Statistics for Environmental Engineers

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Other Ways to Examine a Model


If R2 does not tell all that is needed about how well a model fits the data and how good the model may be for prediction, what else could be examined?


Graphics reveal information in data (Tufte 1983): always examine the data and the proposed model graphically. How sad if this advice was forgotten in a rush to compute some statistic like R2.


A more useful single measure of the prediction capability of a model (including a ^-variate regression model) is the standard error of the estimate. The standard error of the estimate is computed from the variance of the predicted value (y) and it indicates the precision with which the model estimates the value of the dependent variable. This statistic is used to compute intervals that have the following meanings (Hahn, 1973).


•    The confidence interval for the dependent variable is an interval that one expects, with a specified level of confidence, to contain the average value of the dependent variable at a set of specified values for the independent variables.


•    A prediction interval for the dependent variable is an interval that one expects, with a specified probability, to contain a single future value of the dependent variable from the sampled population at a set of specified values of the independent variables.


•    A confidence interval around a parameter in a model (i.e., a regression coefficient) is an interval that one expects, with a specified degree of confidence, to contain the true regression coefficient.


Confidence intervals for parameter estimates and prediction intervals for the dependent variable are discussed in Chapters 34 and 35. The exact method of obtaining these intervals is explained in Draper and Smith (1998). They are computed by most statistics software packages.

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