Unveiling University Groupings: A Clustering Analysis for Academic Rankings ^†

Abstract

The evaluation and ranking of educational institutions are of paramount importance to a wide range of stakeholders, including students, faculty members, funding organizations, and the institutions themselves. Traditional ranking systems, such as those provided by QS, ARWU, and THE, have offered valuable insights into university performance by employing a variety of indicators to reflect institutional excellence across research, teaching, international outlook, and more. However, these linear rankings may not fully capture the multifaceted nature of university performance. This study introduces a novel clustering analysis that complements existing rankings by grouping universities with similar characteristics, providing a multidimensional perspective on global higher education landscapes. Utilizing a range of clustering algorithms—K-Means, GMM, Agglomerative, and Fuzzy C-Means—and incorporating both traditional and unique indicators, our approach seeks to highlight the commonalities and shared strengths within clusters of universities. This analysis does not aim to supplant existing ranking systems but to augment them by offering stakeholders an alternative lens through which to view and assess university performance. By focusing on group similarities rather than ordinal positions, our method encourages a more nuanced understanding of institutional excellence and facilitates peer learning among universities with similar profiles. While acknowledging the limitations inherent in any methodological approach, including the selection of indicators and clustering algorithms, this study underscores the value of complementary analyses in enriching our understanding of higher educational institutions’ performance.

1. Introduction

Finding a distinguished university is a critical step in securing a high-quality education, offering students a sturdy foundation of knowledge, skills, and credentials. Furthermore, it can lead to heightened career prospects post-graduation, as employers tend to favor graduates from prestigious universities, often translating into more enhanced job opportunities. These esteemed universities offer students the chance to participate in cutting-edge research, gain invaluable experiential insights, and cultivate essential skill sets crucial for their forthcoming endeavors.

Evaluating and ranking universities is a complex and multifaceted process that involves careful consideration of various indicators to ensure an objective and precise assessment. Leading ranking organizations, such as the Academic Ranking of World Universities (ARWU) [1], Quacquarelli Symonds (QS) [2], and Times Higher Education World University Rankings (THE) [3], employ sophisticated algorithms to evaluate universities based on a wide range of factors, including academic reputation, faculty–student ratio, academic citations, and international diversity. Each indicator is assigned a specific weight to reflect its relative importance in the university evaluation process. According to [4], an examination of these indicators reveals the following:

• The low similarity of indicators across ranking lists implies that organizations primarily rely on distinct sets of criteria when assessing universities.
• The weight assigned to indicate the significance of an indicator can vary across different ranking lists.
• The research production and impact-related indicators are similar among ranking lists.

University rankings not only aid students in making informed decisions, but also incentivize universities to enhance their performance in critical domains like research, teaching, and internationalization, all of which are essential to their stakeholders. While these rankings offer significant benefits, they are not without limitations, including the following:

• Subjectivity—The choice of indicators and their associated weights may rely on subjective and contentious criteria, resulting in rankings that do not accurately reflect the genuine capabilities and performance of universities.
• Narrow Focus—The indicators employed in ranking systems might fall short of encompassing the entire spectrum of activities and outputs of universities, potentially resulting in a limited and partial assessment of their performance. For instance, rankings primarily centered around research productivity may not adequately gauge the quality of teaching or the broader impact of universities on their local communities.
• Incentive Distortion—Ranking systems can incentivize universities to prioritize specific activities over others, potentially distorting their original missions and objectives.
• Lack of Transparency—The lack of transparency in the methodology and data sources employed by ranking organizations poses a challenge for universities to pinpoint and address areas for improvement. Additionally, this opacity may foster skepticism and mistrust among stakeholders towards the reliability of the rankings.

Recognizing the limitations of existing ranking systems, this study is motivated by the pursuit of a complementary analytical approach. By employing clustering algorithms to group universities based on similar characteristics, we aim to provide a more nuanced perspective on the global higher education landscape. This clustering analysis is not intended to replace traditional rankings but to augment them, offering interested parties an alternative approach for viewing and assessing the strengths and similarities among groups of universities. Such an approach acknowledges the diversity of university missions and the importance of multiple factors in determining institutional success, thus encouraging a more holistic and collaborative understanding of educational excellence. Our motivations are rooted in the belief that a multidimensional analysis can facilitate more informed decision-making for students, educators, policymakers, and institutions alike, promoting a richer dialogue about the qualities that define leading universities worldwide. This approach may also be preferable to a straightforward ranking of universities from top to bottom for several compelling reasons, such as the following:

• A limitation of criteria—Ranking methodologies often rely on a narrow set of criteria for assessing university performance, yet certain universities excel in diverse domains of expertise. Consequently, employing clustering methods can highlight their individual strengths and areas of excellence.
• Diverse needs—Every individual harbors distinct priorities when it comes to selecting a university, encompassing factors like expenses, geographic location, and available academic programs.
• Avoiding Stigma—Establishing a ranking hierarchy where universities are sorted from the best to the worst can result in the marginalization of lower-ranked institutions. To alleviate this effect and promote a more constructive and inclusive view of universities, clustering them based on specific attributes or areas of expertise can be a valuable alternative.

The remainder of the article is organized as follows: In Section 2, we review other related studies, while Section 3 presents the dataset and the data pre-processing phase. Section 4 presents the methodology, and finally, Section 5 concludes our study.

2. Related Work

Clustering is a machine learning technique that belongs to the category of unsupervised learning and aims to group related objects together into distinct clusters. There are several different types of clustering, each with their own benefits and drawbacks. Some of the types of clustering are as follows:

• Hierarchical—Utilizing this methodology, clusters are structured within a dendritic, tree-like arrangement, in which each smaller cluster serves as a subset of a more comprehensive cluster. Two fundamental modes of hierarchical clustering are evident: agglomerative [5] and divisive [6]. Within the agglomerative clustering technique, each data point originates within its own distinct cluster, culminating in the eventual amalgamation of all clusters into a unified entity. Conversely, the divisive clustering method commences with all data points situated within a solitary cluster, subsequently undergoing incremental partitioning into smaller clusters through the algorithm.
• K-Means—One widely employed clustering method is K-Means clustering [7]. In this technique, the dataset is partitioned into K clusters, where K is a modifiable parameter. The objective is to minimize the total sum of squared distances between each data point and its designated cluster center.
• Fuzzy Clustering—This approach enables data points to exhibit multiple degrees of membership across multiple clusters. Among the field, Fuzzy C-Means (FCM) clustering [8] stands as the most widely utilized algorithm for fuzzy clustering.
• Density-Based—The density-based clustering technique involves the grouping of data points that are in close proximity within high-density regions, while being separated by regions of lower density. Among these methods, DBSCAN [9], which stands for “Density-Based Spatial Clustering of Applications with Noise”, is the most widely recognized density-based clustering algorithm.
• Model-Based—This methodology posits a mixture of probability distributions as the origin of the data points. Among model-based clustering techniques, the Gaussian Mixture Model (GMM) algorithm [10] stands as the most prevalent and widely employed approach.

Considerable research has been undertaken in the field of evaluating and ranking academic institutions, spanning universities, departments, and diverse academic domains.

A renowned technique employed for the evaluation of university rankings is rank fusion [11]. Rank fusion, also referred to as meta-ranking, constitutes a procedure for amalgamating the outcomes of multiple university rankings, which include assessments based on multiple factors and criteria, thereby yielding a more encompassing and dependable overall ranking. The process of rank fusion entails the normalization of diverse rankings and their amalgamation through a weighted averaging mechanism, wherein the assigned weights quantify the significance attributed to each respective ranking.

In contrast to individual ranking systems, the process of rank fusion presents several advantages. It fosters a greater degree of stability and dependability in the rating system, diminishing the impact of outliers or inaccuracies within individual rankings. Additionally, it possesses the capacity to consider a broader spectrum of factors or perspectives, thereby furnishing a more comprehensive assessment of the performance or quality of the ranked entities. However, it is imperative to exercise caution when employing rank fusion, as the weights applied to amalgamate the rankings may carry a degree of subjectivity or bias.

A useful method for ranking universities is the Borda Count method [12]. The Borda Count method is a single-winner, voting-based technique for aggregating and consolidating rankings from different sources or criteria. In this method, each university is assigned a score in each individual ranking, typically based on its position in that ranking. The scores are then summed across all rankings, and universities are ranked based on their total scores. The higher a university ranks in an individual ranking, the more points it receives. The Borda Count allows for the integration of diverse indicators into a single composite ranking. It is particularly useful when dealing with university rankings from various organizations, as it provides a way to balance and combine these rankings to create a unified assessment of university performance that reflects the collective preferences of the rankings.

There exists a multitude of scholarly investigations concerning the ranking of universities, drawing inspiration from conventional methodologies such as Data Envelopment Analysis (DEA) [13,14], multicriteria sorting [15], the application of the Pareto Front [16], as well as the aforementioned rank fusion technique. Nevertheless, it is important to acknowledge that each university ranking entity may adopt a distinct ranking framework or devise proprietary methodologies to appraise the performance of individual universities, thereby yielding divergent ranking lists.

For our study, we relied on [17], although [18] is very similar to our analysis. In [17], the authors clustered the top 500 universities from the National Taiwan University (NTU) ranking list. The NTU ranking list is based on eight validated research performance indicators, organized into three main categories, each with a corresponding weight:

• Research productivity—25%
• Research impact—35%
• Research excellence—40%

The outcomes of their clustering experiments, involving 12 and 43 clusters, utilizing the DBSCAN, EM (Expectation-Maximization) [19], and K-Means algorithms, demonstrated a strong similarity to the ranking provided by the NTU ranking list. This similarity was notably pronounced with the EM and K-Means algorithms, while the DBSCAN algorithm did not yield comparable results. Within the 12 and 43 clusters derived from their analysis, a singleton cluster emerged at the top, containing a single university recognized for its exceptional performance. As a result, they concluded that K-Means stands out as the most suitable algorithm for university clustering.

In the study [18], the dataset from Quacquarelli Symonds (QS) for the year 2022 served as the basis for clustering and analysis. The author employed six score indicators, elaborated in Section 3, and opted for GMM as the primary clustering algorithm. Differing from the eight research-oriented indicators employed in [17], the QS indicators encompass a broader spectrum of criteria for evaluating university performance. The determination of the optimal number of clusters, set at four, was accomplished using Akaike’s Information Criteria (AIC) [20] and Bayesian Information Criteria (BIC) [21]. In contrast to QS’s existing ranking system, the study’s findings introduced a novel classification scheme for universities, where each cluster portrayed universities in a manner distinct from their assigned rankings in the QS list.

In [22], the authors classified the top 500 universities into 21 types according to their disciplinary characteristics, using data from the Institute of Higher Education, Shanghai Jiao Tong University. The indicators used for the classification are the percentage of publications in six broad disciplinary areas:

• Arts/Humanities and Social Sciences
• Natural Sciences and Mathematics
• Engineering/Technology and Computer Sciences
• Life Sciences
• Clinical Medicine
• Interdisciplinary and Multidisciplinary Sciences

Universities were categorized based on their focus, priority, and orientation with specific disciplinary groups. The universities that did not fall into any of the aforementioned categories were designated as “balanced”. An examination was conducted to assess the distribution of various types of universities across nations and ranking systems. In the clustering procedure, a customized algorithm was employed, utilizing the Squared Euclidean Distance as the similarity metric.

There are other notable studies related to our work of clustering universities, such as those in [23,24,25,26]. In [23], the object of research is the internal structure of management in universities, its relationship to rating, and the clustering of universities in the Republic of Kazakhstan in order to determine the effectiveness of management. The authors considered three clustering models, each of which presented intriguing results regarding the clustering parameters and their values. In [24], to enable a more suitable and fair comparison of knowledge exchange performance among English institutions, a cluster analysis sought to identify groups of universities based on their structural characteristics that shape knowledge-sharing possibilities and challenges. The study also recognized the criticality of diverse higher educational institutions and the difficulties that arise from clustering them. In [25], the study highlights the universities located in countries that host 90% of the top-ranked universities in Latin America by presenting the findings of a descriptive analysis based on clusters of 85 Latin American universities found in the top 50 positions of the ARWU, SIR Scimago, QS, and Webometrics rankings. For the purposes of performing the descriptive statistical analysis, clusters were constructed by taking into account the frequency of the presence of universities in the top 50 of the four rankings, their location, and the country they belong to. In [26], the purpose of the study was to classify Korean universities according to their research performance, and validate the classifications by comparing their research performance to those that are located in the U.S. As compared to U.S. peers, Korean universities’ research performance was comparable. Furthermore, the study revealed that the classification outcomes produced by the performance-based method were comparable to those of conventional classifications that used predetermined criteria. However, the above studies are limited only to national or regional data.

Compared to the analyses conducted in studies [17,18,22], our study employed K-Means, GMM, Agglomerative, and Fuzzy C-Means algorithms to offer a more comprehensive and nuanced perspective on the dataset. This approach aimed to unveil diverse patterns and structures that may remain concealed when relying on a single algorithm. To ensure a fair and equitable university clustering process, we expanded upon the six indicators utilized in study [18] by incorporating three additional indicators sourced from the ranking list, as detailed in Section 3. In contrast to the methodologies applied in studies [17,22], which predominantly incorporated research-focused indicators, our approach encompassed a broader spectrum of indicators for evaluating university performance. Despite Fuzzy C-Means calculating the probability of a university belonging to multiple clusters, our algorithm ultimately assigned each university to a single cluster.

3. Dataset and Data Pre-Processing

3.1. Dataset

For the purposes of this research, we acquired a dataset from the official website of Quacquarelli Symonds (QS) for the year 2023. In the dataset, there are over 1400 universities from all over the world, including universities from diverse locations in Europe, Asia, and North America. We selected this particular dataset due to the presence of diverse indicators used for the assessment of university performance, in contrast to other university rankings that primarily rely on bibliographic-related metrics.

In total, there are 27 columns in the dataset, but, according to the QS ranking, each university is ranked and assessed using only the six following columns/indicators:

• Academic Reputation (ar score)—Evaluates the teaching and research quality of the university.
• Employer Reputation (er score)—Evaluates how competent, innovative, and effective students and graduates are for the employment market.
• Faculty/student ratio (fsr score)—Evaluates how universities provide students with meaningful access to faculty staff.
• Citations per faculty (cpf score)—Evaluates the total number of academic citations about the papers published in the last five years.
• International student ratio (isr score)—Evaluates the ability of a university to attract foreign students.
• International faculty ratio (ifr score)—Evaluates the ability of a university to attract foreign faculty staff.

The above indicators each, in turn, get a percentage of $40\%,10\%,20\%,20\%,5\%,\mathrm{and}5\%$ of the total score, a proportion specified by QS [2].

In our analysis, we included the following three additional indicators:

• Size—The total number of full-time degree-seeking students.
• Focus—The broad subject areas of each university, e.g., Arts, Humanities, Engineering and Technology, Natural Sciences, etc.
• Age Band—The age of each university.

The selection of the three supplementary indicators was premised on their ability to facilitate the categorization of universities according to their respective values, as shown in Table 1. This enables the establishment of distinct and equitable rankings among the institutions, thus providing a more comprehensive evaluation of their performance across multiple dimensions. Each of the three indicators captures a percentage of the total number of universities. All the above indicators have been selected to be applicable to all universities, regardless of their geographical area.

Table 1. Size, focus, and age band categories’ definitions according to QS [2]. Reprinted with permission from Ref. [27].

	Size	Students	Perc. (%)
XL	Extra Large	More than 30,000	23.8
L	Large	≥12,000	46.6
M	Medium	≥5000	23.8
S	Small	Fewer than 5000	5.8
	Focus	Faculty Area
FC	Full Comprehensive	5 Faculty Areas and Medical School	41.6
CO	Comprehensive	5 Faculty Areas	32.5
FO	Focused	3 or 4 Faculty Areas	22.0
SP	Specialist	2 or Fewer Faculty Areas	3.9
	Classification	Age
5	Historic	100 Years Old and More	37.5
4	Mature	50–99 Years Old	34.4
3	Established	25–49 Years Old	20.4
2	Young	10–24 Years Old	6.7
1	New	Less than 10 Years Old	0.9

3.2. Data Pre-Processing

Following the description of the indicators used in our analysis, it was important to undertake certain data pre-processing steps as a necessary precursor to the clustering process. As an initial step, the exclusion of universities with missing data in any of the aforementioned nine indicators was essential to ensure equitable evaluation. Subsequently, categorical string values within the “size” and “focus” indicators were mapped into categorical numerical values spanning from 0 to 100. This conversion was conducted in order for the clustering algorithms to function properly, as they receive only numerical values. Lastly, to scrutinize the correlation among the indicators, whether categorical or continuous, we employed the Spearman rank correlation coefficient [28]. The Spearman rank correlation coefficient is a statistical measure that assesses the strength and direction of the monotonic relationship between two variables. It is based on the ranks of the data rather than their actual values. The resulting correlation matrix for the nine indicators is provided in Table 2.

Table 2. Spearman rank correlation matrix for the nine indicators.

	size	focus	age	ar	er	fsr	cpf	ifr	isr
size	1
focus	0.44	1
age	0.22	0.21	1
ar	0.34	0.39	0.33	1
er	0.18	0.24	0.23	0.79	1
fsr	−0.22	0.11	0.13	0.32	0.28	1
cpf	0.11	0.26	0.2	0.51	0.32	0.07	1
ifr	−0.068	0.12	−0.003	0.45	0.42	0.21	0.45	1
isr	−0.1	0.05	0.09	0.39	0.33	0.28	0.41	0.69	1

Upon scrutinizing the aforementioned tables, several pertinent observations can be deduced, contributing to the formulation of meaningful conclusions in the context of the present research. The academic and employer reputation indicators have a relatively strong correlation with most of the other indicators and especially with one another, implying that, as the academic reputation of a university increases, employers are more likely to hire students who graduated from that university. Furthermore, a strong correlation is observed in the relationship between the number of international students and the presence of foreign faculty staff. This outcome aligns with expectations, as a university’s acceptance of a substantial number of international students logically corresponds with the recruitment of foreign faculty members. Another strong and obvious correlation emerges between a university’s academic reputation and the quantity of citations per faculty. This alignment is unsurprising, given that a higher volume of citations stemming from a school’s research activities naturally increases the university’s overall academic standing. Noteworthy observations can be made regarding the size, focus, and age band indicators. It is apparent that the size of a university influences its focus, and vice versa, but does not significantly impact the age of the university. Therefore, the presence of an extensive array of faculty areas in a university does not necessarily imply an older age but rather suggests a relatively larger student population. Additionally, upon analyzing their correlation with the other continuous indicators, it is clear that the size, age, and broad subject areas do not exert a significant influence on a university’s overall performance.

Having established the relative significance of the indicators within our cluster analysis, our subsequent action was to define custom indicator weights, which maintain a proportional ratio to the initial weight assignments provided by QS. Before assessing the clustering algorithms, two distinct normalization techniques were considered:

• We normalize the data and apply indicator weights.
• We normalize the data without applying indicator weights.

For normalization, we employed the MinMaxScaler method:

$$x_{\mathrm{scaled}}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}.$$

(1)

Each technique was used once per experimentation cycle. One experimentation cycle involves the following steps:

• Normalizing the data using one of the aforementioned techniques.
• Clustering the data with different numbers of clusters.
• Computing the pairwise similarity between all combinations of clustering algorithms using the Rand Index (RI) [29]. The Rand Index measures the similarity between two clustering results by comparing the agreements and disagreements in pairwise cluster assignments.

By calculating the similarity between a pair of clustering algorithms, we gain insights into the degree of proximity between them. However, our objective was to compare the ranking list generated through the amalgamation of clusters produced by a clustering algorithm, with the QS ranking list. Details of how a ranking list is produced from the clustering algorithms are discussed in Section 4. To facilitate this comparison, we created distinct clusters (named QS clusters) to represent the QS ranking list, separate from those produced by the algorithms. The size of these QS clusters corresponded to the size of the clusters generated by the clustering algorithms. Consequently, the members of each QS cluster were organized to match the order of universities as presented in the QS ranking list. For example, if the first cluster (after applying a clustering algorithm) has a size of 100, then the first QS cluster encompasses the first one hundred universities from the QS ranking list. Respectively, the second QS cluster, irrespective of its size, encompasses the consecutive universities following the top one hundred universities from the first cluster. This approach allowed us to essentially compare the QS ranking list, which is represented by the QS clusters, with the rankings produced by the clustering algorithms.

4. Clustering Analysis

The clustering algorithms that were used for this analysis are the following:

• K-Means—K-Means can be a useful algorithm for university clustering when simplicity, efficiency, and intuitive results are important.
• GMM—GMM was selected due to its capability to model the data as coming from a mixture of Gaussian distributions, which is appropriate given the continuous nature of our data. GMM provides a probabilistic model that not only assigns memberships to clusters but also accounts for the covariance structure of the data, thus allowing for a more nuanced understanding of the relationships between universities. GMM was also utilized in a previous study [18], which is relevant to our own research.
• Agglomerative—Agglomerative clustering can be used when a hierarchy of clusters is needed, or when flexibility and interpretability are important. However, it may not be the best choice for all situations, particularly when the underlying data have complex relationships or non-linear distributions.
• Fuzzy C-Means—Fuzzy C-Means was chosen for its ability to allow each university to belong to multiple clusters with varying degrees of membership. This method is particularly useful in capturing the inherent overlaps in university characteristics, which are too complex to be neatly partitioned into crisp clusters.

K-Means and GMM initialized their centroids using the K-Means++ method, which selects initial cluster centroids using sampling based on an empirical probability distribution of the points’ contribution to the overall inertia. This technique sped up convergence. The algorithm implemented was “Greedy K-Means++”. Fuzzy C-Means initialized the centroids randomly.

To find the optimal number of clusters using the nine indicators we considered the following:

• The elbow method for a maximum of 30 clusters using the K-Means algorithm to calculate the sum of squared distances from each point to its assigned center.
• The AIC and BIC for a maximum of 30 clusters. The maximum number of clusters was chosen arbitrarily. Since AIC and BIC are both statistical measures used for model selection and comparison, we used the GMM algorithm for these measures.

By applying the elbow method, the graph in Figure 1 came up. By taking a closer look at the graph, it is apparent that the elbow method does not work well because, as we will see later on, the data are not very clustered. As a result, there is a smooth curve and the optimal value of K is unclear, thus making this method unfit for use.

Figure 1. Elbow method for a maximum of 30 clusters using the K-Means algorithm.

Using the AIC and BIC measures we came up with the graph that is shown in Figure 2. The number of clusters for the minimum value of BIC is seven, which is identical to the results of the EM algorithm. The number of clusters for the minimum value of AIC is approximately 29 but we used 24, as the difference between the AIC value of 29 and 24 clusters is quite small. In order to visualize the clusters, we used the Multidimensional Scaling (MDS) algorithm [30] to reduce the initial dimensions of the data to two.

Figure 2. AIC and BIC values for a maximum of 30 clusters and a minimum of two. Reprinted with permission from Ref. [27].

The clusters were ordered according to the mean value of each cluster ($CM$). The mean value of a cluster was calculated as follows:

$$CM=\frac{{\sum }_{j=1}^{\mathrm{cluster}\mathrm{size}}\frac{{\sum }_{i=1}^9{\mathrm{indicator}}_{j,i}}{9}}{\mathrm{cluster}\mathrm{size}}.$$

(2)

We calculated a university’s overall performance by summing up its individual indicator values and then dividing the total by nine, the number of indicators, to derive the mean score. Next, by aggregating the means of all universities within a cluster and dividing this sum by the cluster’s size, we obtained the cluster’s mean score. This methodology generates a cluster ranking that subsequently reflects the ranking of universities within each respective cluster.

4.1. Clustering without Weights for Seven Clusters

The clustering results for seven clusters and nine indicators without weights are shown in Figure 3. The sizes of each cluster for every algorithm are shown in Table 3. Table 4 presents the similarity between every pair of algorithms.

Figure 3. Visualization for seven clusters without weights. (a) K-Means, (b) GMM, (c) Agglomerative, and (d) Fuzzy C-Means.

Table 3. Cluster sizes for seven clusters without weights.

	1	2	3	4	5	6	7
KM	112	91	183	126	153	332	311
GMM	441	166	183	130	97	128	163
Agg	99	138	166	183	145	220	357
Fuzzy	138	221	214	161	62	81	431

Table 4. Rand Index for seven clusters without weights.

	KM	GMM	Agg	Fuzzy	QS
KM	1
GMM	0.729509	1
Agg	0.86622	0.736062	1
Fuzzy	0.831163	0.723259	0.819637	1
QS	0.776705	0.760719	0.78335	0.784693	1

By visualizing the results of the clustering algorithms and calculating the Rand Index (see Table 4), we can make a few observations. First of all, the cluster with the highest mean (the red cluster), for every algorithm, has a very low density. That is to be expected given that universities in that cluster tend to have diverse values for their indicators, whereas universities in clusters with lower means tend to have similar values for these indicators. Thus, clusters with a lower mean have a higher density.

It is important to note that the clusters exhibit overlap, meaning that universities belonging to one cluster may appear in others. This overlap is a result of two key factors. Firstly, the clustering process relies on nine distinct indicators, while the visualization phase reduces these indicators to just two dimensions. This dimensionality reduction can lead to errors in accurately representing the data. Secondly, the differentiation between the cluster members of a clustering algorithm and the QS clusters contributes to the observed overlap. For the sake of comparison, we assumed that the cluster members of a clustering algorithm mirrored those of the QS clusters (the clusters created to compare the QS ranking list to the ranking list produced by the clustering algorithms). This assumption implies that, regardless of their size, each cluster of a clustering algorithm will encompass the same members (universities) as its corresponding counterpart of the QS clusters. In practice, however, the cluster members of a clustering algorithm often differed from those of the QS clusters. In other words, a university may find itself within a cluster featuring a lower mean, while, according to the clustering of QS, it should have been placed in a cluster with the highest mean, or vice versa. As a result, some universities may appear in different clusters, according to the QS clusters. In order to compare the clusters derived from the clustering methods with those of the QS, we selected the algorithm that demonstrated the highest similarity to the QS, as defined by the Rand Index.

Table 5 illustrates the difference among the cluster members of Fuzzy C-Means and QS, and Figure 4 provides a visual representation corresponding to the findings presented in Table 5. The Cluster Differentiation (CD) row in Table 5 enumerates the difference between the cluster members of Fuzzy C-Means and QS clusters. For example, the first cluster has a CD of 15, implying that there are 15 universities in the first cluster of QS that are not present in the first Fuzzy C-Means cluster. In Figure 4, the university nodes are delineated by two colors. The left color designates the cluster to which a university belongs according to the results of Fuzzy C-Means, while the right color designates the cluster a university is expected to be a part of based on the clustering of QS. The complete/detailed list of clusters for all clustering algorithms can be found in Table A1.

Table 5. Differentiation between cluster members of Fuzzy C-Means and QS for seven clusters without weights.

Cluster	1	2	3	4	5	6	7
CD	15	130	133	114	56	77	127

Figure 4. Fuzzy C-Means for seven clusters without weights. The color of the sixth cluster in Figure 3 has been altered from its original light green to pink to enhance visual interpretability.

In terms of university distribution within the clusters, the K-Means algorithm and the Agglomerative method exhibit considerable similarity. The outcomes of one align closely with the results of the other, and their Rand Index demonstrates a relatively high level of agreement compared to other algorithms. Fuzzy C-Means shares similarities with K-Means and Agglomerative methods, but upon closer examination through the graph and size table, it becomes evident that Clusters 5 and 6 contain significantly fewer universities than their counterparts in the other two methods. Interestingly, the number of universities in the final cluster of Fuzzy C-Means exceeds those in K-Means and Agglomerative by a considerable margin. GMM stands out distinctly from all other algorithms, characterized by an exceptionally large size of the first cluster, and notably, the smallest size of the last cluster in comparison to corresponding clusters in the other algorithms. Furthermore, GMM reveals that a significant number of universities from the higher mean clusters are intermixed with universities from clusters of lower means to a degree not observed in the other clusters.

To assess the significance of each indicator to the clustering process, we employed the following steps:

• We aggregated the results of K-Means, GMM, Agglomerative, and Fuzzy C-Means to determine the cluster each university belongs to according to the prevailing consensus among the clustering methods. This consensus is determined by identifying the cluster assignment that is most frequently agreed upon by the different clustering algorithms. In other words, we identify the cluster that is most commonly assigned to a university across the various clustering methods.
• We computed the Spearman correlation coefficient between each indicator and the cluster assignment of universities.

Table 6 shows the strength and nature of the correlation between an indicator and a university’s cluster. The nature of the correlation is negative, which indicates a reverse relationship between cluster assignments and indicator values. Some universities have high scores across the nine indicators and are assigned to clusters, such as Cluster 1, characterized by high mean values. Conversely, when universities are assigned to clusters with lower mean values, their individual indicator values tend to decrease. This negative correlation between cluster assignment and indicator values suggests that, as universities are grouped into clusters with lower mean values, such as Cluster 6, their performance, as measured by the indicators, diminishes. Regarding the strength of the correlation, certain indicators, such as focus, academic and employer reputation, citations per faculty, and international faculty and student ratios, showcase a strong correlation with cluster assignments. These indicators likely experience a considerable decrease when universities are assigned to clusters with lower mean values.

Table 6. Correlation of indicators and university cluster assignments for seven clusters without weights.

	Size	Focus	Age	ar	er	fsr	cpf	ifr	isr
Cluster	−0.14	−0.45	−0.2	−0.62	−0.52	−0.37	−0.54	−0.66	−0.57

4.2. Clustering without Weights for 24 Clusters

The clustering results for 24 clusters and nine indicators without weights are shown in Figure 5 and Figure 6. According to Table 7, the Rand Index between every pair of algorithms has significantly increased and the results are now closer to QS. As the number of clusters increases, there are more possible ways to assign universities to clusters. Consequently, the probability of random agreement between two clustering results also increases. Since the Rand Index considers both the number of agreements and disagreements, it tends to increase with the number of clusters.

Figure 5. Visualization for 24 clusters without weights. (a) K-Means, (b) GMM, (c) Agglomerative, and (d) Fuzzy C-Means.

Figure 6. Agglomerative for 24 clusters without weights.

Table 7. Rand Index for 24 clusters without weights.

	KM	GMM	Agg	Fuzzy	QS
KM	1
GMM	0.895197	1
Agg	0.939602	0.897548	1
Fuzzy	0.898327	0.848678	0.885155	1
QS	0.90274	0.872009	0.90325	0.832328	1

Moreover, Table 8 presents the cluster sizes, while Table 9 lists the differentiation between cluster members of Agglomerative and QS clusters for 24 clusters without weights.

Table 8. Cluster sizes for 24 clusters without weights.

	1	2	3	4	5	6	7	8	9	10	11	12
KM	28	34	17	21	24	29	70	30	41	62	24	47
	13	14	15	16	17	18	19	20	21	22	23	24
	42	37	79	60	31	154	37	111	70	100	91	69
GMM	11	8	23	19	196	22	11	11	20	132	37	59
	13	14	15	16	17	18	19	20	21	22	23	24
	33	24	165	80	28	42	28	91	73	69	49	77
Agg	24	19	56	24	48	65	36	30	27	22	44	37
	13	14	15	16	17	18	19	20	21	22	23	24
	33	32	47	57	95	88	42	31	125	120	132	74
Fuzzy	42	62	49	16	3	55	92	34	109	67	4	130
	13	14	15	16	17	18	19	20	21	22	23	24
	1	3	2	2	1	2	258	1	1	2	121	251

Table 9. Differentiation between cluster members of Agglomerative and QS clusters for 24 clusters without weights.

Cluster	1	2	3	4	5	6	7	8	9	10	11	12
CD	8	12	33	22	40	49	33	26	26	21	41	37
	13	14	15	16	17	18	19	20	21	22	23	24
	31	30	43	54	84	75	37	31	106	97	100	65

4.3. Clustering with Weights for Seven Clusters

The clustering results for seven clusters and nine indicators multiplied with weights are shown in Figure 7. The academic reputation indicator carries the highest weight as it exerts the most substantial correlation on the other indicators. In the university categorization process, the size, age, and focus indicators have each been assigned a weight of 0.1. This assignment reflects the fact that these indicators were not considered by the QS in their university rankings. The initial weight values for the faculty-to-student ratio (fsr) and citations per faculty (cpf) indicators were set at half the weight of the academic reputation indicator. By assigning a weight of 0.125, we have maintained this initial proportion. To avoid bias against universities that may not admit or have the legal capacity to accept foreign students or faculty, the international student and faculty ratio indicators were assigned smaller weight values.

Figure 7. Visualization for seven clusters and nine indicators multiplied with weights. (a) K-Means, (b) GMM, (c) Agglomerative, and (d) Fuzzy C-Means.

The Rand Index (Table 10) between the weighted clustering, utilizing nine indicators multiplied by weights, and the unweighted clustering shows a relatively modest change. Specifically, we observe a slight increase in the similarity between the Fuzzy C-Means algorithm and Agglomerative, which results in an improvement in the similarity between Agglomerative and QS. K-Means exhibits a higher similarity with QS.

Table 10. Rand Index for seven clusters with weights [27].

	KM	GMM	Agg	Fuzzy	QS
KM	1
GMM	0.75669	1
Agg	0.843652	0.721143	1
Fuzzy	0.823669	0.733421	0.774022	1
QS	0.783811	0.769252	0.739359	0.7783	1

In terms of the configuration and size of the clusters (Table 11), the graphs reveal that clusters generated by the K-Means, Agglomerative, and Fuzzy C-Means algorithms are more distinctly separated in the weighted approach, making it easier to discern the distribution of universities. However, this improvement does not extend to the GMM algorithm, in which clusters remain mixed. The differentiation between cluster members of K-Means and QS clusters for seven clusters with weights is presented in Table 12 and visualized in Figure 8. The indicators and their corresponding weights are shown in Table 13.

Table 11. Cluster sizes for seven clusters with weights.

	1	2	3	4	5	6	7
KM	85	152	162	166	312	202	229
GMM	30	350	136	189	162	219	222
Agg	77	138	29	117	211	422	314
Fuzzy	78	138	236	193	1	242	420

Table 12. Differentiation between cluster members of K-Means and QS clusters for seven clusters with weights.

Cluster	1	2	3	4	5	6	7
CD	9	53	98	107	216	144	171

Figure 8. K-Means for seven clusters with weights.

Table 13. Weights for the nine indicators. Reprinted with permission from Ref. [27].

Size	Age	Focus	ar	er	fsr	cpf	ifr	isr
0.1	0.1	0.1	0.25	0.1	0.125	0.125	0.05	0.05

Table 14 shows the correlation of the indicators with the cluster assignments using the weights specified in Table 13. The correlations of the size and age indicators have notably increased compared to their correlations in Table 6, when weights were not considered. Additionally, the correlations of the academic reputation, employer reputation, and faculty/student ratio indicators have shown an increase. This suggests that, as the cluster assignment increases, the indicator values now decrease to a greater extent than before, when weights were not applied. However, the correlation of the citations per faculty indicator remains unchanged, while the correlation of the international faculty and student ratio indicators has decreased in relation to the cluster assignment.

Table 14. Correlation of indicators and university cluster assignments for seven clusters with weights.

	Size	Focus	Age	ar	er	fsr	cpf	ifr	isr
Cluster	−0.27	−0.45	−0.45	−0.75	−0.59	−0.44	−0.54	−0.34	−0.35

4.4. Clustering with Weights for 24 Clusters

In the case of clustering with 24 clusters in Figure 9, there are no noteworthy observations regarding the K-Means and Agglomerative methods. In the case of Fuzzy C-Means, it is notable that results are provided for only 20 clusters (Table 15). This is due to the presence of four clusters with zero universities. This is not surprising, given that there are already two clusters that have only one university. The specific identity of these clusters with zero universities does not carry particular significance, as universities are generally assigned to a cluster before the calculation of cluster means. The calculation of the mean value of each cluster is the key determinant of a cluster’s significance, and, in this context, clusters with zero universities are excluded from that process.

Figure 9. Visualization for 24 clusters with weights. (a) K-Means, (b) GMM, (c) Agglomerative, and (d) Fuzzy C-Means.

Table 15. Cluster sizes for 24 clusters with weights.

	1	2	3	4	5	6	7	8	9	10	11	12
KM	40	38	28	48	23	23	44	53	35	41	33	56
	13	14	15	16	17	18	19	20	21	22	23	24
	34	68	80	59	53	38	133	81	96	38	94	72
GMM	17	8	8	10	140	34	136	19	18	43	31	16
	13	14	15	16	17	18	19	20	21	22	23	24
	101	25	39	70	147	27	114	35	60	40	143	27
Agg	38	14	25	44	7	38	56	22	57	35	42	60
	13	14	15	16	17	18	19	20	21	22	23	24
	82	39	58	131	39	64	48	56	60	138	68	87
Fuzzy	28	27	24	39	43	27	38	82	121	98	92	1
	13	14	15	16	17	18	19	20	21	22	23	24
	76	15	12	23	262	1	2	297	0	0	0	0

The differentiation between cluster members of K-Means and QS clusters for 24 clusters with weights is presented in Table 16 and visualized in Figure 10. Finally, according to Table 17, K-Means demonstrates a higher similarity with QS for 24 clusters with weights.

Table 16. Differentiation between cluster members of K-Means and QS clusters for 24 clusters with weights.

Cluster	1	2	3	4	5	6	7	8	9	10	11	12
CD	4	18	21	30	19	22	32	38	31	33	28	45
	13	14	15	16	17	18	19	20	21	22	23	24
	31	64	66	51	47	37	119	72	69	38	67	63

Figure 10. K-Means for 24 clusters with weights.

Table 17. Rand Index for 24 clusters with weights.

	KM	GMM	Agg	Fuzzy	QS
KM	1
GMM	0.904505	1
Agg	0.938935	0.907833	1
Fuzzy	0.879995	0.847311	0.873114	1
QS	0.916053	0.882512	0.908657	0.845611	1

4.5. Discussion

Among the methods tested, Fuzzy C-Means emerged as the most promising method for seven clusters without weights, exhibiting the highest Rand Index (RI). We decided to opt for seven clusters over 24 clusters due to interpretability considerations, as fewer clusters are easier to interpret. Additionally, clustering methods with weights were not chosen as promising methods due to their arbitrary nature. Although the QS ranking system has established predetermined weights for its indicators, there are numerous ways to adjust these weights while maintaining their original ratio. This highlights the importance of selecting appropriate clustering methods and cluster numbers based on both performance metrics and interpretability requirements.

This study’s primary insight lies in revealing the positioning of universities into clusters, as explicitly outlined in Table A1. Interestingly, despite being expected to reside within the same cluster according to the QS ranking, some universities are instead placed in a distant cluster, characterized by a significantly different cluster mean. This understanding underscores that these universities exhibit greater similarity to the universities within their current cluster than to those they were expected to be grouped with according to the QS ranking. This information can aid universities in assessing their performance by offering a different perspective on their rankings.

While the current study has provided valuable insights into clustering universities, it is important to acknowledge its limitations. A significant limitation lies in the use of only nine indicators. With a broader range of indicators, we could obtain more accurate and comprehensive data results. Another limitation arises from the time-dependent nature of university rankings. QS publishes university rankings annually, potentially assigning different ranks to individual universities, and different weights to the indicators each year. Consequently, the clustering process must be conducted every year. Finally, a limitation is related to the visualization of the data and, more precisely, the case where different universities seem to overlap across different clusters. This is because of the transition from nine to two dimensions.

5. Conclusions

Rankings offer a clear and straightforward way to compare universities, while clustering provides a more nuanced understanding of the diversity and similarities within the higher education landscape. Instead of a strict numerical ranking, clustering provides a more comprehensive view by highlighting similarities and differences between groups of universities, allowing for better comparisons and assessments. Clustering analysis introduces a dynamic aspect to rankings, highlighting that institutions can move between clusters based on their performance over time. Moreover, grouping universities may impact their reputation positively or negatively, depending on the cluster they belong to. Institutions in higher-ranked clusters may experience an improvement in their global reputation, attracting more students, faculty, and research collaborations.

In this article, we conducted a thorough analysis of university ranking data and concluded that clustering methods can be applied to group institutions with similar characteristics and provide a ranking between different groups. The latter is achieved by ordering clusters based on the performance of institutions within them. When examining the distribution of institutions within each cluster, discernible differences emerged among clusters generated through different algorithms, varied cluster numbers, or the inclusion of adjusted weights, as opposed to those formed without incorporating weights. Moreover, a comprehensive analysis of the features utilized in the clustering analysis revealed the degree of correlation and importance of each feature in assessing an institution’s performance. Overall, creating a hierarchical ranking that ranks universities from best to worst or relying on a narrow set of indicators to assess an institution’s performance may restrict the richness of results. Alternatively, leveraging clustering methods allows us to overcome these limitations and achieve a more nuanced understanding.

Table A1 illustrates the distribution of universities across clusters based on their profile similarities. It demonstrates that our method groups institutions not by their ordinal rankings, but by their shared attributes and challenges, enhancing the comprehension of what constitutes institutional excellence beyond mere numerical ranking. The consistency in cluster membership, as shown in Table A1, provides empirical evidence that our clustering approach enables a deeper and more stable understanding of universities’ characteristics, aligning with our goal of facilitating peer learning among similarly profiled institutions.

Opportunities for future research abound in refining and expanding upon the current study’s findings. One avenue for exploration involves further refinement of distance functions used in assessing university similarity or proximity, with a focus on tailoring these functions to better capture nuances in university performance. Moreover, future research could delve into the exploration of additional indicators and weights influencing university clustering, such as geographical location, academic disciplines, or institutional characteristics. To enrich the clustering process, we might consider incorporating supplementary indicators from alternative sources or organizations. By integrating these elements into the clustering framework, researchers can gain deeper insights into the complex dynamics of university groupings and their implications for various stakeholders in the higher education landscape. A more comprehensive analysis could extend beyond our clustering process to delve into the question of which indicators and their corresponding values lead certain universities to be grouped into clusters with means different from those anticipated based on QS rankings. Moving forward, we intend to consider the implications of the Variance Inflation Factor (VIF) more thoroughly in future work, exploring methods to integrate this important aspect of data pre-processing. This will enable us to refine our model further and enhance the interpretability and validity of our findings.