Construction of maximin and non-elitist composite indices - alternatives to elitist indices obtained by the principal components analysis

On many occasions we need to construct an index that represents a number of variables. Cost of living index, general price index, human development index, index of level of development, etc are some of the examples that are constructed by a weighted (linear) aggregation of a host of variables. The weights are determined by the importance assigned to the variables to be aggregated. The criterion on which importance of a variable (vis-à-vis other variables) is determined may be varied and usually has its own logic. In many cases the analyst does not have any preferred means or logic to determine the relative importance of different variables. In such cases, weights are assigned mathematically. One of the methods to determine such mathematical weights is the Principal Components analysis. In the Principal Components analysis weights are determined such that the sum of the squared correlation coefficients of the index with the constituent variables is maximized. The method has, however, a tendency to pick up the subset of highly correlated variables to make the first component, assign marginal weights to relatively poorly correlated subset of variables and/or to relegate the latter subset to construction of the subsequent principal components. If one has to construct a single index, such an index undermines the poorly correlated set of variables. The index so constructed is elitist in nature that has a preference to the highly correlated subset over the poorly correlated subset of variables. Further, since there is no dependable method available to obtain a composite index by merging two or more principal components, the deferred set of variables never finds its representation in the further analysis. In this paper we suggest a method to construct an index by maximizing the sum of the absolute correlation coefficients of the index with the constituent variables. We also suggest construction of an alternative index by maximin principle. Our experiments suggest that the indices so constructed are inclusive or egalitarian. They do not prefer the highly correlated variables much to the poorly correlated variables.