Correlation, Method of Least Squares, Linear Regression and the General Linear Model - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 105

Chapter 4
Correlation, Method of Least Squares,
Linear Regression and the General
Linear Model
Abstract The scatterplot in which two variables are plotted against one another is a
basic tool in all branches of science. The ordinary correlation coefficient quantifies
degree of association between two variables for the same object of study. In some
software packages, the squared correlation coefficient ( R 2 ) is used instead of the
correlation coefficient to express degree of fit. A best-fitting straight-line obtained
by the method of least squares can represent underlying functional relationship if one
variable is completely or approximately free of error. When both variables are subject
to error, use of other methods such as reduced major axis construction is more
appropriate. A useful generalization of major axis construction in which individual
observations all have different errors in both variables is Ripley's Maximum
Likelihood for Functional Relationship a (MLFR) fitting method. Kummell's equation
( cf . Agterberg 1974) for linear relationship between two variables that are both subject
to error can be regarded as a special case of MLFR.
Multiple regression can be used for curve-fitting if the relationship between two
variables is not linear but other explanatory variables have to be considered as well.
The general linear model is another logical extension of simple regression analysis.
It is useful in mineral resource appraisal studies. Although this approach can be too
simplistic in some applications such as estimation of probabilities of occurrence of
discrete events, it remains useful as an exploratory tool. During the late 1970s and
early 1970s a probabilistic regional mineral potential evaluation was undertaken at
the Geological Survey of Canada ( cf . Agterberg et al. 1972) to estimate probabil-
ities of occurrence of large copper and zinc orebodies in the Abitibi area on the
Canadian Shield. These predictions of mineral potential made use of the general
linear model relating known orebodies in the area to rock types quantified from
geological maps and regional geophysical anomaly maps. About 10 and 40 years
later, after more recent discoveries of additional copper ore, two hindsight studies
were performed to evaluate accuracy and precision of the mineral potential pre-
dictions previously obtained by multiple regression. This topic will be discussed in
detail because it illustrates problems encountered in projecting known geological
relations between orebodies and geological framework over long distances both
horizontally and vertically.