A CORRELATION AND REGRESSION ANALYSIS APPLIED TO RURAL FARM POPULATION DENSITIES IN THE GREAT PLAINS ARTHUR H. ROBINSON, JAMES B. LINDBERG, LEONARD W. BRINKMAN University of Wisconsin INTRODUCTION HE geographers interest in modern de- T scriptive and analytical statistical methods is growing, especially with respect to their po- tential usefulness in regional analysis. Corre- lation techniques, including multiple correla- tion and regression, are particularly suited to aiding the geographer in his traditional study of the areal variations of related phenomena, since the variables always exist in complex interconnection.2 One may properly employ these statistical-cartographic techniques after he has established tentative descriptive hy- potheses regarding the mutuality that may exist among the distributions of an area, in- ferred through the study of individual maps and other sorts of data. Coefficients of cor- relation and related indices provide general quantitative statements of the degree to which each hypothesis is valid. These may be sharpened through the use of partial correla- tion techniques which statistically hold con- stant designated variables while investigating any two. Regression mapping portrays the areal distribution of the degree of correspond- ence, shows the locations of departures from the average relationship, and provides a basis for formulating additional hypotheses. In this paper attention is focused on the areal variations in the density of the 1950 rural farm population in the Great Plains. Several distinct phenomena are believed to vary in a reciprocal fashion with rural farm population density; these interrelationships are examined, and statements are made about 1 Portions of this study were supported by the Graduate School Research Committee, University of Wisconsin. As demonstrated, for example, in: Harold H. McCarty, J. C. Hook, and D. S . Knos, The Measure- ment of Association in Industrial Geography (State University of Iowa, Iowa City, 1956); George W. Hartman and J. C. Hook, “Substandard Urban Hous- ing in the United States, a Quantitative Analysis,” Economic Geography, Vol. 32( 1956), pp. 95-114; E. J. Taaffe, “Trends in Airline Passenger Traffic: a Geographic Case Stndy,” Annals, Association of American Geographers, Vol. 49( l959), pp. 393-408. the closeness of correspondences among them. Average annual precipitation, distance to ur- ban centers, and percent of cropland are sev- eral of the factors that are considered to be important “determinants” in the variations from place to place of farm population density. But in the Great Plains just how significant are they individually and relative to one another? If there is a general pattern of covariation, where are the areas that depart from it? Such questions, in an areal setting, form the core of much of modern geographical study, and the techniques that are presented here as partial answers are believed to have wide applica- bility. This paper is also concerned with several questions of statistical-cartographic method- ology which are as yet only partially solved. Among these are: (1) the question of what sampling method to use when a map of uni- form validity is desired as opposed to a single summary measure; (2) the problems that arise when one wishes to employ both point data and unit-area data in a statistical computa- tion; ( 3 ) the problem occasioned by the un- equal sizes of unit areas; and (4) the question of how best to map departures from certain mean relationships (regression residuals ) .3 If modern statistical tools that are appropri- ate to geographical analysis are to find their way into the research repertoire of geogra- phers, they must be subjected to the rigors of repeated testing in a variety of geographical contexts. It is only through such testing that the usefulness, and the limitations, of these tools can be understood. Since the research reported in this paper was completed a good technical summary of tlic use of residuals from linear regression has appeared: Eclwin N. Thomas, Maps of Residuals from Regresion: Their Characteristics and Uses in Geographic Reseurch, No. 2, Department of Geography, State University of Iowa, Iowa City, 1960. Although carried fonvard without what would have been the considerable aid of Thomas monograph, this paper may be considered as an illustration of some of the methods he treated. The technically interested rcadcr should refer to tlie monograph for a fuller description of the methods. 211 212 ROBINSON, LINDBERC, AKD BRINKMAN Jllne FIG. 1. The area dealt with in the study. The pattern of dots sliows the arrangement of control points usctl for computation and mapping purposes. THE AREA AND THE HYPOTHESES The focus of interest is on the area in the United States generally designated as the Great Plains and its marginal areas extending from the Canadian border to central Texas. It was arbitrarily defined by centering a rec- tangle on the intersection of the 40th par