Using the Regression Equation for Prediction. The fitted regression equation may be used for two kinds of predictions:

(a) To estimate the true value of y associated with a particular value of x, e.g., given x = x' to estimate the value of y' = Bo+ Bix'; or, βο

(b) To predict a single new observed value Y corresponding to a particular value of x, e.g., given x = x' to predict the value of a single measurement of y'.

Which prediction should be made? In some cases, it is sufficient to say that the true value of y (for given x) lies in a certain interval, and in other cases we may need to know how large (or how small) an individual observed Y value is likely to be associated with a particular value of x. The question of what to predict is similar to the question of what to specify (e.g., whether to specify average tensile strength or to specify minimum tensile strength) and can be answered

only with respect to a particular situation. The difference is that here we are concerned with relationships between two variables and therefore must always talk about the value of y, or Y, for fixed x.

The predicted y' or Y' value is obtained by substituting the chosen value (x') of x in the fitted equation. For a particular value of x, either type of prediction ((a) or (b)) gives the same numerical answer for y' or Y'. The uncertainty associated with the prediction, however, does depend on whether we are estimating the true value of y', or predicting the value Y' of an individual measurement of y'. If the experiment could be repeated many times, each time obtaining n pairs of (x, Y) values, consider the range of Y values which would be obtained for a given x. Surely the individual Y values in all the sets will spread over a larger range than will the collection consisting of the average Y's (one from each set).

5-4.1.2 What are the Confidence Interval Estimates for: the Line as a Whole; a Point on the Line; a Future Value of Y Corresponding to a Given Value of x?

Once we have fitted the line, we want to make predictions from it, and we want to know how good our predictions are. Often, these predictions will be given in the form of an interval together with a confidence coefficient associated with the interval-i.e., confidence interval estimates. Several kinds of confidence interval estimates may be made:

(a) A confidence band for the line as a whole. (b) A confidence interval for a point on the line-i.e., a confidence interval for y' (the true value of y and the mean value of Y) corresponding to a single value of x = x'.

If the fitted line is, say, a calibration line which will be used over and over again, we will want to make the interval estimate described in (a). In other cases, the line as such may not be so important. The line may have been fitted only to investigate or check the structure of the relationship, and the interest of the experimenter may be centered at one or two values of the variables.

Another kind of interval estimate sometimes is required:

(c) A single observed value (Y') of Y corresponding to a new value of x = x'.

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Suppose that we repeated our experiment a large number of times. Each time, we obtain n pairs of values (x,, Y.), fit the line, and compute a confidence interval estimate for y' Bo + B1x', βο βιχ', the value of y corresponding to the particular value x x'. Such interval estimates of y' are expected to be correct (i.e., include the true value of y') a proportion (1 − a) of the time. If we were to make an interval estimate of y” corresponding to another value of x = x", these interval estimates also would be expected to include y" the same proportion (1 - a) of the time. However, taken together, these intervals do not constitute a joint confidence statement about y' and y" which would be expected to be correct exactly a proportion (1 - a) of the

ORDP 20-110

time; nor is the effective level of confidence (1 - a)2, because the two statements are not independent but are correlated in a manner intimately dependent on the values x' and x" for which the predictions are to be made.

The confidence band for the whole line (a) implies the same sort of repetition of the experiment except that our confidence statements are not now limited to one x at a time, but we can talk about any number of x values simultaneously-about the whole line. Our confidence statement applies to the line as a whole, and therefore the confidence intervals for y corresponding to all the chosen x values will simultaneously be correct a proportion (1a) of the time. It will be noted that the intervals in (a) are larger than the intervals in (b) by the ratio

√2F/t. This wider interval is the "price" we pay for making joint statements about y for any number of or for all of the x values, rather than the y for a single x.

Another caution is in order. We cannot use the same computed line in (b) and (c) to make a large number of predictions, and claim that 100 (1 - a) % of the predictions will be correct. The estimated line may be very close to the true line, in which case nearly all of the interval predictions may be correct; or the line may be considerably different from the true line, in which case very few may be correct. In practice, provided our situation is in control, we should always revise our estimate of the line to include additional information in the way of new points.

Procedure

(1) Choose the desired confidence level, 1 — a

Example

a = .95

Procedure

Example

ORDP 20-110

(7) To draw the line and its confidence band, plot Y. at two of the extreme selected values of X. Connect the two points by a straight line. At each selected value of X, also plot Y. + W, and Y. - W1. Connect the upper series of points, and the lower series of points, by smooth curves. If more points are needed for drawing the curves for the band, note that, because of symmetry, the calculation of W1 at n values of X actually gives W1 at 2n values of X.

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