Ill‐conditioned information matrices, generalized linear models and estimation of the effects of acid rain

Abstract

The problem of acid rain deposition has generated much interest in the modelling and estimation of the effects of acid rain. Recent studies in the northeastern United States have focused on the question of trends in lake acidity and the effects on aquatic organisms, especially fish. One approach has been to model the presence or absence of fish species as a function of relevant environmental variables. As the number of these explanatory variables may be large, there is concern about redundancies and collinearities. Because the model used is a special case of generalized linear models, standard approaches to assessment and adjustment for collinearity may be misleading. Estimation of parameters in the generalized linear model involve an iterative method of solution. The important parameter is the information matrix. Illconditioning of this matrix, as caused by collinearity has severe effects on parameter and variance estimates. To asssess the effects of collinearities, some new diagnostics are presented. Two techniques for estimating parameters in the presence of multicollinearity; the ridge estimator and the principal component method, are extended to the generalized linear model.

Methodology for the Development of Empirical Models Relating 13C NMR Spectral Features to Fuel Properties

“…111 Although the effects of multicollinearity on model parameter estimates for the rest of the GLM family are not as well studied, there are reasons to believe it will affect them in similar ways. 112,113 A possible way to reduce the influence of multicollinearity when estimating a model is through a transformation of the data. Centering the predictors (by subtracting the mean) or standardizing them to fit in a range of fixed length (between [−1, 1] e.g.)…”

The r-k class estimator in generalized linear models applicable with simulation and empirical study using a Poisson and Gamma responses

“…Here, we utilize from the singular value decomposition which was used by [24] for proposing the PCR in GLMs. The linear predictor η = Xβ can be expressed as…”

“…From [24], we approximately write the expected value and variance of the first-order approximated PCR estimator as…”

“…An overwhelming number of alternative techniques have been proposed to address multicollinearity problem in GLMs. Smith and Marx [24] developed a principal component regression (PCR) estimator to estimate the effects of acid rain. Nyquist [17] explored ML estimation of the GLMs by using a penalty function approach under linear restrictions on the parameters.…”

A window on The International Environmetrics Society: The first 25 international conferences

“…Presentations were divided between those directed to a particular specialty and those to a more interdisciplinary audience. On a statistical spectrum, giving some examples to illustrate, the range was from the theoretical statistics viewpoint as in the Special Lecture of D. A. S. Fraser, “Statistical Inference in Science” (Fraser & Reid, ); the statistical method development for an environmental problem as in “Ill‐Conditioned Information Matrices, The Generalized Linear Model and Estimation of the Effects of Acid Rain”, by Smith and Marx (); the statistical application as in “Experimental Design and Parameter Estimation for Carbon Adsorption Kinetics”, by D. K. Stevens; and the evaluation of the practice of environmental regulation as in “Microbiological Monitoring of Drinking Water Quality: US Practices”, by W. O. Pipes.…”