# Exploratory factor analysis

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Date added: 17-06-26

**Type:** Descriptive essay

**Category:** Science

### 1. Introduction

Hirsch (2005) introduced a new indicator for the assessment of the research performance of scientists. The proposed h-index is intended to measure simultaneously the quality and sustainability of scientific output, as well as, to some extent, the diversity of scientific research. The specific index attracted interest immediately and received great attention in the scientometrics literature. Not only it has found a wide use in a very short time, but also a series of articles were subsequently published either proposing modifications of the original h-index for its improvement, or implementations of the newly proposed index.

The h-index (sometimes called the Hirsch index or the Hirsch number) is based on the distribution of citations received by a given researcher's publications. By definition:

“A scientist has index h if h of his Np papers have at least h citations each, and the other (Np - h) papers have at most h citations each”.

The index is designed to improve simpler measures such as the total number of citations or publications, to distinguish truly influential (in terms of citations) scientists from those who simply publish many papers. Among the advantages of this index is its simplicity, the fact that it encourages researchers to produce high quality work, the fact that it can combine citation impact with publication activity and that is also not affected by single papers that have many citations.

Besides its popularity, a lot of criticism has been raised, too (see, e.g., Adler, Ewing, & Taylor, 2009; Schreiber, 2007a; Vinkler, 2007; Meho, 2007), and various modifications and generalizations have appeared (see, e.g., Egghe, 2006a; Jin, Liang, Rousseau, & Egghe, 2007; Schreiber, 2007b, 2008b; Sidiropoulos, Katsaros, & Manolopoulos 2006; Tol, 2009).

The h-index is robust to the numbers of citations received by the papers belonging to the h-core (i.e. the papers receiving h or more citations). To relax this “robustness”, various modifications have appeared in the literature, e.g. the g-index (Egghe, 2006), the A-index (Jin, 2006), the R-index (Jin et al., 2007), and the hw-index (Egghe and Rousseau, 2008). Since the suggestion of the Hirsch index a lot more h-type variants have been devised in order to overcome this “robustness” [e.g. the g-index (Egghe, 2006), the A-index (Jin, 2006), the R-index (Jin et al., 2007), and the hw-index (Egghe and Rousseau, 2008)]. However, more and more voices argue against the usefulness of all these measures (see e.g. Bornmann et al., 2009b; Adler, Ewing, & Taylor, 2009; Schreiber, 2007a; Vinkler, 2007; Meho, 2007). In the same vein, van Noorden (2010) states that “many metrics correlate strongly with one another, suggesting that they are capturing much of the same information about the data they describe”.

After a comparison of some of the more important variants, Bornmann, Mutz, and Daniel (2008) by performing exploratory factor analysis on a set of some of the most important h-type indices, including the h-index, conclude that indices can be categorized into two basic categories: those that came to the conclusion that essentially there are two types of indices, one type of indices that “describe the most productive core of the output of a scientist and tell us the number of papers in the core” (p. 836) while and those that “the other indices depict the impact of the papers in the core” (p. 836). In particular, theAccording to the authors, h-index and the g-index were classified as belonging to the first category, while the A-index and the R-index certainly belong to the second group. However, Bornmann et al. (2008) recommended a more thorough validation of their factor analysis results by using other data sets, especially from different fields of research.

Schreiber, Malesios and Psarakis (2011) have shown that the distinction is not so evident for the citation records of 26 physicists, which were previously analyzed (Schreiber, 2008a and 2010b). Specifically, the authors utilized 7 bibliometric indices - similar to the analysis of Bornmann et al. (2008), with the addition of standard indicators of quantity and impact, namely total number of publications n, total number of citations S and average number of citations . In particular, the nearly equal factor loadings for g in the exploratory factor analysis (EFA) of the raw data seemed to confirm verify the assumption (Schreiber, 2010a) that the g-index measures both, the quantity and the impact. However, this was not substantiated by the more comprehensive FA. Significant differences to previous analysesthe findings of Bornmann et al. (2008, 2009a, 2009b) have also been found. On the other hand, the results were mostly in agreement with those of Costas & Bordons (2007; 2008) and Hendrix (2008).

In the current article, we expand the previous analysis of Schreiber et al. (2011), by once again utilizing EFA using this time an augmented database consisting of a set of 17 indicators - in addition to the h-index - that have been proposed in recent years to improve the h-index, illustrated in detail by Schreiber (2010b). The actual values of these indices and some standard bibliometric indicators can be found in Appendix A and a short description in Appendix B. By this we attempt to clarify the properties and behaviour of the latter indices, by coming up with categorizations to latent items provided by the factor analysis. Moreover, we attempt to interpret the categorization of those indices based on previous research and the properties shared by the indices.

In addition we investigate the claim that the g-index can be considered to measure both the actual scientific productivity and the scientific impact of a scientist that the g-index can be classified as a bibliometric index that can measure both the “quantity of the productive core” and the “impact of the productive core”, a property not shared by the majority of the other indices.

### 2. Data

The data are from 26 present or former members of the Institute of Physics at Chemnitz University of Technology, including all full and associate professors as well as scientists who have been working as assistants or senior assistants (see Table A1). Data collection period covers the time period between January and February 2007, and were collected Data for the subsequent analysis were compiled between January and February 2007 from the ISI Thomson Web of Science (WoS) database Science Citation Index provided by Thomson Scientific in the Web of Science (WoS) (Schreiber, 2007a). The 26 datasets include the citation records of present or former members of the Institute of Physics at Chemnitz University of Technology, including all full and associate professors as well as scientists who have been working as assistants or senior assistants (see Table A1). The datasets for each researcher are indexed A, B, C, ....., Z in conformity with the previous analysis (Schreiber, 2007a).

In the current article we utilize 18 Hirsch-type indices, namely w, h(2), h, , A, f, t, g, , m, hw, R, Ä§, Ï€, e, s, hT and x (Maxprod). In parallel to the h- and g-indices we also utilize the interpolated and in compliance with the analysis of Schreiber (2010b). In addition the standard bibliometric indicators n, n1, S, c1, and for each dataset are also used.Â

### 3. Methodology - Overview

The statistical methodology of EFA can be used to examine for latent associations to identify the latent structure present in a set of observed variables, called the factors or latent variables. In this way, EFA and reduces dimensionality of the data to a few representative factors. , and therefore summarizes the multivariate information in a simpler form.

Our aim with the specific paper, is to provide a valid In this paper we employ EFA in order to derive categorizations of the h-index and some of its variants, by employing EFA. Although the sample size used for the factor analysis can be regarded as relatively small (N=26), recent studies based on simulations have shown that when certain conditions exist the small sample size does not play a very important role and reliable FA results can be obtained. Specifically, presence of high communalities, when combined with a relatively small number of factors, tends to alleviate the effects of small sample sizes (Preacher & MacCallum, 2002). (For more on this see Schreiber et al. 2011). even with very small sample sizes (e.g. N=10), when certain conditions exist. Specifically, presence of high communalities, when combined with a relatively small number of factors, tends to alleviate the effects of small sample sizes (Preacher & MacCallum, 2002). Our analysis is a typical example of the above, since communalities are extremely high (way above 0.9 in almost all variables) and the number of factors is very small (2 factors), indicating that the analysis can produce valid and robust results.

Bornmann et al. (2008) have utilized a logarithmic transformation to make their data more suitable for the factor analysis, since EFA techniques require that the variables should be approximately normally distributed. In our case there is no need for such transformation, since the non-parametric Kolmogorov-Smirnov test for normality has shown that only 3 out of the 18 items deviate from normality at a 5% level of statistical significance (see Table 1).

Due to the small number of datasets one would expect that the index values are better described by Student's t -distribution. We have performed the respective Kolmogorov-Smirnov test and the results in Table 1 confirm that untransformed data are even better described by the t -distribution than by the normal distribution.

One possible reason for which - in contrast to the data of Bornmann et al. (2008) - our datasets of most of the 18 indices are approximately normally distributed is the diversity of the status of the selected researchers. Indeed, among the 26 researchers of our dataset there are young researchers with comparatively low index scores as well as senior professors with high values of most of their indices.Â On the other hand, Bornmann et al. (2008) study the data of young researchers, whose index values are small and are concentrated within a very narrow field of values, with the direct consequence of giving extremely skewed distributions.

Bornmann et al. (2008) have applied a logarithmic transformation to the raw data before utilizing FA, due to that EFA techniques require that all variables should be approximately normally distributed. To test for normality of our data, Table 1Â presents results of Kolmogorov-Smirnov test for normality, which indicate that data are adequately normally distributed - hence can be forced for conducting FA - although are better described by the t-distribution than by the normal distribution .

However, it is of interest to check if there are any discrepancies in the results between the raw data and the transformed ones, and thus additionally to the raw data x the logarithmically transformed shifted data (ln(x+1)) and the square-root transformed data were also utilized. The latter transformation was applied in this context by Costas & Bordons (2008).

### Table 1: One-sample Kolmogorov-Smirnov test

normal distribution | Student distribution | ||||||

Mean | Median | Std. Dev. | D | p | D | p | |

w | 3.54 | 3.5 | 1.84 | 0.285 | 0.029* | 0.215 | n.s |

h(2) | 5.00 | 5 | 1.60 | 0.230 | n.s | 0.188 | n.s |

h | 14.88 | 14 | 6.92 | 0.186 | n.s | 0.100 | n.s |

| 15.05 | 14 | 6.89 | 0.194 | n.s | 0.087 | n.s |

A | 33.55 | 29.5 | 17.8 | 0.217 | n.s | 0.096 | n.s |

f | 19.23 | 18 | 9.59 | 0.196 | n.s | 0.096 | n.s |

t | 20.92 | 20 | 10.44 | 0.192 | n.s | 0.120 | n.s |

g | 23.96 | 22 | 11.99 | 0.202 | n.s | 0.094 | n.s |

| 24.40 | 22.4 | 12.00 | 0.197 | n.s | 0.095 | n.s |

m | 25.58 | 23.25 | 12.95 | 0.198 | n.s | 0.107 | n.s |

hw | 19.03 | 17.75 | 9.20 | 0.186 | n.s | 0.092 | n.s |

R | 22.18 | 20.2 | 10.82 | 0.199 | n.s | 0.090 | n.s |

Ä§ | 19.80 | 17.55 | 10.17 | 0.247 | n.s | 0.246 | n.s |

Ï€ | 4.55 | 2.95 | 4.93 | 0.273 | 0.041* | 0.273 | 0.041* |

e | 16.26 | 14.3 | 8.69 | 0.199 | n.s | 0.088 | n.s |

s | 12.60 | 10.9 | 6.64 | 0.252 | n.s | 0.252 | n.s |

hT | 24.72 | 22.35 | 12.32 | 0.247 | n.s | 0.247 | n.s |

x | 336.7 | 231 | 341.3 | 0.319 | 0.01* | 0.250 | n.s |

*significant at a 5% significance level

n.s.: non-significant

### 3.1 Exporatory Factor Analysis - Results

We used a least squares factor extraction procedure since it has been argued that the least squares method performs betterwell for small sample sizeswhen using small datasets in comparison to other factor extraction methods such as maximum likelihood (see Ihara and Okamoto, 1985) and a rotated varimax transformation.

In order to confirm the suitability of implementing EFA for the specific data and items selected, the EFA gave a value of 0.828 for the Kaiser-Meyer-Olkin (KMO) measure of model adequacy was used (Kaiser, 1974), indicating that the 18 indices are suitable for the factor analysis. It gave an adequate value of 0.828 for the raw data (see Table 2). The results gave also and similar values for the transformed data.

### Table 2: KMO test

Raw indices x | ln(x+1) | âˆšx | |

## KMO | 0.828 | 0.822 | 0.841 |

p-value | < 0.001 | < 0.001 | < 0.001 |

Both the eigenvalue criterion (according to which one drops any factors with an eigenvalue of less than one) and the scree plot criterion indicated the existence of two major latent structures (factors) as the best solution for explaining the variability in the data. The two factors extracted accounted for 97.64%, 96.48% and 97.11% of the total variance in the raw, the log-transformed, and the square-root transformed data, respectively. For the raw data we see that the first factor accounted accounts for the 53.9% of the variance, the second factor for 43.7%.

The factor loading matrix of factor loadings for the three models with the 18 indices can be found in Table 3. The corresponding communalities shared by the items are presented in Table 4.

### Table 3: Varimax rotated loading matrices (applying least squares extraction and Kaiser normalization) for the 3 EFA models with values above 0.7 given in bold face

Indices | Raw indices x | ln(x+1) | âˆšx | |||

Component | Component | Component | ||||

1 | 2 | 1 | 2 | 1 | 2 | |

w | ## 0.711 | 0.629 | 0.688 | 0.588 | ## 0.702 | 0.597 |

h(2) | ## 0.736 | 0.629 | ## 0.749 | 0.615 | ## 0.748 | 0.618 |

h | ## 0.827 | 0.553 | ## 0.864 | 0.492 | ## 0.848 | 0.520 |

| ## 0.827 | 0.555 | ## 0.866 | 0.493 | ## 0.849 | 0.521 |

A | 0.499 | ## 0.863 | 0.444 | ## 0.895 | 0.471 | ## 0.880 |

f | ## 0.816 | 0.572 | ## 0.850 | 0.519 | ## 0.835 | 0.543 |

t | ## 0.784 | 0.619 | ## 0.809 | 0.585 | ## 0.799 | 0.599 |

g | 0.685 | ## 0.727 | 0.675 | ## 0.735 | 0.682 | ## 0.730 |

| 0.691 | ## 0.722 | 0.685 | ## 0.726 | 0.690 | ## 0.723 |

m | ## 0.706 | 0.649 | 0.650 | 0.607 | 0.686 | 0.624 |

hw | 0.691 | ## 0.721 | 0.677 | ## 0.734 | 0.686 | ## 0.726 |

R | 0.678 | ## 0.733 | 0.675 | ## 0.734 | 0.678 | ## 0.733 |

Ä§ | ## 0.798 | 0.587 | ## 0.786 | 0.599 | ## 0.792 | 0.592 |

Ï€ | 0.675 | ## 0.704 | 0.640 | ## 0.753 | 0.659 | ## 0.735 |

e | 0.549 | ## 0.836 | 0.494 | ## 0.867 | 0.523 | ## 0.852 |

s | ## 0.831 | 0.540 | ## 0.836 | 0.531 | ## 0.834 | 0.534 |

hT | ## 0.835 | 0.550 | ## 0.851 | 0.523 | ## 0.844 | 0.534 |

x | ## 0.770 | 0.591 | ## 0.744 | 0.619 | ## 0.762 | 0.599 |

Eigenvalues | 9.701 | 7.873 | 9.614 | 7.752 | 9.717 | 7.763 |

### Table 4: Variance explained by the 3 EFA models

Indices | Raw indices x | ln(x+1) | âˆšx |

w | 0.901 | 0.819 | 0.849 |

h(2) | 0.938 | 0.939 | 0.942 |

h | 0.990 | 0.989 | 0.989 |

| 0.993 | 0.994 | 0.993 |

A | 0.994 | 0.998 | 0.997 |

f | 0.994 | 0.991 | 0.993 |

t | 0.997 | 0.997 | 0.997 |

g | 0.999 | 0.996 | 0.998 |

| 0.999 | 0.997 | 0.998 |

m | 0.919 | 0.791 | 0.860 |

hw | 0.999 | 0.997 | 0.998 |

R | 0.998 | 0.995 | 0.996 |

Ä§ | 0.981 | 0.976 | 0.978 |

Ï€ | 0.951 | 0.977 | 0.975 |

e | 0.999 | 0.996 | 0.998 |

s | 0.982 | 0.981 | 0.981 |

hT | 0.999 | 0.997 | 0.998 |

x | 0.942 | 0.937 | 0.940 |

A possible interpretation is complicated, when choosing a value of 0.6 as a cut-off threshold for the factor loadings. Then for the raw data 9 items load on both factors, and only h, , Ä§ , f, s, hT, x load on only the first factor, while A and e load strongly on the second factor. This confirms from another viewpoint the observation of Schreiber (2010) that A and e are closely related. This could be so, because these indices are the only ones solely based on h and total number of h-core citations S(h) (The related index R is based entirely on S(h)).Â

The observation of Schreiber (2010b) that the rank orders for w and h(2) are not very different, is reflected in the FA as both indicators share similar loadings on the two dimensions. Both indices - along with h - are based directly on citation counts for different core sizes. However, in the current analysis, h exhibits different behavior in comparison to w and h(2), since it loads solely on the first factor.Â

While A and g are both based on the average number of citations in the FA they appear different since A loads highly on the second factor whereas g loads more evenly on both latent structures. For the indices m, f, t and g depending on different average citation numbers we observe that three of them load on both dimensions, while f loads only on the first dimension. Similarly, the Ä§-index seems to differ from g, R and hw although all of them depend on the square root of the summed number of citations.

The results of applying EFA to the transformed indices are very similar to the categorizations given for the raw data (using a cut-off value of 0.6), except that now w and t have shifted and fall into the first category, too.

Choosing a threshold level 0.7 leads to a clear separation of all indices to the two dimensions for the raw data. Now, besides A and e, also g, , hw, R, Ï€ fall into the second category, the others into the first category. This is also true for the transformed data with the exception of m which is no more attributed to any of the factors. In contrast Bornmann et al. (2008) assign h and g to the same factor (measuring quantity of the research output).

We cannot conclude - in the wayas Bornmann et al. (2008) did - that the first factor relates to the number of papers in the productive core of the researchers' outputs, because indices like f and Ä§ load on that factor, but are based on the number of citations in the core. On the other hand, all the indices loading on the second factor reflect the impact of the papers in that core, i.e. the quality dimension.

The varimax rotation method is an orthogonal rotation method which assumes that the factors in the analysis are uncorrelated. We have additionally to the varimax orthogonal rotation method, utilized an oblique rotation method (specifically promax oblique rotation with least squares extraction) which - in contrast to varimax - does not require the factors to be uncorrelated. Such oblique rotation techniques have been favored against the use of orthogonal rotations There are several studies proposing the use of oblique rotation instead of orthogonal rotation methodology (see e.g. McCroskey and Young, 1979). The value of the promax rotation exponent k was set to 4 since that value provided more interpretable results (Tataryn, Wood and Gorsuch, 1999).

### Table 5: Promax oblique rotated loading matrices for the 3 EFA models with values above 0.5 given in bold face

Indices | Raw indices x | ln(x+1) | âˆšx | |||

Component | Component | Component | ||||

1 | 2 | 1 | 2 | 1 | 2 | |

w | ## 0.599 | 0.384 | ## 0.595 | 0.347 | ## 0.612 | 0.343 |

h(2) | ## 0.642 | 0.361 | ## 0.669 | 0.336 | ## 0.669 | 0.336 |

h | ## 0.866 | 0.148 | ## 0.963 | 0.037 | ## 0.922 | 0.085 |

| ## 0.864 | 0.151 | ## 0.966 | 0.037 | ## 0.923 | 0.085 |

A | 0.022 | ## 0.978 | -0.067 | ## 1.055 | -0.026 | ## 1.020 |

f | ## 0.829 | 0.190 | ## 0.916 | 0.093 | ## 0.879 | 0.135 |

t | ## 0.731 | 0.298 | ## 0.794 | 0.234 | ## 0.769 | 0.260 |

g | 0.463 | ## 0.574 | 0.445 | ## 0.595 | 0.458 | ## 0.581 |

| 0.478 | ## 0.559 | 0.470 | ## 0.571 | 0.477 | ## 0.563 |

m | ## 0.571 | 0.424 | ## 0.517 | 0.410 | ## 0.561 | 0.403 |

hw | 0.479 | ## 0.559 | 0.450 | ## 0.590 | 0.468 | ## 0.571 |

R | 0.446 | ## 0.591 | 0.447 | ## 0.593 | 0.449 | ## 0.589 |

[1] Ä§ | ## 0.784 | 0.233 | ## 0.744 | 0.277 | ## 0.764 | 0.255 |

Ï€ | 0.468 | ## 0.545 | 0.374 | ## 0.654 | 0.416 | ## 0.611 |

e | 0.132 | ## 0.885 | 0.039 | ## 0.964 | 0.085 | ## 0.926 |

s | ## 0.884 | 0.123 | ## 0.883 | 0.125 | ## 0.887 | 0.120 |

hT | ## 0.881 | 0.135 | ## 0.915 | 0.098 | ## 0.902 | 0.112 |

x | ## 0.733 | 0.266 | ## 0.659 | 0.347 | ## 0.707 | 0.294 |

Eigenvalues | 16.462 | 15.516 | 16.018 | 14.998 | 16.269 | 15.233 |

Applying a threshold value 0.5 the results in Table 5 provide a clear distinction of the indices, in full compliance with the results of varimax rotation (when using the threshold 0.7).

### 3.2 Expanded Set

In an effort to further categorize h-type variants into indices based on quantity and quality Bornmann, Mutz, Daniel, Wallon and Ledin (2009) have re-run the EFA of Bornmann et al. (2008) including the standard bibliometric measures n and S. Along the same lines, we re-ran our EFA including besides n also other bibliometric measures, as in Schreiber (2010), namely the number of cited publications n1, the average number of citations per article = S/n, the highest number of citations c1, and the average number of citations in the elite set defined by Vinkler (2009) as the most cited nÏ€=âˆšn papers.

In this way we intend - similarly to Bornmann et al. (2009b) - a categorization of the indices to the quantity dimension (expressed by n and n1) and the impact dimension (expressed by and c1). The results of the EFA using the least squares extraction method and the varimax rotation with Kaiser normalization are presented in Tables 6, 7 and 8. Once again, the results suggested a factor structure with only two factors having an eigenvalue larger than 1, which both explain 96.1% of the variability in the data.

### Table 6: KMO test

Raw indices x | ln(x+1) | âˆšx | |

## KMO | 0.66 | 0.716 | 0.657 |

p-value | < 0.001 | < 0.001 | < 0.001 |

### Table 7: Varimax rotated loading matrices for the 3 EFA models with values above 0.685 given in bold face

Indices | Raw indices x | ln(x+1) | âˆšx | |||

Component | Component | Component | ||||

1 | 2 | 1 | 2 | 1 | 2 | |

w | ## 0.685 | 0.648 | 0.678 | 0.591 | ## 0.687 | 0.605 |

h(2) | ## 0.696 | 0.665 | ## 0.730 | 0.627 | ## 0.721 | 0.640 |

h | ## 0.767 | 0.618 | ## 0.833 | 0.521 | ## 0.804 | 0.565 |

| ## 0.769 | 0.617 | ## 0.837 | 0.520 | ## 0.807 | 0.563 |

A | 0.496 | ## 0.857 | 0.473 | ## 0.869 | 0.491 | ## 0.859 |

f | ## 0.763 | 0.629 | ## 0.827 | 0.539 | ## 0.799 | 0.580 |

t | ## 0.738 | 0.665 | ## 0.790 | 0.601 | ## 0.769 | 0.627 |

g | 0.658 | ## 0.753 | 0.682 | ## 0.731 | 0.676 | ## 0.737 |

| 0.662 | ## 0.750 | ## 0.689 | ## 0.725 | 0.681 | ## 0.733 |

m | 0.659 | ## 0.692 | 0.623 | 0.624 | 0.658 | 0.648 |

hw | 0.666 | ## 0.745 | 0.682 | ## 0.730 | 0.681 | ## 0.732 |

R | 0.647 | ## 0.763 | 0.674 | ## 0.738 | 0.666 | ## 0.746 |

Ä§ | ## 0.806 | 0.590 | ## 0.821 | 0.565 | ## 0.819 | 0.569 |

Ï€ | 0.674 | ## 0.707 | 0.675 | ## 0.720 | ## 0.686 | ## 0.710 |

e | 0.541 | ## 0.835 | 0.516 | ## 0.848 | 0.535 | ## 0.837 |

s | ## 0.834 | 0.550 | ## 0.862 | 0.504 | ## 0.853 | 0.519 |

hT | ## 0.812 | 0.580 | ## 0.856 | 0.515 | ## 0.840 | 0.540 |

x | ## 0.788 | 0.585 | ## 0.781 | 0.584 | ## 0.799 | 0.569 |

n1 | ## 0.949 | 0.195 | ## 0.950 | 0.195 | ## 0.951 | 0.190 |

n | ## 0.958 | 0.149 | ## 0.966 | 0.143 | ## 0.964 | 0.142 |

c1 | 0.346 | ## 0.843 | 0.302 | ## 0.853 | 0.316 | ## 0.847 |

0.369 | ## 0.926 | 0.377 | ## 0.927 | 0.378 | ## 0.926 | |

0.107 | ## 0.938 | 0.157 | ## 0.952 | 0.139 | ## 0.953 | |

Eigenvalues | 11.13 | 10.975 | 11.742 | 10.188 | 11.614 | 10.423 |

### Table 8: Variance explained by the 3 EFA models

Indices | Raw indices x | ln(x+1) | âˆšx |

w | 0.889 | 0.809 | 0.837 |

h(2) | 0.926 | 0.926 | 0.930 |

h | 0.970 | 0.965 | 0.965 |

| 0.972 | 0.971 | 0.969 |

A | 0.981 | 0.979 | 0.979 |

f | 0.977 | 0.974 | 0.974 |

t | 0.987 | 0.985 | 0.986 |

g | 0.999 | 0.999 | 0.999 |

| 0.999 | 0.999 | 0.999 |

m | 0.914 | 0.778 | 0.852 |

hw | 0.998 | 0.998 | 0.998 |

R | 0.999 | 0.999 | 0.999 |

Ä§ | 0.998 | 0.993 | 0.995 |

Ï€ | 0.954 | 0.974 | 0.976 |

e | 0.990 | 0.986 | 0.987 |

s | 0.998 | 0.997 | 0.997 |

hT | 0.996 | 0.998 | 0.997 |

x | 0.963 | 0.951 | 0.961 |

n1 | 0.938 | 0.941 | 0.940 |

n | 0.940 | 0.953 | 0.950 |

c1 | 0.830 | 0.819 | 0.817 |

| 0.993 | 0.999 | 0.999 |

| 0.891 | 0.931 | 0.927 |

From Table 7 we see that by selecting a threshold between 0.674 and 0.685, we get a clear distinction of all the raw indices, with the first dimension of the EFA comprising w, h(2), h, , f, t, Ä§, s, hT, x, n1, n while A, g, , m, hw, R, Ï€, e, c1,Â , load on the second factor. The high loadings of n and n1 on the first factor and and on the second factor, mean that by including these standard bibliometric indicators into the analysis we have successfully enforced a distinction separation of between the quantity and the quality dimension.

Results of the promax oblique rotation (with k = 3 and least squares extraction) in Table 9 show once again a more distinct separation to the two dimensions.

### Table 9: Promax oblique rotated loading matrices for the raw indices with values above 0.54 given in bold face

Indices | Raw indices x | |

Component | ||

1 | 2 | |

w | ## 0.548 | 0.481 |

h(2) | ## 0.554 | 0.497 |

h | ## 0.667 | 0.402 |

| ## 0.670 | 0.401 |

A | 0.206 | ## 0.839 |

f | ## 0.656 | 0.419 |

t | ## 0.607 | 0.477 |

g | 0.462 | ## 0.628 |

| 0.469 | ## 0.621 |

m | 0.493 | ## 0.550 |

hw | 0.477 | ## 0.613 |

R | 0.443 | ## 0.645 |

Ä§ | ## 0.730 | 0.348 |

Ï€ | 0.506 | ## 0.561 |

e | 0.273 | ## 0.789 |

s | ## 0.785 | 0.284 |

hT | ## 0.742 | 0.333 |

x | ## 0.710 | 0.350 |

n1 | ## 1.104 | -0.222 |

n | ## 1.139 | -0.284 |

c1 | 0.023 | ## 0.895 |

0.012 | ## 0.988 | |

-0.326 | ## 1.134 | |

Eigenvalues | 17.577 | 17.441 |

The obtained results suggest that the g-index (accordingly also ) contributes more in measuring the quality dimension, whereas the h-index (and accordingly ) measures mostly the quantity dimension.

To achieve an even clearer categorization of the indices we have performed the analysis including also the total number of citations S, as this specific metric has been also utilized by Bornmann et al. (2008). The contribution of the indices to the two factors shown in Table 10 yields a clear distinction in full agreement with Table 7, if again the threshold value 0.685 is used.

### Table 10: Varimax rotated loading matrices for the raw indices with values above 0.685 given in bold face

Indices | Raw indices x | |

Component | ||

1 | 2 | |

w | ## 0.686 | 0.646 |

h(2) | ## 0.694 | 0.664 |

h | ## 0.767 | 0.616 |

## 0.769 | 0.616 | |

A | 0.499 | ## 0.855 |

f | ## 0.763 | 0.627 |

t | ## 0.739 | 0.663 |

g | 0.659 | ## 0.752 |

0.663 | ## 0.748 | |

m | 0.661 | ## 0.691 |

hw | 0.668 | ## 0.743 |

R | 0.648 | ## 0.761 |

Ä§ | ## 0.807 | 0.588 |

Ï€ | 0.681 | ## 0.703 |

e | 0.543 | ## 0.834 |

s | ## 0.835 | 0.548 |

hT | ## 0.813 | 0.579 |

x | ## 0.794 | 0.581 |

n1 | ## 0.951 | 0.192 |

n | ## 0.959 | 0.146 |

S | ## 0.782 | 0.581 |

c1 | 0.354 | ## 0.840 |

0.372 | ## 0.924 | |

0.106 | ## 0.939 | |

Eigenvalues | 11.795 | 11.261 |

A rather surprising result is that S exhibits higher loading on the first factor, rather than on the second factor on which the other indicators that are based on the citations load strongly. That was already observed by Schreiber et al. (2011), and might be explained by the assumption that S correlates more strongly with n than with , since more papers attract more citations. This may also be an indication that S is not the best indicator for measuring quality. The same argument applies to Ä§, because it is proportional to . Thus it loads strongly on the first factor just like S.

Most distinctive (except from the standard bibliometric indices) in terms of very high loadings are A and e belonging clearly in the group of indices measuring the impact of the productive core and Ä§, s, x and hT measuring the number of papers in the productive core.

### 4. Conclusions

In this paper we have examined the relationship of the h-index with other related indices measuring research performance using exploratory factor analysis. We have shown, that for our dataset consisting of a wide variety of bibliometric indices, for most of the investigated indices a distinction was evident to one of the two basic dimensions of scientific performance, namely the quality and quantity of scientific output. In summary, two different groups of indices were identified according to the results of EFA. Generally, there was strong indication based on the results of the conducted EFA that most of the indices cannot be fully categorized in any of the two factors. However, for some of the indices there is a stronger tendency to describe the quantity of the productive core. Among these indices are the w, h(2), h, , f, t, Ä§, s, hT, and x. Especially for the h-index, both quantity and impact of articles are taken into account, however the analysis suggests that quantity of publications plays the most important role. In the same manner, for other indices there is a stronger tendency to describe the impact of the productive core, including the A, g, , m, hw, R, Ï€ and e. These results also confirm the results of Schreiber (2010a), who based on theoretical arguments suggests that g, A and R belong to the same category of indices, and contrast the different classifications between g and A, R by Bornmann et al. (2008).

Nevertheless, the present investigation adds to the results derived by Schreiber et al. (2011), by generalizing the preliminary findings obtained using a set of 7 indices, this time by including most of the important h-type indices proposed to correct insufficiencies of the Hirsch index.