Role of Range and Precision of the Independent Variable in Regression of Data

Regression of the experimental data of one independent variable, y vs. a linear combination of functions of an independent variable of the form y = Σ β_jf_j(x) is considered. Inherent collinearity among the terms of such functions may prevent obtaining a model of a desired accuracy. Traditional collinearity indicators, condition number of the normal matrix, variance inflation factor, and a new indicator (truncation-error-to-noise ratio) are used to investigate the effects of the range and precision of the independent-variable data on collinearity among functions in a regression model. Statistical confidence intervals are used to demonstrate harmful effects of collinearity. The harmful effects increase by reducing the range of the independent variable data and/or its precision. Using only independent variable data, the new collinearity indicator allows the identification of the point where the number of terms in a particular regression model becomes larger than can be justified on statistical grounds. The use of the new criterion can improve experimental design in order to minimize the harmful effects of collinearity and enable a rapid screen of correlations published in the literature for identifying those that include more parameters than can be justified.