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Ill‐conditioned information matrices, generalized linear models and estimation of the effects of acid rain
Author(s) -
Smith Eric P.,
Marx Brian D.
Publication year - 1990
Publication title -
environmetrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.68
H-Index - 58
eISSN - 1099-095X
pISSN - 1180-4009
DOI - 10.1002/env.3170010107
Subject(s) - collinearity , multicollinearity , generalized linear model , estimator , linear model , mathematics , statistics , econometrics , design matrix , principal component analysis , linear regression , estimation theory , variance (accounting) , estimation , generalized additive model , accounting , business , management , economics
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 interative 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.