Suppose that a new cereal (say, the Chocolate Frosted Sugar Bombs loved by Calvin, the comic strip character written by Bill Watterson) arrives on the market with a very high sugar content of 30 grams per serving. Let us use our estimated regression equation to estimate the nutritional rating for Chocolate Frosted Sugar Bombs: y = 59.4 ? 2.42(sugars)=59.4 ? 2.42(30) = ?13.2. In other words, Calvins cereal has so much sugar that its nutritional rating is actually a negative number, unlike any of the other cereals in the data set (minimum = 18) and analogous to a student receiving a negative grade on an exam. What is going on here?
The negative estimated nutritional rating for Chocolate Frosted Sugar Bombs is an example of the dangers of extrapolation. Analysts should confine the estimates and predictions made using the ERE to values of the predictor variable contained within the range of the values of x in the data set. For example, in the cereals data set, the lowest sugar content is zero grams and the highest is 15 grams, so that predictions of nutritional rating for any value of x (sugar content) between zero and 15 grams would be appropriate. However, extrapolation, making predictions for xvalues lying outside this range, can be dangerous, since we do not know the nature of the relationship between the response and predictor variables outside this range.
Extrapolation should be avoided if possible. If predictions outside the given range of x must be performed, the end user of the prediction needs to be informed that no x-data is available to support such a prediction. The danger lies in the possibility that the relationship between x and y, which may be linear within the range of x in the data set, may no longer be linear outside these bounds.
Consider Figure 4.4. Suppose that our data set consisted only of the data points in black but that the true relationship between x and y consisted of both the black (observed) and the gray (unobserved) points. Then, a regression line based solely on the available (black dot) data would look approximately similar to the regression line indicated. Suppose that we were interested in predicting the value of y for an x-value located at the triangle. The prediction based on the available data would then be represented by the dot on the regression line indicated by the upper arrow. Clearly, this prediction has failed spectacularly, as shown by the vertical line indicating the huge prediction error. Of course, since the analyst would be completely unaware of the hidden data, he or she would hence be oblivious to the massive scope of the error in prediction. Policy recommendations based on such erroneous predictions could certainly have costly results.
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