A Comparative Study of Various Inclusive Indices and the Index Constructed by the Principal Components Analysis

Construction of (composite) indices by the PCA is very common, but this method has a preference for highly correlated variables to the poorly correlated variables in the data set. However, poor correlation does not entail the marginal importance, since correlation coefficients among the variables depend, apart from their linearity, also on their scatter, presence or absence of outliers, level of evolution of a system and intra-systemic integration among the different constituents of the system. Under-evolved systems often throw up the data with poorly correlated variables. If an index gives only marginal representation to the poorly correlated variables, it is elitist. The PCA index is often elitist, particularly for an under-evolved system. In this paper we consider three alternative indices that determine weights given to different constituent variables on the principles different from the PCA. Two of the proposed indices, the one that maximizes the sum of absolute correlation coefficient of the index with the constituent variables and the other that maximizes the entropy-like function of the correlation coefficients between the index and the constituent variables are found to be very close to each other. These indices alleviate the representation of poorly correlated variables for some small reduction in the overall explanatory power (vis-à-vis the PCA index). These indices are inclusive in nature, caring for the representation of the poorly correlated variables. They strike a balance between individual representation and overall representation (explanatory power) and may perform better. The third index obtained by maximization of the minimal correlation between the index and the constituent variables cares most for the least correlated variable and in so doing becomes egalitarian in nature.