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Combining non‐linear regressions that have unequal error variances and some parameters in common
Author(s) -
Hamilton David C.,
Knop O.
Publication year - 1998
Publication title -
journal of the royal statistical society: series c (applied statistics)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.205
H-Index - 72
eISSN - 1467-9876
pISSN - 0035-9254
DOI - 10.1111/1467-9876.00105
Subject(s) - mathematics , logarithm , least squares function approximation , standard error , statistics , linear regression , extension (predicate logic) , variance (accounting) , regression analysis , bromine , generalized least squares , chemistry , computer science , mathematical analysis , accounting , estimator , business , programming language , organic chemistry
Methods of estimation and inference are presented for the situation where two non‐linear regression models with unequal error variances contain some parameters in common. Such a situation arises in structural chemistry, when bond lengths are available for three nearly collinear atoms in crystals and a model is required to quantify the extent and form of the relationship between the longer and the shorter bond. Some atomic triples are symmetric and require a different model and error variance from those required by the asymmetric triples. The profile likelihood for the regression parameters is a weighted sum of the logarithms of the sums‐of‐squares functions from each model, and the estimates can be obtained by using a simple modification to a standard non‐linear least squares program. A likelihood ratio test for assessing whether the parameters in common are equal is described. When these techniques are applied to two data sets consisting of bond lengths for bromine–tellurium–bromine and sulphur–tellurium–sulphur triples, there is no evidence against the equality hypothesis. An extension to the model to allow for a non‐constant variance is required for proper analysis of the sulphur–tellurium–sulphur data.