1CORRELATION IN SIMPLE LINEAR REGRESSION

Many people who do data analysis take only a few classes in statistics and hence, in general, get introduced only to classical methods of statistics; i.e., least squares, normal theory, and possibly maximum likelihood methods. The motivation for most of these techniques is the maximizing of a function of the data with respect to some parameter—mean, variance, slope, e.g. Calculus is used by taking derivatives and solving an equation set to equal zero. So many analyzers of data do not know about robust, nonparametric, or alternative statistical methods that are probably better suited to avoiding misinterpreting one’s data. Many of these latter methods cannot be developed by Calculus! A possible way to avoid this dilemma is to offer a method of instruction that allows both classical and other estimation techniques to be developed simultaneously. Correlation coefficients offer a very general method of estimating parameters and hypothesis testing. The motivation for their use is based on n-dimensional geometry and the generalization of the concept of the parallelogram law and perpendicularity in a Hilbert Space. Classical statistical methods are represented by Pearson’s Correlation Coefficient and the Cosine function. Other methods, such as median methods, are represented by Kendall’s tau or by one or more absolute value correlation coefficients, still more techniques, equal area or volume, by the Greatest Deviation Correlation Coefficient. The regression approach is first shown using Pearson’s and Kendall’s correlation coefficients. Then n-dimensional geometry and orthogonality are used for motivation, a third correlation is introduced, the Greatest Deviation, and finally, some simple linear regression examples illustrate these ideas. After the development of simple linear regression, these techniques can be broadened for location and scale estimation