AN APPLICATION OF LINEAR-PROGRAMMING TO ECOLOGICAL INFERENCE - AN EXTENSION OF AN OLD PROCEDURE

Nearly four decades ago, Duncan and Davis (1953) developed a simple ec ological inference technique. They noted that, given two dichotomized individual-level variables, the minimum and maximum possible number of individuals having each combination of these variables could be deriv ed, with certainty, from the marginals of the cross-tabulation of thei r aggregate-level counterparts. Shively (1975, 1991) expanded this tec hnique to allow the use of side information to constrain further the n umber of individuals having each combination of the individual-level v ariables. This approach to ecological inference avoided many of the pi tfalls of ecological regression. Despite this, the technique has been little used because there has not been any easy, mechanical way to emp loy this technique and to apply this procedure to larger tables has be en difficult. Finally, there was the feeling that the whole approach w as ad hoc and without a mathematical foundation. In this paper, we rem edy the limitations of this technique. Our remedy is based on a simple observation: both the original Duncan and Davis approach and Shively' s extension of it can be seen as an application of linear programming (LP). As such, a large body of mathematical machinery and software exi sts to solve these problems. Thus, the minimum and maximum possible va lue for any linear combination of the unobserved internal cells, given any set of linear constraints, for any size table can easily be calcu lated.