitting a polynomial model to your data (statistical
software can jump in and fit the models for you after you tell it which ones
to fit):
1. Try to fit a first-degree polynomial (straight line) to the data first:
y = b 0 + b 1x.
This model is for a straight line. If it doesn’t fit (using both the correla-
tion coefficient, r, and the scatterplot), move to step two.
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2. Try to fit a second-degree polynomial (parabola): y = b 0 + b 1 x + b 2 x .
If the data fits the model well, stop here (see the section on assessing
model fit). If the model still doesn’t fit well, go to step three.
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3. Try to fit a third-degree polynomial: y = b 0 + b 1 x + b 2 x + b 3 x .
If the data fits the model well (check out the section on assessing model
fit), don’t go on to the next polynomial. If the model still doesn’t fit well,
go to step four.
4. Continue trying to fit higher-order polynomials until you find one that
fits or until the order of the polynomial (largest exponent) is simply
getting too large to find a reliable pattern.