Ordinary least-squares weights

A.1 Ordinary least-squares weights for linear contrasts... 

Ordinary least squares regression requires constant variance across the range of data. This has typically not been satisfied with chromatographic data ( 4,9,10 ). Some have adjusted data to constant variance by a weighted least squares method ( ) The other general adjustment method has been by transformation of data. The log-log transformation is commonly used ( 9,10 ). One author compares the robustness of nonweighted, weighted linear, and maximum likelihood estimation methods ( ). Another has... 

I Ordinary least squares regression Weighting variable = none ... 

Assuming that we have measured a series of concentrations over time/ we can define a model structure and obtain initial estimates of the model parameters. The objective is to determine an estimate of the parameters (CLe, Vd) such that the differences between the observed and predicted concentrations are comparatively small. Three of the most commonly used criteria for obtaining a best fit of the model to the data are ordinary least squares (OLS)/ weighted least squares (WLS)/ and extended least squares (ELS) ELS is a maximum likelihood procedure. These criteria are achieved by minimizing the following quantities/... 

In case of a bilinear model the above reads (X — AB0W 2 = XW — AB W 2 = XW — AH 2. Thus, by fitting the bilinear model AH to the data scaled within the second mode, XW, in an ordinary least squares sense, the weighted loss function is automatically optimized. This is the basic mathematical rationale behind scaling. 

Kiers HAL, Weighted least squares fitting using ordinary least squares algorithms, Psychometrika, 1997, 62, 251-266. 

When the variance of an observation is large, less weight is given to that observation. Conversely, when an observation is precisely measured with small variance, more weight is given to that observation, a beneficial property. Obviously, ordinary least-squares is a special case of weighted least-squares in that all the observations have weights equal to one. 

Ordinary least squares regression Weighting variable = none Dep. var. = G Mean= 100.7008114, S.D.= 14.08790232... 

Discrete least square Ordinary least square Regular least square Weighted least square... 

---