Hierarchical Cluster Analysis (HCA)

Suppose a local government runs 12 libraries. Some libraries lend a lot of materials, some have high participation in children's programs, and others see heavy use of digital services. How could you group the libraries into a few operating types based on these differences?

You could use cluster analysis, which groups libraries with similar average characteristics. But you may not want to fix the number of clusters at three or four from the start. If you want to see the order in which similar libraries merge and then choose a suitable level of division, hierarchical cluster analysis (HCA) is useful.

HCA builds a stepwise cluster structure using the similarity or distance between observations. Results are usually shown as a dendrogram, and the height at which branches merge lets you see the distance or merging criterion between clusters.

In this article, we use a single case to look at distance measurement, linkage methods, standardization, dendrograms, choosing the number of clusters, cluster profiles and cautions for the analysis.

In practice, agglomerative clustering, which starts with each observation as its own cluster and merges the closest ones first, is most common.

For example, if libraries A and B are the most similar, they are grouped first. Then, if C is similar to the AB cluster, an ABC cluster is formed, and this process repeats until all libraries are merged into one cluster. The result is not a classification table fixed at one number of clusters but a structure you can cut at various heights to examine several cluster solutions.

Conversely, divisive clustering starts with everything in one cluster and splits it step by step. Both approaches belong to hierarchical cluster analysis, but their specific algorithms and computational costs differ.

Agglomerative HCA, by contrast, first builds the order in which observations merge, so you can examine several numbers of clusters in a single dendrogram.

You cannot generalize that one method is better. Choose based on data size, cluster shape, the meaning of distance and explainability.

The goal of the analysis is not simply to rank operating performance but to group libraries with similar service characteristics.

Larger libraries can show higher counts on every measure, so in a real analysis you should also consider converting to indicators per population or per registered member, or accounting for a size variable separately.

Here, to compare differences in operating profiles, we assume the four variables are standardized and then Euclidean distance and Ward linkage are applied.

The figures below are examples for understanding the concepts of the analysis.

For example, if annual loans are in the tens of thousands and the share of children's programs ranges from 0 to 100%, differences in loans could determine almost all of the Euclidean distance on the raw data.

The standard way to reduce this problem is Z-score standardization.

Here x_ij is library i's value on variable j, and x̄_j and s_j are the mean and standard deviation of variable j. Standardized values make different units comparable.

However, standardizing does not automatically make all variables equally important. You may give weights to variables that matter for the research purpose, and if there are many extreme values you can consider robust scaling methods.

Also, if variables overlap heavily, the same characteristic can be counted repeatedly in the distance calculation. So it is a good idea to look at the variable definitions and their correlations before standardizing.

Depending on the purpose of the analysis, you can use various measures such as Euclidean distance, Manhattan distance or correlation-based distance.

If the standardized values of two libraries are A = (1, 0) and B = (0, 1), the Euclidean distance is √2 ≈ 1.414. These two coordinates are a simplified calculation example of the libraries' full data.

Because the order of the nearest libraries can differ with each distance measure, you should state the reason for your choice. In particular, Ward linkage is generally based on the increase in squared Euclidean error, so it should not be combined with an arbitrary distance measure and interpreted the same way.

The rule that defines the distance between clusters is called the linkage method.

Ward linkage merges based on how much within-cluster variation increases when two clusters are combined. So it can be useful for forming clusters of similar values. However, if clusters are very heterogeneous in size or shape, or there are many extreme values, the expected result may not appear.

In this example, we use the four standardized operating indicators and Ward linkage.

The horizontal axis shows the individual libraries or clusters, and the vertical axis shows the level at which clusters merge.

For example, if two libraries merge at a low height, their operating characteristics can be considered very similar under that linkage criterion. If two different large clusters merge at a high level, on the other hand, the difference between those clusters is relatively large.

Note that the vertical axis of a dendrogram does not always mean the original Euclidean distance itself. With Ward linkage in particular, depending on how the software defines it, the merge height shown may be related to the increase in the sum of squares. So check the axis label and the analysis settings together.

Cutting the dendrogram just below a merge where the height jumps sharply gives clusters that are distinct from each other. But a single visible gap does not objectively settle the number of clusters.

You can consider together the changes in merge height on the dendrogram, the silhouette score, the interpretability of the clusters and the minimum sample size per cluster. The silhouette score summarizes how close an observation is to its own cluster and how far it is from other clusters.

With these results, you can choose three clusters. However, since there are only 12 libraries, you should not rely too much on small differences in the silhouette score. How much the results change with the distance measure or standardization method also matters.

The values below are cluster means in the original units to aid understanding, and are distinct from the standardized scores used in the analysis.

Cluster 1 has relatively high loan counts and can be interpreted as lending-focused; Cluster 2 has high program participation and a high share of children's programs and can be interpreted as community-program-focused; and Cluster 3 has high digital service use and can be interpreted as digital-service-focused.

However, these names are not objective types assigned automatically by the algorithm. They are the researcher's interpretation after reviewing actual operating indicators and policy goals. Also, the fact that cluster means differ does not by itself let you conclude that the differences are statistically significant in the population or caused by operating strategies.

For example, if just one library has extremely high loans or digital service use, that library may merge late with the other clusters in the dendrogram.

This may signal that a genuinely distinctive type of library exists, but it could also be due to a unit entry error or differences in how data were tallied. So check extreme and missing values before the analysis.

For example, you can run sensitivity analyses like these: change the standardization method, apply average linkage in addition to Ward, or change which variables are included, and then check how similar the cluster memberships are. You can also exclude some observations or rebuild the clusters on bootstrap samples and examine cluster stability.

When comparing cluster results, you can use indices such as the Adjusted Rand Index (ARI). With very small samples, however, this index can also be highly variable, so along with the number it is good to check directly which libraries changed membership.

If the goal is to propose operationally meaningful types, it also helps to review the shared characteristics within each cluster and the differences from other clusters together with library staff.

The distance measure decides what counts as similar, and the linkage method decides by what criterion clusters are merged. In particular, do not apply an arbitrary non-Euclidean distance with Ward linkage and interpret it as the same variance-minimizing procedure.

Check the units of variables and their overlap.

If you skip standardization, variables with large scales can dominate the distance, and including several variables carrying similar information can overweight a particular characteristic.

Do not decide the number of clusters from the dendrogram's shape alone.

Check not only merge heights but also silhouette scores, cluster sizes, interpretability and stability under other analysis settings.

Do not confuse closeness in the dendrogram with causation.

The fact that two libraries have similar operating indicators does not mean one library influenced the other.

Do not use cluster names as if they were objective diagnoses.

Cluster names are interpretations to describe observed characteristics, not evaluation scores that directly indicate the quality or relative performance of the libraries.

If possible, it is good to provide both the dendrogram and a table of cluster profiles.

In real research, you should also report the distribution of variables in each cluster and the results of stability checks.

Unlike K-Means, which fixes one number of clusters from the start, it lets you examine several levels of division in the dendrogram.

But the results can change with the distance measure and linkage method, standardization of variables, outliers and sample composition. So rather than settling the meaning of clusters from the dendrogram's shape alone, evaluate each cluster's characteristics and stability together.