Introduction to Linear Regression Analysis. Douglas C. Montgomery

      Example 2.6 The Rocket Propellant Data

Consider finding a 95% CI on E(y|x₀) for the rocket propellant data in Example 2.1. The CI is found from Eq. (2.43) as

If we substitute values of x₀ and the fitted value at the value of x₀ into this last equation, we will obtain the 95% CI on the mean response at x = x₀. For example, if , then , and the CI becomes

Table 2.6 contains the 95% confidence limits on E(y|x₀) for several other values of x₀. These confidence limits are illustrated graphically in Figure 2.4. Note that the width of the CI increases as increases.

       TABLE 2.6 Confidence Limits on E(y|x0) for Several Values of x0

Lower Confidence Limit	x 0	Upper Confidence Limit
2438.919	3	2593.821
2341.360	6	2468.481
2241.104	9	2345.836
2136.098	12	2227.942
2086.230		2176.571
2024.318	15	2116.822
1905.890	18	2012.351
1782.928	21	1912.412
1657.395	24	1815.045

Figure 2.4 The upper and lower 95% confidence limits for the propellant data.

Many regression textbooks state that one should never use a regression model to extrapolate beyond the range of the original data. By extrapolation, we mean using the prediction equation beyond the boundary of the x space. Figure 1.5 illustrates clearly the dangers inherent in extrapolation; model or equation error can severely damage the prediction.

      Equation (2.43) points out that the issue of extrapolation is much more subtle; the further the x value is from the center of the data, the more variable our estimate of E(y|x₀). Please note, however, that nothing “magical” occurs at the boundary of the x space. It is not reasonable to think that the prediction is wonderful at the observed data value most remote from the center of the data and completely awful just beyond it. Clearly, Eq. (2.43) points out that we should be concerned about prediction quality as we approach the boundary and that as we move beyond this boundary, the prediction may deteriorate rapidly. Furthermore, the farther we move away from the original region of x space, the more likely it is that equation or model error will play a role in the process.

      This is not the same thing as saying “never extrapolate.” Engineers and economists routinely use prediction equations to forecast a variable of interest one or more time periods in the future. Strictly speaking, this forecast is an extrapolation. Equation (2.43) supports such use of the prediction equation. However, Eq. (2.43) does not support using the regression model to forecast many periods in the future. Generally, the greater the extrapolation, the higher is the chance of equation error or model error impacting the results.

      The probability statement associated with the CI (2.43) holds only when a single CI on the mean response is to be constructed. A procedure for constructing several CIs that, considered jointly, have a specified confidence level is a simultaneous statistical inference problem. These problems are discussed in Chapter 3.

2.5 PREDICTION OF NEW OBSERVATIONS

      An important application of the regression model is prediction of new observations y corresponding to a specified level of the regressor variable x. If x₀ is the value of the regressor variable of interest, then

      (2.44)

      is the point estimate of the new value of the response y₀.

      Now consider obtaining an interval estimate of this future observation y₀. The CI on the mean response at x = x₀ [Eq. (2.43)] is inappropriate for this problem because it is an interval estimate on the mean of y (a parameter), not a probability statement about future observations from that distribution. We now develop a prediction interval for the future observation y0.

      Note that the random variable

is normally distributed with mean zero and variance

      because the future observation y₀ is independent of . If we use to predict y₀, then the standard error of is the appropriate statistic on which to base a prediction interval. Thus, the 100(1 − α) percent prediction interval