Concerning bivariate least squares linear regression, the classical approach
pursued for functional models in earlier attempts is reviewed using a new
formalism in terms of deviation (matrix) traces. Within the framework of
classical error models, the dependent variable relates to the independent
variable according to the usual additive model. Linear models of regression
lines are considered in the general case of correlated errors in X and in Y for
heteroscedastic data. The special case of (C) generalized orthogonal regression
is considered in detail together with well known subcases. In the limit of
homoscedastic data, the results determined for functional models are compared
with their counterparts related to extreme structural models. While regression
line slope and intercept estimators for functional and structural models
necessarily coincide, the contrary holds for related variance estimators even
if the residuals obey a Gaussian distribution, with a single exception. An
example of astronomical application is considered, concerning the [O/H]-[Fe/H]
empirical relations deduced from five samples related to different stars and/or
different methods of oxygen abundance determination. For selected samples and
assigned methods, different regression models yield consistent results within
the errors for both heteroscedastic and homoscedastic data. Conversely, samples
related to different methods produce discrepant results, due to the presence of
(still undetected) systematic errors, which implies no definitive statement can
be made at present. A comparison is also made between different expressions of
regression line slope and intercept variance estimators, where fractional
discrepancies are found to be not exceeding a few percent, which grows up to
about 20% in presence of large dispersion data.Comment: 56 pages, 2 tables, and 2 figures. New Astronomy, accepte
