Statistics for Environmental Engineers

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•    The null model (y = во + et) has residual sum of squares X (yi — y)2 = RSSnull model.


The comparison of the residual sums of squares (RSS) defines:


R2 = 1



RSS model



RSS null model


This shows that R2 is a model comparison and that large R2 measures only how much the model improves the null model. It does not indicate how good the model is in any absolute sense. Consequently, the common belief that a large R2 demonstrates model adequacy is sometimes wrong.


The definition of R2 also shows that comparisons are made only between nested models. The concept of proportionate reduction in variation is untrustworthy unless one model is a special case of the other. This means that R2 cannot be used to compare models with an intercept with models that have no intercept: у = во is not a reduction of the model у = Plx. It is a reduction of у = во + Plx and у = во + Pix + в2Х2.


2

A High R Does Not Assure a Valid Relation


Figure 39.1 shows a regression with R = 0.746, which is statistically significant at almost the 1% level of confidence (a 1% chance of concluding significance when there is no true relation). This might be impressive until one knows the source of the data. X is the first six digits of pi, and Y is the first six Fibonocci numbers. There is no true relation between x and у. The linear regression equation has no predictive value (the seventh digit of pi does not predict the seventh Fibonocci number).

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