Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 43

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 on a future observation at x₀ is

(2.45)

The prediction interval (2.45) is of minimum width at and widens as increases. By comparing (2.45) with (2.43), we observe that the prediction interval at x₀ is always wider than the CI at x₀ because the prediction interval depends on both the error from the fitted model and the error associated with future observations.