Minimizing the Effects of Collinearity in Polynomial Regression

Abstract

Data transformation for obtaining the most accurate and statistically valid correlation is discussed. It is shown that the degree of a polynomial used in regression is limited by collinearity among the monomials. The significance of collinearity can best be measured by the truncation to natural error ratio. The truncation error is the error in representing the highest power term by a lower degree polynomial, and the natural error is due to the limited precision of the experimental data. Several transformations for reducing collinearity are introduced. The use of orthogonal polynomials provides an estimation of the truncation to natural error ratio on the basis of range and precision of the independent variable data. Consequently, the highest degree of polynomial adequate for a particular set of data can be predicted. It is shown that the transformation which yields values of the independent variable in the range of [-1,1] is the most effective in reducing collinearity and allows fitting the highest degree polynomial to data. In an example presented, the use of this transformation enables an increase in the degree of the statistically valid polynomial, thus yielding a much more accurate and well-behaved correlation.

Reproductive rate, not dominance status, affects fecal glucocorticoid levels in breeding female meerkats

“…marginal error structure in R) are presented to control for co-variation among fixed terms (Pinheiro and Bates, 2000). Orthogonal polynomials were tested to control for collinearity among polynomial degrees (Shacham and Brauner, 1997). All statistical tests were two-tailed.…”

Analysis of the major drivers of the ecological footprint using the STIRPAT model and the PLS method—A case study in Henan Province, China

“…From this point of view, collinearity (multicollinearity) may not be a serious problem. However, other scientists (Wang, 1999(Wang, , 2006Naes and Mevik, 2001;Tu et al, 2004;Shacham and Brauner, 1997) consider that collinearity is a problem that should receive more attention. For example, if there is multicollinearity in the regression model, regression coefficients will no longer have the meaning for general interpretation (Wang, 1999); further, collinearity can sometimes lead to serious instability of the variables' coefficients (Naes and Mevik, 2001).…”

The SROV program for data analysis and regression model identification

“…Dividing elements of every data column with the maximal (absolute) value in the same column normalizes the independent and dependent variables. Shacham and Brauner (1997) found the transformation to the range of [(/1, '/1] especially useful in polynomial regression, as it minimizes the interdependency between the model parameter values. This transformation is defined by:…”

“…Also, the derivatives of the dependent variable are not represented correctly and extrapolation outside the region, where the measurements were taken, yields absurd results even for a small range of extrapolation. Shacham and Brauner (1997), Brauner and Shacham (1998a,b) provide several examples where regression models published in the chemical engineering literature are grossly inaccurate and/or unstable.…”