Prediction following Regression
The popular notion is that predicting the future is impossible, and that attempts at prediction are nothing more that crystal-gazing. However, all branches of applied science rely upon prediction. These predictions may be based on extensive experimentation (as in engineering or agriculture) or they may be based on detailed, long-term observations (as in astronomy or meteorology). In all cases, however, the main issue to be confronted in prediction is how to deal with uncertainty: uncertainty about the suitability of the fitted model, uncertainty about the representativeness of the data used to parameterize the model, and uncertainty about future conditions (in particular, uncertainty about the future values of the explanatory variables).
There are two kinds of prediction, and these are subject to very different levels of uncertainty. Interpolation, which is prediction within the measured range of the data, can often be very accurate and is not greatly affected by model choice. Extrapolation, which is prediction beyond the measured range of the data, is far more problematical, and model choice is a major issue. Choice of the wrong model can lead to wildly different predictions (see p. 411).
Here are two kinds of plots involved in prediction following regression: the first illustrates uncertainty in the parameter estimates; the second indicates uncertainty about predicted values of the response. We continue with the tannin example: