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On the Significance of Properly Weighting Sorption Data for Least Squares Analysis

Bolster, Carl H., Tellinghuisen, Joel
Soil Science Society of America journal 2010 v.74 no.2 pp. 670
soil chemistry, phosphorus, sorption, data analysis, least squares, accuracy, statistical analysis, variance, correlation, Monte Carlo method, soil types, uncertainty
In this study, we examined the role of proper weighting in the least squares (LS) analysis of P sorption data when both the dependent (y) and independent (x) variables contain heteroscedastic errors. We compared parameter estimates and uncertainties obtained with unweighted LS (ULS) regression with those obtained using two different weighted LS (WLS) regression methods. In the first WLS method, we weighted the data by the inverse of the variance in y. In the second WLS method, we included the variance in x when calculating the weights. This method, commonly referred to as the effective variance method, has primarily been applied to data with uncorrelated errors in x and y, conditions not representative of sorption studies where values of y are calculated from measured values of x. Therefore, in this study we tested a modified version of the effective weighting function that specifically accounts for correlated errors in x and y. The accuracy of the different weighting methods was assessed using Monte Carlo simulations and high-replication sorption data obtained for three different soil types. Our findings show that the effective variance weighting method provides superior parameter estimates and uncertainties compared with ULS or traditional WLS methods, although the differences between the weighting methods were not always large enough to be of practical concern. We also found that weighting by the effective variance allowed improved assessments of model fits. Our findings are applicable to sorption studies where the dependent variable is calculated from measured values of the so-called independent variable.