Prediction and Extrapolation in the Simple Linear Model

The goal of regression analysis is usually two-fold. First, a model is needed to explain the data. Second, using the model, predictions about mean responses or future observations may be needed. The distinction between mean responses and future observations must be clarified. Mean responses are based on already observed 

The standard error for Y(x0) is interpreted as the standard error the mean response conditional on xo. Thus, the variance of Y(xq) is 

Note that the standard error of prediction is not a constant for all values of xo. but reflects where xo is collected in relation to the mean. Observations removed from the mean of x will have larger standard errors of prediction than values close to the mean. Equation (2.34) is developed as the confidence interval for a single observation measured at x0. If more than one observation is made at xo, the term 1/n in Eqs. (2.33) and (2.34) is substituted with the term m/n, where m is the number of observations at x0. Note that m is contained within n. If the confidence interval is made for all points on the regression line, the result would be a confidence band. 

The confidence interval for a future response, one not in the original data set, must be more variable due to the additional uncertainty in its measurement. Thus, Eq. (2.33) is modified to 

Clearly prediction intervals are wider than corresponding confidence intervals. 

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