Statistics for Environmental Engineers

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The least squares parameter estimates for a general linear model n = во + Pi x + в2 x2 + ••• + Pnxn are obtained from:


minimize S = ^w;(y; n)2 = ^w;[y; — (в0 +    x; +—+Pnxn)]2


The analytical solution for a straight-line model applied to calibration is given in Gibbons (1994), Otto (1999), and Zorn et al. (1997, 1999).

Determining the Appropriate Weights


If the variance is not constant, the magnitude of the weights will depend somehow on the magnitude of the variance. We present two ways in which the weights might be assigned.

Method 1


The weights are inversely proportional to the variance of each observation (w; = 1/ of) where s2 is used as an estimate of o2. Obviously this method can only be used when there are replicate measurements to calculate the s2. The weights may increase smoothly from low to high levels of the analyte, as shown in the left-hand panel of Figure 37.5, or they might be as shown in the other two panels. Using weights that are inversely proportional to the variance will deal with any of these cases.


1 10 100


У

Method 2


The weights are a smooth function of the dependent variable, such as:


0 = ayb


and


_-_ = _1_


2 = b


ayt

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