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 probabilities 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 predictions 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.