Combining non-linear regressions that have unequal error variances and some parameters in common
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.