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BEHAVIORAL INTERPRETATIONS OF THE GRAVITY MODEL Robert E. Lloyd INTRODUCTION. The gravity model has been viewed as a useful empirical tool for predicting spatial interaction (Equation 1, Fig. I)-(I) It has also been categorized by some as having little theoretical importance. The necessity to recalibrate the model for data relating to different locations in space and different periods of time is often cited as a major drawback of the model. The most immediate problem in the gravity model formulation lies in the inadequacy of its assumptions. In discussing the behavioral assumptions of the gravity model in a migration context, Shaw states: The first and most highly questionable assumption is that all places are populated by standard people with identical needs, tastes, and contacts. A second is that interaction intensity decreases over distance systematically in all directions. Finally, it is apparent that not all migrants seek advantages that are a function of population size as is demonstrated by return migration, compulsory moves, and migration of the highly specialized. (2) If the researcher is able or willing to make these behavioral assumptions , he is still faced with a number of important methodological decisions that may influence his results. A recent debate in the literature has pointed out some potential problems that researchers using gravity models should consider when calibrating their models and interpreting their results. (3) Selecting the appropriate measures to serve as the independent variables is the first major consideration. Population size and linear distance are the most commonly used variables, but many others could be selected for specialized problems . (4) If the model is to be calibrated using linear regression, care must be taken to assure that a linear relationship exists between the dependent variable and the independent variables. If linear relationships do not exist, it is sometimes possible to produce them by transforming one or more of the variables. If the researcher wishes only to produce a model that predicts Dr. Lloyd is Assistant Professor of Geography at The University of South Carolina, Columbia, SC 29208. 80 Southeastern Geographer GENERAL MODEL I;;= a. Pjbi U REGRESSION MODEL LOG(D= a +b., LOG(Pj) - b2 LOG(Djj) NORMAL EQUATIONS IN MATRIX FORM S?? IPD IDP IDD S?? IDI PROPORTION OF THE TOTAL VARIANCE EXPLAINED BY POPULATION ___ "1 D1TPI S?" 100 PROPORTION OF THE TOTAL VARIANCE EXPLAINED BY DISTANCE b2 «IDI 100 PROPORTION OF THE TOTAL VARIANCE EXPLAINED BY THE GRAVITY MODEL _ b,· IPI + b2.rDI 100 (D (2) (3) (4) (5) (6) Figure 1. Equations and indices related to the gravity model. well, then he may be satisfied if the proportion of the total variance explained by the gravity model is relatively high (Equation 6, Fig. 1). If the researcher wishes to compare the relative importance of the independent variables for a given model, however, or to compare models calibrated from different data, he must be concerned with the reliability of the estimated regression coefficients (bi and b_¡ in Fig. 1)· Curry has questioned the validity of estimated regression coeffi- Vol. XVII, No. 2 81 cients for gravity models on the grounds that the coefficients measure the influence of map pattern among the origins and destination of a system, as well as the true friction of distance effect. (5) This confounding effect would manifest itself as spatial autocorrelation in the variable representing mass in the gravity model. Because most geographic distributions form some spatial pattern, it has been suggested that spatial autocorrelation may be the rule rather than the exception. (6) That states with large populations tend to be closer together on the average than would have occurred if the population of the United States were randomly distributed is an obvious example . Curry concluded that the confounding of map pattern and true friction of distance implied that the estimated regression coefficient has: nothing to do with friction and everything to do with map pattern. . . . It is only in the unlikely event of zero autocorrelation among the population values that the distance exponent may be read directly as a friction term. Otherwise any "calibration" is specific to a particular pattern of origins and destinations and may be substantially meaningless . Different degrees of clustering will exhibit different frictional terms...

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