Introduction to Linear Regression Analysis. Douglas C. Montgomery

Introduction to Linear Regression Analysis - Douglas C. Montgomery


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      Example 2.6 The Rocket Propellant Data

ueqn31-3 ueqn32-1

       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 in32-5 2176.571
2024.318 15 2116.822
1905.890 18 2012.351
1782.928 21 1912.412
1657.395 24 1815.045
image

      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.

      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) image

      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

ueqn33-1 ueqn34-1

      because the future observation y0 is independent of in34-1. If we use in34-2 to predict y0, then the standard error of in34-3 is the appropriate statistic on which to base a prediction interval. Thus, the 100(1 − α) percent prediction interval


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