Ranking of research output of universities on the basis of the multidimensional prestige of influential fields: Spanish universities as a case of study

Abstract

A university may be considered as having dimension-specific prestige in a scientific field (e.g., physics) when a particular bibliometric research performance indicator exceeds a threshold value. But a university has multidimensional prestige in a field of study only if it is influential with respect to a number of dimensions. The multidimensional prestige of influential fields at a given university takes into account that several prestige indicators should be used for a distinct analysis of the influence of a university in a particular field of study. After having identified the multidimensionally influential fields of study at a university their prestige scores can be aggregated to produce a summary measure of the multidimensional prestige of influential fields at this university, which satisfies numerous properties. Here we use this summary measure of multidimensional prestige to assess the comparative performance of Spanish Universities during the period 2006–2010.

Appendices

Appendix 1: Set of axioms

A first axiom states that a field of study at the given university which is not multidimensionally prestigious should not influence a summary measure of the overall prestige of multidimensionally influential fields.

Axiom 1

Given two configurations of dimension-specific scores X and ${\bf X}^{\prime}$ of the same size n × d where the scores of multidimensionally influential fields at the university are the same in both cases, the summary measure of the multidimensional prestige of influential fields measured on either configuration should give the same value.

A second axiom can be justified on the idea that small changes in the configuration of dimension-specific scores for multidimensionally influential fields of study shall not lead to discontinuously large changes in the summary measure of multidimensional prestige.

Axiom 2

The summary measure of the multidimensional prestige of influential fields at a given university should be a continuous function of dimension-specific scores for multidimensionally influential fields.

A third axiom states than an increment in some dimension-specific score (above the corresponding threshold z _j ) for a multidimensionally influential field of study shall increase the summary measure.

Axiom 3

An index of multidimensional prestige of influential fields should increase whenever some dimension-specific score (above threshold z _j corresponding to that dimension) rises for a multidimensionally influential field of study.

Next another axiom states a property of subgroup decomposability. That is, the index has to be additively decomposable, i.e., the index of overall prestige is a weighted sum over several subgroups of fields of study in which the complete set U can be partitioned.

Axiom 4

The overall prestige of multidimensionally influential fields can be decomposed into the weighted sum of subgroup-prestige indices.

And the following axiom requires that the summary measure of multidimensional prestige of influential fields shall increase after a progressive transfer (from a more influential field of study to a less prestigious one) of domain-specific scores above the corresponding threshold z _j between two multidimensionally influential fields at the university.

Axiom 5

An overall prestige index should increase when a rank-preserving progressive transfer (above the corresponding domain-specific threshold) between two multidimensionally influential fields at a given university takes place.

Appendix 2: Proof of Theorem 1

Here we follow the proof given in (Garcia et al. 2012a).

Proof

Given a configuration X, let MPIF be a normalized weighted sum of the dimension-specific scores in X using weighting function f

$$ {\text{MPIF}}\,=\,\frac{1}{n \times d} \sum_{i=1}^n \sum_{j=1}^d f \left( \frac{x_{ij}}{z_j}\right) $$

(13)

where we have that f should be a continuous function for multidimensionally influential fields of study in order to satisfy Axiom 2, i.e., to verify that small changes in the configuration of dimension-specific scores (for multidimensionally influential fields at the university) shall not lead to discontinuously large changes in the summary measure MPIF.

But also it follows that weighting function f should be a strictly increasing function for multidimensionally influential fields of study at the university, since Axiom 3 states that an increment in some dimension-specific score (above the corresponding threshold z _j ) for a multidimensionally influential field shall increase the summary measure of multidimensional prestige MPIF.

From Axiom 1, a field of study which is not multidimensionally prestigious should not influence the overall prestige MPIF, i.e., MPIF is independent of the dimension-specific scores for fields of study at the given university which are not multidimensionally influential. Hence to fulfill Axiom 1 we have that

$$ f \left( \frac{x_{ij}}{z_j}\right)\,=\,0 $$

(14)

for all i such that ϕ _i (z; k)  =  0; where ϕ _i (z; k) equals to one if field s _i is multidimensionally prestigious and zero otherwise, as given in Eq. (4).

Now, from Axiom 4, the summary measure MPIF can be decomposed into the weighted sum of subgroup prestige indices. Thus it follows that the measure MPIF has to be additively decomposable.

Finally, following Axiom 5, the summary measure of multidimensional prestige MPIF should increase after a progressive transfer (from a more influential field of study to a less prestigious one) of domain-specific scores above the corresponding threshold z _j between two multidimensionally influential fields at the university under consideration. Hence we have that weighting function f has to be concave for multidimensionally influential fields, and thus, the relative dimension-specific scores $\frac{ x_{ij} }{z_j}$ then have to be transformed by a function that is concave on $(1,\infty)$ for multidimensionally influential fields of study.

For example, given a multidimensionally influential field s _i , we have that

$$ f \left(\frac{x_{ij}}{z_j}\right)\,=\,\left(1 - \left( \frac{z_j}{x_{ij}} \right)^{\beta}\right) \cdot \phi_{i} ({\bf z}; k) $$

is concave for x _ij  > z _j and β > 0.

To sum up, following Axiom 1 through Axiom 5, the summary measure MPIF

$$ {\text{MPIF}}\,=\,\frac{1}{n \times d} \sum_{i=1}^n \sum_{j=1}^d f \left( \frac{x_{ij}}{z_j}\right) $$

(15)

shall satisfy that $f : R_{+} \rightarrow [0,1]$ is a strictly increasing and concave function on $(1, \infty)$ for multidimensionally influential fields s _i at the given university.

Following (Peichl and Pestel 2010), if we define weighting function f as:

$$ f \left( \frac{x_{ij}}{z_j}\right)\,=\,\left( 1 - \left( \frac{z_j}{x_{ij}} \right)^{\beta} \right)_{+} \cdot \phi_{i} ({\bf z}; k) $$

(16)

where $\left( v \right)_{+}\,=\,\max (v, 0)$, we obtain a summary measure of the multidimensional prestige of influential fields, that resembles Eq. (10) satisfying Axiom 1 through Axiom 5, since f being defined as given in Eq. (16) it is a strictly increasing and concave function $f : R_{+} \rightarrow [0,1]$ on $(1, \infty)$ for multidimensionally influential fields s _i . □