670 SSSAJ: Volume 74: Number 2 • March–April 2010 Soil Sci. Soc. Am. J. 74:670–679 Published online 22 Jan. 2010 doi:10.2136/sssaj2009.0177 Received 8 May 2009. *Corresponding author (carl.bolster@ars.usda.gov). © Soil Science Society of America, 677 S. Segoe Rd., Madison WI 53711 USA All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher. On the Signiicance of Properly Weighting Sorption Data for Least Squares Analysis Nutrient Management & Soil & Plant Analysis T he sorption behavior of P to soils is generally interpreted through the use of sorption curves where a known mass of soil is mixed with a solution having a known concentration of P (Barrow, 2008; Nair et al., 1984). Ater mixing, the con- centration of P remaining in solution is measured and used to calculate the amount of P sorbed to the soil. he data are commonly analyzed by itting the Langmuir or Freundlich models to the data to estimate soil sorption parameters and their uncertainties. Because these values are oten used to further our understanding of the underlying mechanisms controlling P sorption and for making management decisions regarding P application rates, it is imperative that these values be truly representative of the soils tested. his requires that the data be analyzed using prop- er statistical methods and that the data be it with an appropriate sorption model. he most common method for analyzing P sorption data is through LS regres- sion. he objective of this method is to determine the values of the model param- eters that provide the best it to the data. his is most oten done by minimizing the sum of the squared errors (SSE) between observation and model prediction of the dependent variable, a method known as ULS regression. A signiicant drawback to this method, however, is that it tacitly assumes that the data are homoscedastic (i.e., have constant measurement uncertainty), an assumption unlikely to be valid for P sorption data. With heteroscedastic data (i.e., data with heterogeneous measure- ment uncertainty), the data should be weighted inversely by their variances using WLS regression, whereupon minimum-variance estimates of the model parameters and reliable estimates of their standard errors can be obtained (Draper and Smith, Carl H. Bolster* USDA-ARS 230 Bennett Lane Bowling Green, KY 42104 Joel Tellinghuisen Dep. of Chemistry Vanderbilt Univ. Nashville, TN 37235 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 diferent weighted LS (WLS) regression methods. In the irst 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. his method, commonly referred to as the efective 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. herefore, in this study we tested a modiied version of the efective weighting function that speciically accounts for correlated errors in x and y. he accuracy of the diferent weighting methods was assessed using Monte Carlo simulations and high-replication sorption data obtained for three diferent soil types. Our indings show that the efective variance weighting method provides superior parameter estimates and uncertainties compared with ULS or traditional WLS methods, although the diferences between the weighting methods were not always large enough to be of practical concern. We also found that weighting by the efective variance allowed improved assessments of model its. Our indings are applicable to sorption studies where the dependent variable is calculated from measured values of the so-called independent variable Abbreviations: LS, least squares; MC, Monte Carlo; NLS, nonlinear least squares; ULS, unweighted least squares; WLS, weighted least squares.