It is relatively easy to misuse the no-intercept model, particularly in situations where the data lie in a region of x space remote from the origin. For example, consider the no-intercept fit in the scatter diagram of chemical process yield (y) and operating temperature (x) in Figure 2.12a. Although over the range of the regressor variable 100°F ≤ x ≤ 200°F, yield and temperature seem to be linearly related, forcing the model to go through the origin provides a visibly poor fit. A model containing an intercept, such as illustrated in Figure 2.12b, provides a much better fit in the region of x space where the data were collected.
Frequently the relationship between y and x is quite different near the origin than it is in the region of x space containing the data. This is illustrated in Figure 2.13 for the chemical process data. Here it would seem that either a quadratic or a more complex nonlinear regression model would be required to adequately express the relationship between y and x over the entire range of x. Such a model should only be entertained if the range of x in the data is sufficiently close to the origin.
Figure 2.12 Scatter diagrams and regression lines for chemical process yield and operating temperature: (a) no-intercept model; (b) intercept model.
Figure 2.13 True relationship between yield and temperature.
The scatter diagram sometimes provides guidance in deciding whether or not to fit the no-intercept model. Alternatively we may fit both models and choose between them based on the quality of the fit. If the hypothesis β0 = 0 cannot be rejected in the intercept model, this is an indication that the fit may be improved by using the no-intercept model. The residual mean square is a useful way to compare the quality of fit. The model having the smaller residual mean square is the best fit in the sense that it minimizes the estimate of the variance of y about the regression line.
Generally R2 is not a good comparative statistic for the two models. For the intercept model we have
Note that R2 indicates the proportion of variability around
so that the no-intercept model analogue for R2 would be
The statistic
There are alternative ways to define R2 for the no-intercept model. One possibility is
However, in cases where
Example 2.8 The Shelf-Stocking Data
The time required for a merchandiser to stock a grocery store shelf with a soft drink product as well as the number of cases of product stocked is shown in Table 2.10. The scatter diagram shown in Figure 2.14 suggests that a straight line passing through the origin could be used to express the relationship between time and the number of cases stocked. Furthermore, since if the number of cases x = 0, then shelf stocking time y = 0, this model seems intuitively reasonable. Note also that the range of x is close to the origin.
The slope in the no-intercept model is computed from Eq. (2.50) as
Therefore, the fitted equation is
This regression line is shown in Figure 2.15. The residual mean square for this model is MSRes = 0.0893 and
We may also fit the intercept model to the data for comparative purposes. This results in
The t statistic for testing H0: β0 = 0 is t0 = −0.65, which is not significant, implying that the no-intercept model may provide a superior fit. The residual mean square for the intercept model is MSRes = 0.0931 and R2 = 0.9997. Since MSRes for the no-intercept model is smaller than MSRes for the intercept model, we conclude that the no-intercept model is superior. As noted previously, the R2 statistics are not directly comparable.
TABLE 2.10 Shelf-Stocking Data for Example 2.8
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Times,
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