The 100 most prolific economists using the p-index

Abstract

In this paper, a new indicator called the performance index (p-index) is used to rank a 100 most prolific economists. The p-index strikes the best balance between activity (total citations C) and excellence (mean citation rate C/P). The surprise is that the h-index, which is now universally accepted almost as a canonical tool for research assessment of individuals, research faculties and institutions and even for comparing performance of journals and countries, is actually a poor indicator of performance.

Introduction

The h-index (Hirsch 2005, 2007) has rapidly captured the imagination of scientometricians and bibliometricians. It was presumed to be an easy to compute composite indicator which attempted to combine in a single number, the quality/excellence and quantity/activity of an individual (or at higher levels of aggregation, of journals, groups of scientists, institutions and even countries).

Along with acceptance, its shortcomings have also been discussed and this has led to an alphabet soup of new variants which have tried to improve on the original (Liu and Rousseau 2009). However, what has been established so far, in a self-fulfilling prophetic way, is that all these new variations correlate well with the h-index. Recently, a mock h-index has been proposed (Prathap 2009), which appears to be an ideal performance indicator. It is clear now that what one looks for in an ideal single number indicator that can capture the multi-dimensionality of the problem (i.e. evaluating activity/quantity and excellence/quality), is that it must be intuitive (Tol 2009), and must increase when the quantity (total papers P or total citations C) increases and the quality increases (mean citation rate C/P). It must not discount highly-cited papers, and must be sensitive to the number of uncited papers (Tol 2009). Also, it must be allowed even to exceed the number of papers, which the h-index cannot (Vinkler 2007). Another recent stipulation was that the new index must be “strictly monotonic”, being able to “assign a positive score to each new citation as it occurs” (Anderson et al. 2008).

A new performance indicator

Glänzel (2008) showed that h could be related to traditional bibliometric measures like publication activity (total number of papers P), citation impact (total citations C) and quality expressed as a mean citation rate (x = C/P), there being a strong correlation between h and x ^a/(a+1) P ^1/(a+1) (Schubert and Glänzel 2007). Much earlier, a similar composite indicator for journal impact was already suggested (Lindsey 1978), which allowed a reconciliation between the number of publications P and the received citations C by using a geometric mean as the balancing correction factor for the mean citation rate (C/P). The Corrected Quality Ratio (CQ) is then defined as CQ = (C/P) · (C · P)^1/2 = (C³/P)^1/2. However, using dimensional analysis (Prathap 2009) showed that the composite indicator (C³/P)^1/2 will then have the dimensions of h ^5/2. Since CQ thus defined does not have the dimensionality of h, it can be brought back to the dimensionality of h by introducing a transformation CQ → CQ ^0.4 (Glanzel 2008) leading to the indicator (C ³/P)^1/5. An alternative approach to this based also on a Glanzel model has been offered (Csajbók et al. 2007), connecting the h-index with the number of papers and the mean citation rate per paper: h = cn ^1/3 x ^2/3. The composite indicator in this case will be (C ²/P)^1/3. This is seen immediately to have the dimensions of h. Note that c is a constant of proportionality that has to be empirically determined from a fit of data and would therefore vary from case to case. Indeed, when applied to the research performance of countries in all fields (Csajbók et al. 2007), the relationship h = 0.932n ^1/3 x ^2/3 (= 0.932 (C ²/P)^1/3) is obtained with R ² = 0.988.

The composite indicator (C ²/P)^1/3 has interesting properties, yet there is a cautious warning (Glanzel 2008) not to use it as a substitute the h-index. Prathap (2009) proposes that Glanzel’s well meaning caveat should be disregarded and instead the indicator should be treated as a substitute or mock h, say h _m  = (C ²/P)^1/3. The ratio (h _m /h) now reflects the sensitivity to the citation numbers in the tall core (citations for each paper significantly exceeding h) and the long tail (P much larger than h, and significantly, where there is a lot of uncitedness) that h by itself fails to capture. Many examples of a theoretical and empirical nature showed that while h alone permits no discrimination for many cases, h _m allows ranking to be done on a more rational basis. Indeed, it appears to deserve a legitimacy as an ideal performance index (perhaps, adding to the alphabet soup of indicators, as a p-index).

We can now re-examine the performance of this new p-index using the example of the hundred most prolific economists (Tol 2009). What is remarkable now is that Robert F. Engle rises effortlessly to the top (Table 1). The h-index is not able to do this because his output of 83 papers restricts his h-index to a low value although his mean citation rate is the highest in this list. Only the p-index captures this well. Similarly, Robert Barro benefits from this new classification, rising to third place.

Table 1 The number of the papers, cited papers and citations; the average number of citations per paper; and the h-, g-, f-, t- and r-numbers of the 100 most prolific economists according to IDEAS/REPEC in May 2007

Tables 2 and 3 reveal the correlation matrix connecting the various indices. It appears that the main contenders are h, g and p while the others are only of academic interest, “adding work but no insight” (Tol 2009). The p-index gives the best balance between quality (C/P) and quantity (C). This is not surprising because by definition, the performance index is based on the substitute or mock index, p = h _m  = (C · (C/P))^(1/3) and has the significance of a “geometric mean” that is consistent with the dimensions of h, and therefore should give the best balance between C and C/P for any non-linear process governed by random multiplicative processes.

Table 2 Correlation matrix for the various indices

Table 3 Correlation matrix emphasizing that p gives the best balance between quantity (C) and quality (C/P) and that h is the worst of the three main contenders

Conclusions

In this paper, a corrected quality measure leading to a performance index derived from a substitute or mock h _m index which can be easily computed from traditionally compiled bibliometric indicators is used to rank a list of 100 leading economists. The newly proposed p-index should be more versatile than the h-index in that it gives the best balance between quantity and quality; indeed perhaps better than all the other variants proposed so far (Liu and Rousseau 2009).