Effectiveness of the T-test in Multiple Linear Regression Modeling of Environmental Systems

Abstract

In previous works that used multiple linear regression (MLR) modeling, some researchers tended to extend the t-test results on the full model to infer a generalized conclusion that some explanatory variables influence the response more significantly than others. We found through a model re-parameterization that such generalization is only valid when the transformed design matrix is orthonormal, in which case the resulting t-ratios of the full model have a consistent order for the entire system (all reduced models). Otherwise, for practical data sets, t-ratios are all interdependent and subjected to changes in reduced models. Significance of the variables inferred from the full model therefore has no general meaning about the physics of the environmental system. In observation of the restriction of the t-test to a single model and the frequent need for deducing a general comparison of variable significance in environmental research, the significance test for explanatory variables can be conducted by a model selection process with, for example, Mallows' Cp, Akaike's Information Criterion (AIC), or Bayesian Information Criterion (BIC), which are insensitive to the orthonormality of the transformed design matrix, as the selection criterion. Once the parsimonious model is identified, the variables in the parsimonious model can be considered to be significant for the response. This is a general inference about the significance of the explanatory variables, which is consistent for the entire system.