The importance of dimensional analysis and dimensional homogeneity in bibliometric studies is always overlooked. In this paper, we look at this issue systematically and show that most h-type indices have the dimensions of [P], where [P] is the basic dimensional unit in bibliometrics which is the unit publication or paper. The newly introduced Euclidean index, based on the Euclidean length of the citation vector has the dimensions [P3/2]. An empirical example is used to illustrate the concepts.
Evaluative and descriptive bibliometrics provide a quantitative focus on citations and/or publications (Andersen 2016). At the simplest level of aggregation, we study the performance of an individual scholar. At this micro-level, a controversial usage of indicators is to perform ranking and hard impact assessment to inform critical decisions about funding, promotion and tenure and the allocation of billions of dollars of research funding (Perry & Reny 2016).
There is therefore a pressing need that indices (e.g., the h-index) go beyond heuristic rules of thumb and instead are founded on axiomatic principles. Five natural properties are considered and Perry & Reny (2016) propose a unique new index, the Euclidean index iE, the Euclidean length of an individual’s citation list. In this paper we shall show that following these five rules is not sufficient for robustness. There is a need to introduce one more requirement, that of dimensional homogeneity or consistency. We show that iE is not dimensionally commensurable with other citation indices like the impact i, or the h-index or many of its variants.
Dimensional homogeneity is a well known principle in theoretical physics and engineering analysis. It requires that an equation must have quantities of same units on both sides. An equation is meaningful only it is homogeneous, with equality being applied between quantities of different nature. [M], [L] and [T] are the first such fundamental units to be encountered in physics , where they stand for the units of mass, length and time. Velocity or speed combines length and time as
Extending this idea to bibliometrics, the basic dimensional unit in bibliometrics is the minimum unit of publication, namely the paper, say [P]. This has the same role as [M], [L] and [T] in physics. Leydesdorff (2009) and Bollen et al. (2009) have identified size and impact as the main categories in which the majority of bibliometric indicators can be arranged into (Andersen 2016). Size is measured as the number of papers P (a numerical quantity) and its corresponding dimensional unit, in this case [P]. Impact is derived from the impact of all the P papers in the portfolio. Thus if the k-th paper has ck citations, this means that this paper has been cited by ck papers. This is also the impact ik of the k-th paper.
Here, ck or ik is the numerical quantity and the fundamental unit is again [P]. The total citations C = Σ ck then has the units [P 2 ] since the individual impact of each paper is summed over the total number of papers in the portfolio. The specific impact i = C/P of the portfolio also has the units [P].
The best-known bibliometric indicator beyond the count of papers P, the impact i or the total citation count C is the h-index (Hirsch 2005). A scholar’s h-index is the number, h, of his/her papers that each have at least h citations. Fortuitously, this definition makes the hindex commensurable with P and i; i.e. h has the same dimensions as number of papers and the impact of the papers. Most of the variants of the h-index, such as the g-index (Egghe 2006a,b) have the same dimension and can be directly compared to each other. If indeed they had different dimensions, they are incommensurable and cannot be directly compared. In this paper we shall show that the newly proposed Euclidean index iE has a different dimension and so is not an alternative to nor can be compared to any of the other h-type indices.
The three-dimensional nature of a citation distribution Leydesdorff (2009) and Bollen et al. (2009) see bibliometrics through a two-dimensional prismquantity/size and impact (which can be interpreted as a proxy for quality or excellence) are the main categories in which most of bibliometric indicators can be arranged into (Andersen 2016). Prathap (2011a,b), proposed that comparative evaluation needs at least three dimensions: quantity/size, quality/excellence and consistency/balance or evenness. The quality-quantity-consistency parameter space leads to the evolution of second order indicators for any portfolio of papers (Prathap 2014a,b).
For any portfolio of publications, the total number of papers or articles, P, and the total number of citations, C, are often taken as indicators or proxy measures for the size of output of a group and the impact of its published research respectively (Katz 2005). The total impact, C, is size-dependent, and a specific impact, i, defined as C/P is size-independent.
The journal impact factor was defined in such a manner as a size-independent indicator to select journals for inclusion in the Science Citation Index. It was not originally intended to be a direct measure or proxy of quality (Pendlebury and Adams 2012), but since then has been accepted as a proxy or indirect measure of the quality or scholarly influence of a journal in a size-independent manner. In the same way, the scientific output of an individual or an entity can be measured using the following three-dimensional parameter
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