Statistics for Environmental Engineers

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The calibration model may not be a straight line, and the variance of the signal may increase as the values of x and у increase. This is the case study in Chapter 37 which deals with weighted regression.


There are other interesting problems associated with calibrations. In some analytical applications (e.g., photometric titrations), it is necessary to locate the intersection of two regression lines. In the standard addition method, the concentration is estimated by extrapolating a straight line down to the abscissa. References at the end of the chapter provide guidance on these problems.


A special case is where there are errors in both x and y. Errors in x can be ignored if a2x < O2 . Carroll and Spiegelman (1986) examine this criteria in some detail. The effect of errors in x is to pull the regression line down so the estimated slope is less than would be estimated if errors in x were taken into account. In going from у to x, this could badly overestimate x. This emphasizes the importance of using accurate standards in preparing calibration curves.


Values of the correlation coefficient (r) and the coefficient of determination (R2) are often cited as evidence that the calibration relation is strong and useful, or that the calibration is in fact a straight line. An R2 value near 1.00 does not prove these points and in the context of calibration curves it has little meaning of any kind. Values of R2 = 0.99+ are to be expected in calibration curves. If the relation between standard and instrumental response is not clean and strong, there is simply no useful measurement method. Second, the value of R2 value can be increased without increasing the precision of the measurements or of the predictions. This is done simply by expanding the range of concentrations covered by the standards. Third, R2 can be large (>0.98) although the curve deviates slightly from the linear. The coefficient of determination (R2) is discussed in Chapter 39.

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