Grade of Membership and Principal Components Analysis: A Comparative Empirical Study
Abdul Suleman
Quantitative Methods Department, ISCTE - IUL University Institute of Lisbon, Lisbon, Portugal

Wachter (1999) tested empirically the close relationship between two statistical tools for reducing data, namely principal components (PC) analysis and typological grade of membership (GoM) analysis. In his empirical work, the author considered two typologies based on dichotomous variables, and realized that individual GoM scores and PC scores were highly correlated. In addition, both analysis proved to recover the underlying individual ordering present in the dataset. Our purpose is to contribute to the same discussion using a most complex dataset.

We investigate the parallelism between PC and GoM analysis using a dataset that came from an original survey (Suleman, 2007), specially designed to study the reward of skills of retail bankers in Portugal. In this survey supervisors were asked to assess skills of each retail banker from a list of thirty skill items on a Likert scale from 1 to 5. As Wachter, we also have submitted this dataset to both analysis and we found a linear relationship between first principal component scores and GoM scores. Moreover, GoM analysis unveiled an ordered skill structure for which the first component gave the same account as well. Consequently, in this particular study the first principal component may further be regarded as an ordering device.

For GoM analysis purpose, we assume that the universe under study is structured by a latent fuzzy K-partition. So, there exists K>1 skill typologies and each individual is allowed to share more than one typology at the same time. The heterogeneity of individuals is represented by a unit sum vector of non-negative K GoM scores. We fixed a priori the value of K to three and showed that the particular distribution of retail bankers in the estimated fuzzy partition makes possible to rank them by skills (Suleman and Suleman, 2011). An utility function based on a linear combination of GoM scores is used for ranking purpose that assigns each individual a number between 0 and 1. So, the greater the number more skilled the individual. To apply PC analysis to the dataset, we consider the same thirty skill items as vectors of R30. The result achieved shows a correlation of 0.96 between the first principal component and the above referred utility function.


Suleman, F. (2007), O valor das competências: um estudo aplicado ao sector bancários, Livros Horizonte, Lisboa.

Suleman A., and Suleman F. (2011) “Ranking by Competence using a Fuzzy Approach”, Quality & Quantity (DOI: 10.1007/s11135-010-9357-1).

Tolley, H.D. and Manton, K.G. (1992) “Ranking by Competence using a Fuzzy Approach”, Annals of Institute of Statistical Mathematics, Vol. 44, 85-95.

Wachter, K.W. (1999), “Ranking by Competence using a Fuzzy Approach”, Statistical Papers, 40, 439-457.

Keywords: Grade of membership; Principal component analysis; Ordering

Biography: Professor at University Institute of Lisbon, teaching Linear Algebra, Calculus and Discrete Computer Simulation.

Main reserach area of interest is the statistical applications of fuzzy sets.