The Assumptions of the Linear Regression Model

The paper is prompted by certain apparent deficiences both in the discussion of the regression model in instructional sources for geographers and in the actual empirical application of the model by geographical writers. In the first part of the paper the assumptions of the two regression models, the 'fixed X' and the 'random X', are outlined in detail, and the relative importance of each of the assumptions for the variety of purposes for which regression analysis may be employed is indicated. Where any of the critical assumptions of the model are seriously violated, variations on the basic model must be used and these are reviewed in the second half of the paper. THE rapid increase in the employment of mathematical models in planning has led R. J. Colenutt to discuss 'some of the problems and errors encountered in building linear models for prediction'.1 Colenutt rightly points out that the mathematical framework selected for such models 'places severe demands on the model builder because it is associated with a highly restrictive set of assumptions . . . and it is therefore imperative that, if simple linear models are to be used in planning, their limitations should be clearly understood'.2 These models have also been widely used in geography, for descriptive and inferential purposes as well as for prediction, and there is abundant evidence that, like their colleagues in planning, many geographers, when employing these models, have not ensured that their data satisfied the appropriate assumptions. Thus many researchers appear to have employed linear models either without verifying a sufficient number of assumptions or else after performing tests which are irrelevant because they relate to one or more assumptions not required by the model. Furthermore, many writers, reportipg geographical research, have completely omitted to indicate whether any of the assumptions have been satisfied. This last group is ambiguous, and it is clearly not possible, unless the values of the variables are published, to judge whether the correct set of assumptions has been tested or, indeed, to ascertain whether any such testing has been performed at all. This problem partially arises from certain shortcomings in material which has been published with the specific objective, at least inter alia, of instructing geographers on the use of quantitative techniques. All of these sources make either incomplete or inaccurate specifications of the assumptions underlying the application of linear models, although it is encouraging to note that there has been a considerable improvement in the quality of this literature in recent years. Thus, there were four books and two articles published in the early and mid-Ig60s which may be classified as belonging to this body of literature,3 yet, in five of these six sources, only one of the assumptions of the model is mentioned and, even