Construction of Pena's Synthetic Indices


A synthetic indicator (or composite index), Z[n], is oftentimes a linear combination of partial indicators (or the constituent variables of Z), X[n,m], such that Z=Xw, where w[m] is the weight vector. In many cases, w is extraneously determined (e.g. on the basis of expert opinion), but applications abound when is determined intrinsically from X itself. Factor Analysis or Principal Component Analysis is one of the most popular methods of obtaining Z (which is called the factor scores). However it has been found that the factor scores thus obtained are highly elitist; often ignoring the poorly correlated partial indicators (OECD, 2003; Munda & Nardo, 2005; Somarriba & Pena, 2009; Mishra, 2011). In response to the limitations of the Principal Component (or Factor) Analysis as a tool to construct the synthetic indicators, a new method based on Pena's DP2 distance (Pena, 1977) was devised (Zarzosa, 1996; Somarriba & Pena, 2009).

The computer programs provided here (there are two programs - the one for cardinal variables and the other for ordinal rank scores) work out cardinal/ordinal Pena-DP2-based index (normal cardinal numbers or alternatively, rank scores) when all the constituent variables (or partial indicators) are cardinal/ordinal (usual real numbers or rank scores). The program (CARDP2 or ORDP2) computes the composite (synthetic) /cardinal/rank scores from m (>1) given partial variables (indicators)/rank scores (Z[n,m]), i=1,2, ..., n and j=1,2, ..., m each of which is a rank score obtained by the individual (case) in a particular dimension. There are n individuals.

Data is fed in an input file (filename.txt) which contains n rows (cases or individuals) in:

(a) m columns - if there is no reference index (HDI/REFI) to compare. In that case the program sums the given m real variables/rank scores, and then standardizes them / orders them and ranks this total to make HDI/REFI.

(b) m + 1 columns - if there is an extraneous reference index (HDI/REFI), (in the last m+1 column)

The program asks to choose certain parameters (in case of ORDP2 - regarding type of correlation, ranking rule, random number seed), and information regarding input and output files, etc which are self explanatory.

The program (codes/exe files) and the input data file must be in the same folder/directory. Output file also is stored in the same directory. The FORCE Fortran 77 compiler may be used to compile the codes. FORCE is free downloadable Fortran 77 compiler.

The Fortran Codes may be downloaded from here: Cardp2 (for cardinal data) and Ordp2 (for ordinal data).

References :

  • Mishra, S. K. (2012) "Construction of Pena's DP2-Based Ordinal Synthetic Indicator When Partial Indicators are Rank Scores", Available at SSRN: http://ssrn.com/abstract=2069500 (download).

 

Other Fortran Computer Programs

 

 

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