The use of weighted least squares is common in analytical chemistry, but the evaluation of uncertainty is generally poorly documented. Do you know any international standard or guide that addresses the evaluation of uncertainty consistent with GUM (JCGM 100) when interpolated values are used in a curve obtained by weighted least squares.
I think this paper may help:
Article A Unified Formula for Uncertainty Estimation in Interlaborat...
Abstract: A variety of statistical methods are available for estimating consensus values in interlaboratory studies (including key comparisons). These methods, although developed based on different approaches and ranging from simple to complex, produce largely compatible estimates of consensus values. However, the formulas or methods for estimating the standard uncertainty (SU) associated with the consensus value estimators are different and may produce incompatible results. This causes confusion among practitioners about the performance of consensus value estimators. To resolve the incompatibility problem and to eliminate potential confusion, we propose using a unified formula for estimating the SU. The unified formula is an unconditional SU estimator that accounts for both within- and between-laboratory variances. It is applicable to all weighted-average (WA) type estimators. The unified formula makes it possible to evaluate the performance of WA-type consensus value estimators on the same methodological basis. A case study of estimating the Newtonian constant of gravitation is presented to demonstrate the effectiveness of the unified formula.
Dear Pablo,
You raise a very interesting question. The use of least squares solution is very famous in Surveying or Geomatics engineering and it is known as Adjustment Computations. So many interesting ideas and approaches were developed within the framework of adjustment computations. Please check books in adjustment computations.
Regards,
Gamal
Uncertainty in your example is evaluated through Monte Carlo simulations :)
Thanks Andrew Paul McKenzie Pegman . I have studied many aspects of estimating uncertainties from the perspective of GUM, as well as from Monte Carlo. Please check my latest paper at
Article MCM Alchimia Methods: solved examples on computer-aided unce...
where I address most of JCGM 100 and 101 examples from both approaches. However, I have not found coherent documentation on weighted least squares neither from GUM, nor from MCM, apart from the fact that in my work in Physical Metrology the use of weights in least squares is not common, so I do not know in depth its use.
Pablo Constantino -
The top of page 5 in the following may show you what you want to know about the estimated variance of the prediction error:
https://www.researchgate.net/publication/261586154_Using_Prediction-Oriented_Software_for_Survey_Estimation.
The paper in general, however, handles some special problems and that paper showed how they were solved.
To obtain the regression coefficients, simultaneous equations have to be solved, one for each predictor and one more for the intercept term, IF one is needed. So the easiest case is for one predictor and no intercept term. See the bottom half of page 2 in https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys. For a case of two predictors, see https://www.researchgate.net/publication/261534907_WEIGHTED_MULTIPLE_REGRESSION_ESTIMATION_FOR_SURVEY_MODEL_SAMPLING. For one predictor and an intercept term, an intermediate level of effort is needed. However, your software should do this for you.
The task then is to determine the regression weight, w.
It works well to use the regression weight in the form
w = (predicted y)^(-2gamma)
where gamma is the coefficient of heteroscedasticity.
Note the following:
Nature and magnitude of heteroscedasticity regarding V(y|y*),
in regressions of form y=y*+e,
for finite populations:
https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as bounded by Ken Brewer,
https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity,
and
https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and the various updates there, in reverse chronological order. Note the preprint for an upcoming paper in the Pakistan Journal of Statistics.
Your software should let you input the regression weight 'formula' (based on the coefficient of heteroscedasticity, gamma) which is used in the regression coefficients and in finding what you want: the estimated variance of the prediction error.
I assume you know the 'correct' model. If you leave out an important predictor, you could have substantial bias, but I assume that you have just one predictor, so that might not happen. Therefore we can just look at the estimated variance of the prediction error, not worrying about bias. And you did mention interpolation, not extrapolation.
However, if there are model specification problems, that will add to "uncertainty." If the y-value data are a bit dirty, that impacts the estimated variance of the prediction errors, but I think it will make it larger to the degree that is appropriate. However, if the predictor (or predictors?) are "dirty," you might want errors-in-variables regression, which requires multiple measures in each case.
I hope that some of the above is helpful. This is not my subject matter area, however, so there may be some other considerations of which I am not aware. However, they should all fall somewhere in the notes above, such as model specification error.
Cheers - Jim
What are your variables? You must have some idea of the measurement precision and hence the uncertainty.
You might find useful examples here:
www.lerenisplezant.be/fitting.htm
It's been a long time since you asked this question. But, in case it is still of interest to you, I leave you a link that might be useful to you:
https://www.e-medida.es/numero-11/sobre-los-ajustes-por-minimos-cuadrados-en-metrologia/
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