Hierarchical Cluster Analysis
Hierarchical Cluster Analysis (HCA) widely used in Agricultural research … Read more …
Hierarchical Cluster Analysis (HCA) is a statistical method widely used in agricultural and social-science research to group similar observations into clusters based on their characteristics, building a tree-like structure (a dendrogram) to show how these groups form. In RAISINS you can perform HCA, explore it through a heatmap, run Hierarchical Clustering on Principal Components (HCPC), compare dendrograms with a tanglegram, and chat with your results using RA-One, all without writing a single line of code. This tutorial walks through the whole workflow using one worked example, from preparing your data to interpreting every table and plot the module produces.
1 What is Hierarchical Cluster Analysis?
Imagine you visit a field with 30 genotypes of a crop. At first glance every genotype looks different, but some are clearly more alike than others in yield, height, or seed characters. Instead of visually inspecting every pair, you measure a set of traits and organize genotypes into groups with similar trait patterns. Hierarchical Cluster Analysis (HCA) does exactly this: it starts by treating every observation as its own cluster, then repeatedly merges the two most similar clusters together, one merge at a time, until everything has been grouped into a single cluster. The order in which groups merge, and the similarity level at each merge, is shown in a tree-like diagram known as a dendrogram
Hierarchical agglomerative cluster analysis groups similar genotypes, treatments, or respondents based on their measured characters, and shows the grouping as a dendrogram. It is exploratory: nothing forces a “correct” number of clusters on you, you decide where to cut the tree, guided by the optimal-cluster methods covered in Section 5.
Unlike many statistical tests, HCA has no null hypothesis and does not produce a significance test. It always creates a dendrogram, and once you choose a cut height, the observations are divided into clusters. Since clusters are always formed, you should evaluate whether they make practical sense by examining the dendrogram-fit diagnostics (Section Section 12) and the optimal number of clusters plots (Section Section 5), instead of relying on a significance threshold.
A little history: Sokal, Sneath, and the birth of numerical taxonomy
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Agglomerative hierarchical clustering grew out of numerical taxonomy, a movement led in the late 1950s by biologists Robert Sokal and Peter Sneath, who wanted to classify organisms using measured characters and explicit, repeatable rules rather than subjective expert judgement. The core linkage methods used today were formalised soon after: Joe H. Ward Jr. published the variance-minimising method that bears his name in 1963, and average and complete linkage were already in circulation from the same taxonomic literature. Because the method needs no assumption about the number of groups in advance, and the dendrogram it produces is easy to read even without statistical training, it remains one of the most widely used exploratory tools across biology, agriculture, and the social sciences.
2 Choosing how to scale your variables
HCA measures similarity using distance, and distance is dominated by whichever variable happens to have the largest numbers. If Assault is measured in the hundreds of arrests per 100,000 residents while UrbanPop is a percentage between 0 and 100, the raw distance calculation will be almost entirely driven by assault, drowning out the contribution of urban population. To keep every variable contributing fairly, RAISINS lets you scale the clustering variables before computing distances.
| Method | What it does | When to use it |
|---|---|---|
| Z-score (default) | Centres each variable at mean 0 and rescales to standard deviation 1 | The safe general-purpose default for most datasets |
| Center | Subtracts the mean only, keeps the original spread | When the original units and relative variability should be preserved |
| Min-Max | Rescales every variable to the 0–1 range | When you want all variables to have exactly the same range |
| Unit length | Rescales each variable to length 1 | When only the pattern across variables matters, not magnitude |
| Robust | Centres on the median and scales by the IQR | When your data contain outliers that would distort a mean-based scaling |
| None | Uses the raw values as measured | Only when all variables already share the same meaningful unit and scale |
Leave the scaling method at the default, Z-score. Switch to Robust if a quick look at your data shows one or two extreme values, and choose None only if every clustering variable you selected already shares the same meaningful unit.
See the exact formula and a worked example for each scaling method
The table above summarises what each method does. To see exactly how each one is calculated, take a single clustering variable measured on five genotypes: x = 10, 12, 14, 18, 26. Its mean is \(\mu = 16\), sample standard deviation \(\sigma = 6.3246\), median \(= 14\), \(Q_1 = 12\), \(Q_3 = 18\) (so \(IQR = 6\)), minimum \(= 10\), maximum \(= 26\), and Euclidean norm \(\lVert x \rVert_2 = \sqrt{1440} = 37.947\). Values below are rounded to 3 decimals.
None: \(x' = x\) → 10, 12, 14, 18, 26 (unchanged)
Center: \(x' = x - \mu\) → −6.000, −4.000, −2.000, 2.000, 10.000
Z-score (default): \(x' = \dfrac{x - \mu}{\sigma}\) → −0.949, −0.632, −0.316, 0.316, 1.581
Min-Max: \(x' = \dfrac{x - \min(x)}{\max(x) - \min(x)}\) → 0.000, 0.125, 0.250, 0.500, 1.000
Unit length: \(x' = \dfrac{x}{\lVert x \rVert_2}\) → 0.264, 0.316, 0.369, 0.474, 0.685
Robust: \(x' = \dfrac{x - \text{median}(x)}{IQR(x)}\) → −0.667, −0.333, 0.000, 0.667, 2.000
The most extreme observation (26) scales to 1.581 under z-score but only 2.000 under robust scaling, relative to the tighter middle spread of the data, because robust scaling never uses the mean or standard deviation. This is why robust scaling is the recommended choice when your clustering variables contain outliers.
3 Distance Metrics
Once your variables are scaled, RAISINS needs a rule for turning a row of numbers into a single “how far apart are these two genotypes” value. Five distance metrics are available: Euclidean, Manhattan, Maximum, Canberra, and Minkowski.
Let's understand the different distance metrics
Formula
\[\sqrt{\sum_i (x_i - y_i)^2}\]
Explanation
The straight-line distance between two observations, calculated using the Pythagorean theorem. It is the most familiar and widely used distance metric.
Formula
\[\sum_i |x_i - y_i|\]
Explanation
Also known as City Block Distance, it measures the total distance travelled by summing the absolute differences across all traits.
Formula
\[\max_i |x_i - y_i|\]
Explanation
Also called Chebyshev Distance, it considers only the largest difference among all measured traits.
Formula
\[\sum_i \frac{|x_i-y_i|}{|x_i|+|y_i|}\]
Explanation
A weighted distance measure that gives greater importance to differences when the values are small.
Formula
\[\left(\sum_i |x_i-y_i|^p\right)^{1/p}\]
Explanation
A generalised distance metric controlled by the parameter p.
- p = 1 → Manhattan Distance
- p = 2 → Euclidean Distance
- Higher p gives more weight to larger differences.
Real-world analogy
Think of Minkowski distance as a flexible distance measure that can behave like different metrics depending on the value of p.
4 Linkage Methods
Distance tells you how far apart two individual genotypes are. Linkage tells you how far apart two whole clusters are, once each already contains more than one genotype, and it is this rule that decides which pair of clusters merges next as the dendrogram is built. RAISINS offers seven linkage methods: single, complete (the default), average, ward.D2, mcquitty, median, and centroid.
Let's understand the different linkage methods
Formula
\[L(R,S) = \min_{i \in R,\; j \in S} D(i,j)\]
Explanation
For two clusters R and S, single linkage returns the minimum distance between any pair of points, one from each cluster. This method creates long, chain-like clusters because it is sensitive to outliers and can connect clusters based on a very small number of close points.
Where
- D(i, j): distance function between points i and j
Formula
\[L(R,S) = \max_{i \in R,\; j \in S} D(i,j)\]
Explanation
For two clusters R and S, complete linkage returns the maximum distance between any pair of points. It tends to create compact and spherical clusters because it is more sensitive to outliers and tries to make sure the clusters are not too far apart.
Where
- D(i, j): distance function between points i and j
Formula
\[L(R,S) = \frac{1}{n_R \times n_S} \sum_{i=1}^{n_R} \sum_{j=1}^{n_S} D(i,j)\]
Explanation
Average linkage returns the average distance between all pairs of points from two clusters. This method maintains a balance between single and complete linkage by considering all pairs of points, not just the closest or farthest point. It usually results in clusters that are moderately compact.
Where
- nR: number of data points in R
- nS: number of data points in S
Formula
\[L(R,S) = \frac{n_R \times n_S}{n_R + n_S} \sum_{i=1}^{n_R} \sum_{j=1}^{n_S} D(i,j)\]
Explanation
Ward’s linkage calculates the distance between two clusters by looking at the total spread, or variance increase, when the clusters are combined. This method creates compact, well-separated clusters by making sure that data within each cluster stays as similar as possible.
Where
- nR and nS: sizes of clusters R and S
- D(i, j): distance between points i ∈ R and j ∈ S
A variant of average linkage that uses a simple average of distances weighted by cluster size, giving a compromise that accounts for clusters of different sizes as they merge.
Formula
\[L(R,S) = \underset{i \in R,\; j \in S}{\text{median}}\; D(i,j)\]
Explanation
For two clusters R and S, median linkage returns the median distance between a variable in one cluster and a variable in the other cluster, calculated from the pairwise distance matrix between the two clusters. It is more robust to skewed data than average linkage, but can produce non-monotonic merges (reversals) in the dendrogram.
Where
- D(i, j): distance function between points i and j
Formula
\[L(R,S) = D(\bar{R}, \bar{S})\]
Explanation
Centroid linkage calculates the distance between two clusters based on the distance between their central points, i.e. the average of all points in the cluster. This method works well when clusters are round or evenly shaped, but it may not be the best choice for irregularly shaped clusters.
Where
- R̄ and S̄: the centroids (mean points) of clusters R and S
- D(R̄, S̄): the distance between the centroids of clusters R and S
5 Finding the optimal number of clusters (k)
The dendrogram alone does not tell you where to “cut” the tree. In RAISINS, the default optimal number of clusters is determined using the Elbow Method, which identifies the point at which increasing the number of clusters yields diminishing improvements in within-cluster homogeneity. However, the choice of the final number of clusters should ultimately be guided by the researcher’s domain knowledge, study objectives, and interpretability of results. Users are therefore encouraged to explore alternative validation approaches, such as the Silhouette Method and the Gap Statistic, available together in the Optimal Clusters tab (Section 14), each downloadable and each drawn against a maximum number of clusters that you control with a slider.
5.1 The Elbow Method
Plots the within-cluster sum of squares (WSS) against increasing k. WSS always falls as k grows, but the rate of improvement slows sharply once you pass the natural number of groups, showing up as a bend, or elbow, in the curve.
In RAISINS: The bend is detected automatically and pre-fills the Number of clusters box with it, always free to override.
5.2 The Silhouette Method
Measures, for each observation, how much closer it sits to its own cluster than to the next-nearest one, averaged into a score from −1 to 1, higher meaning tighter, more separated clusters.
In RAISINS: The peak is identified automatically and presented as an alternative recommended number of clusters.
5.3 The Gap Statistic
Compares your data’s clustering tightness against what you would get from data with no real structure at all (a reference distribution). A larger “gap” at a given k means your data cluster far more convincingly than random noise would.
In RAISINS: The recommended k from the Gap Statistic is displayed alongside the other two to help you decide.
RAISINS pre-fills the Number of clusters box using the elbow method as a fast default, but the silhouette and gap-statistic plots in Section 14 are there so you can cross-check that choice. Disagreement between the three is common and expected; let your subject-matter knowledge about how many groups you actually expect make the final call.
6 Assumptions and things to check before clustering
| Requirement | What it means | What if it fails? |
|---|---|---|
| Numeric variables | Every clustering variable must be a proper number, not text or a category | Recode categorical variables, or exclude them from the clustering variables |
| No missing values | Distances cannot be computed to a missing value | Clean or impute the dataset before uploading |
| Comparable scales | Variables in very different units can dominate the distance calculation | RAISINS handles this through the scaling methods in Section 2 |
| Unique observation labels | Each row needs its own identifier so results can be traced back to it | Remove duplicate labels before uploading |
RAISINS validates the label and variable columns automatically once selected, before it lets you run the analysis. If the dendrogram or plots in Section 13 suggest genuinely round, evenly-sized groups rather than a chain-like or nested structure, RAISINS’s K-Means Cluster Analysis module is a companion tool built for exactly that case (Section 21).
7 Getting to the module
Visit RAISINS and navigate to Cluster Analysis under Data Analysis, as shown in Figure 8.
7.1 Computational Provenance & Reproducibility Record
CPRR (Computational Provenance & Reproducibility Record) provides a transparent, comprehensive record of the computational workflow behind this module, the R version and exact version of every package used, the specific function behind each reported result, every default parameter and decision rule applied, and fully runnable R code that reproduces each analytical step so you can independently verify the results. It carries its own DOI.
To cite the platform itself in a paper, thesis, or report, use the RAISINS citation, available in APA, Harvard, and BibTeX formats at www.raisins.live/citation.html. That is the primary reference, and for most manuscripts it is all you need.
The CPRR for Cluster Analysis is at www.raisins.live/module_record/cluster.html.
8 Preview mode and Quick Tour
Before subscribing, you can explore the whole module using Preview mode, available from the Welcome page. Preview mode loads built-in datasets so you can try clustering, all eight dendrogram plot types, the Heatmap, HCPC, Cluster Comparisons without uploading your own data. First-time users are also offered a Quick Tour, an interactive walkthrough that highlights each control and explains what it does; you can retake it at any time from the Quick Tour tab.
9 A working example
We will explain HCA using one worked example that we will carry through the rest of this tutorial. Consider the classic USArrests dataset: 50 U.S. states described by four variables, Murder, Assault, and Rape (arrests per 100,000 residents), and UrbanPop (percent of the population living in urban areas). The goal is to group the states based on these characters. If you work in agriculture, your goal is the same, grouping genotypes or treatments based on the variables under study.
Two terms you will see throughout this tutorial: Labels are the row identifiers (here, the state names), and Variables are the measured characters used for clustering (here, the four traits above).
10 How to prepare your data
Arranging data for upload is simple: prepare a single-sheet file exactly like Figure 9, with no blank rows above the data and every column named. You have four routes to get there:
- Create your dataset in MS Excel
- Build your dataset directly within the RAISINS app (Create Data)
- Use the Model datasets in RAISINS (Datasets) as a reference
- Create your dataset using the RA-One chat assistant
10.1 Preparing data in MS Excel
Open a new blank sheet in MS Excel, using a single sheet only, and begin entering data from cell A1 with no blank rows above it. The first row must contain the column names. Column 1 holds the treatment or label ID, and there should be no repetition in the label IDs (if you have replications, use the mean values). From Column 2 onward, enter each variable under study as its own numeric column, for example Murder, Assault, UrbanPop, Rape. You can name the columns however you like, but avoid spaces and special characters.
If you have any doubt about saving a file as CSV or the basics of data preparation, see our tutorial on getting started here.
10.2 Prepare using Create Data in RAISINS
Navigate to the Create Data tab, enter the levels of Factor A, Factor B, Factor C, and the number of characters under study, then click Create. A model data-entry template appears, which you can fill in directly or paste from an existing Excel file. Once complete, click Download CSV and upload the file under the Analysis tab.
10.3 Download Model Datasets
If you are unsure of the required format, or want to explore the module before uploading your own data, navigate to the Datasets tab and download the reference CSV there. Use it as a layout guide, or upload it directly to try the analysis end to end.
10.4 Creating a dataset using RA-One chat
RA-One, the built-in chat assistant, can build a properly formatted dataset through a short conversation. Open RA-One from the chat icon or the RA-One tab, tell it how many observations and clustering variables you need, and it generates a dataset in the required format that you can review, download as CSV, and upload under Analysis.
11 The Analysis tab
Figure 12 walks through the whole Analysis tab, from uploading your file to reading the first result, follow the numbered callouts on the image as you read:
- Upload file - click Browse in the sidebar and select your prepared CSV or Excel file; the sidebar shows Upload complete once it has loaded.
- Select Labels - choose your unique label or observation column from the Select the Labels or Observations column picker (
Locationin our worked example). - Select Clustering Variables - choose at least two numeric variables from the Select the Clustering Variables picker (
Murder,Assault,UrbanPop,Rapein our worked example). - Click on Run Analysis! - the red Run Analysis! button starts the clustering.
- Select Linkage Method - once results appear, the control panel on the right lets you change the Linkage method (Section 4) from the default
complete. - Select Distance Method - the Distance dropdown next to it lets you change the distance metric (Section 3) from the default
euclidean. - Select Method - the Scaling dropdown lets you change the scaling method (Section 2) from the default
zscore; Decimal points and Number of clusters sit alongside it, the latter pre-filled with the elbow method’s suggestion from Section 5 but always editable. - Results - the auto-generated Hierarchical Clustering summary and the results tables (Section 12) appear directly below the control panel.
For our worked example, the default complete linkage and euclidean distance on the 50-state USArrests dataset, scaled with z-score, produced a dendrogram whose elbow-method suggestion was two clusters, shown in Figure 13. You are always free to change the number of clusters, the linkage, the distance metric, or the scaling method and re-examine the result.
Step-by-step: how to read a dendrogram
- Start at the bottom - each leaf represents a single observation (here, a state).
- Follow the lines upward - the first merges connect the most similar items; a low connection height means those points were very close on the chosen distance measure.
- Merging clusters - as you go higher, clusters combine with other clusters, each branching a further step of grouping, like building a family tree for your data.
- Branch heights - the height at which two branches merge shows how similar the merged groups were; lower connections mean more similar groups.
- Deciding the number of clusters - “cutting” the tree horizontally at a chosen height divides the data into clusters, one per branch below the cut line. RAISINS’s Number of clusters box does this cut for you.
12 Analysis Results
Directly below the dendrogram, RAISINS writes an automatically generated plain-language summary titled Hierarchical Clustering. For our worked example it reads: clustering was performed with the complete linkage method and euclidean distance metric; the number of clusters was identified through the elbow method, resulting in 2 clusters; the uploaded dataset was initially scaled using the z-score method; and each cluster groups similar observations based on the selected numeric variables, with branches in the dendrogram colour-coded to reflect cluster membership. Beneath it, five results tables appear in the same tab, each with an info button explaining what it shows.
Distance Matrix
The pairwise dissimilarity between every pair of states on the chosen distance metric, computed on the scaled clustering variables (Section 2), the raw material every later merge in the dendrogram is built from. Smaller values mean two states are more alike on the clustering variables, for example Alabama and Georgia sit at just 1.13, while Alabama and California sit at 3.26.
Cluster Means
The average of each clustering variable within each cluster, computed on the original, unscaled units even though the clustering itself runs on scaled data, summarising the typical characteristics of that cluster. In our run, cluster 1 averages Murder = 12.33, Assault = 259.32, UrbanPop = 68.32, Rape = 29.22, while cluster 2 averages Murder = 5.00, Assault = 116.48, UrbanPop = 63.84, Rape = 16.34, a high-crime cluster and a comparatively low-crime cluster.
Intra-cluster Statistics
The average pairwise distance among members of the same cluster, a measure of how compact or homogeneous each cluster is. Lower values mean the states inside that cluster closely resemble one another, here cluster 1 has an intra-cluster distance of 2.09 and cluster 2 of 1.84, so cluster 2 is the slightly tighter of the two.
Inter-cluster Statistics
The average distance between every pair of different clusters, one row per cluster pair, a measure of how well separated the clusters are from each other. Higher values mean the clusters are more distinct, in our run the inter-cluster distance between cluster 1 and cluster 2 is 3.34, comfortably larger than either intra-cluster distance above, a good sign the two-cluster split is meaningful.
Dendrogram fit: Cophenetic Distance and Correlation
Two further tables report the cophenetic distance, the distance implied between every pair of states purely by where they merge in the dendrogram, and the single cophenetic correlation between that implied distance and the original distance matrix above. A cophenetic correlation close to 1 means the dendrogram faithfully represents the real distance structure in your data; a low value is a signal to try a different linkage method (Section 4) before trusting the tree. Our worked example returns a cophenetic correlation of 0.70, a reasonably faithful, if not perfect, summary of the underlying distances.
As with the dendrogram itself, none of these five tables carries a significance test. Use the cophenetic correlation to judge whether the tree is a faithful summary of the distances, and the cluster means and intra-/inter-cluster statistics to describe what the clusters actually contain.
13 Cluster plots
The Plots & Graphs tab offers eight complementary dendrogram styles, all built from the same underlying clustering. Each plot comes with a gear icon at the top-left corner, allowing you to customise titles, fonts, colours, and sizes. You can also download these plots in high-quality PNG (300 dpi), JPEG, TIFF, PDF, and SVG formats for use in reports and publications.
From Figure 16 to Figure 23, you can see the eight dendrogram styles available in the Plots & Graphs tab of the Cluster Analysis module in RAISINS. Each is visually illustrated and accompanied by a clear description below — hover over any plot thumbnail to read it.
Colour Coded Clusters (k = 2) — Branches and tip labels are coloured by cluster assignment without drawing rectangles, an easy first look at which states ended up in which group.
Cluster Dendrogram (k = 2) — The same colour-coded dendrogram, but with a rectangle drawn around each cluster’s branches, making the grouping easier to trace visually for large datasets.
Rectangular Dendrogram (k = 2) — Clusters are enclosed in rectangular boxes drawn on the dendrogram to make the cluster boundaries explicit at a glance.
Vertical Dendrogram (k = 2) — The classic, user-friendly layout with observations along the bottom and merges represented by upward vertical lines.
Horizontal Dendrogram (k = 2) — The same structure rotated 90 degrees, observations run down the vertical axis and branches extend left to right, useful for datasets with many labels.
Base Dendrogram (k = 2) — The traditional base-graphics tree, showing the nested grouping of states from individual leaves up to the full merged tree.
Ward’s Dendrogram (k = 2) — The same tree drawn specifically to highlight Ward’s linkage, which minimises the increase in within-cluster variance at each merge, producing compact, evenly sized groups.
Circular Dendrogram (k = 2) — A radial, space-efficient layout that arranges the branches in concentric circles, giving a 360-degree view of the grouping, useful for datasets with many observations.
The default cluster number is calculated using the elbow method (Section 5). Since cluster analysis is exploratory, you can change Number of clusters at any time, that change is reflected in every one of these eight plots.
14 Choosing k: the Optimal Clusters tab
The Optimal Clusters tab lets you set Select maximum cluster to display with a slider (1 to 50 states, defaulting to 5), then shows the three diagnostic plots from Section 5 on demand, each downloadable in PNG, JPEG, TIFF, PDF, or SVG format.
Read the elbow plot by looking for the bend where the curve stops falling steeply; that is the suggested k. Read the silhouette plot by looking for the highest peak. Read the gap statistic by looking for the smallest k whose gap value is within one standard error of the largest gap observed.
15 Heatmap
A heatmap represents the clustering-variable values as a colour-coded matrix, with rows and columns reordered according to the hierarchical clustering result so that similar states and similar variables sit next to one another. Open the Heatmap tab (Figure 25) after running the analysis; the gear icon lets you customise the colour scheme, cell borders, cell size, value labels, and whether rows and columns are individually clustered.
Blocks of similar colour indicate states or variables with related profiles, while the accompanying row and column dendrograms show their hierarchical relationships. By default the heatmap uses the same scaling, linkage, and distance chosen in the Analysis tab, for our worked example that is z-score scaling with complete linkage and euclidean distance, resulting in the same 2 clusters seen in Section 12. Download the heatmap in high-quality PNG, JPEG, TIFF, PDF, or SVG format for reports or presentations.
16 HCPC: Hierarchical Clustering on Principal Components
HCPC is a hybrid approach combining Principal Component Analysis (PCA) with hierarchical clustering. PCA first summarises the correlated clustering variables into a smaller set of uncorrelated components that retain most of the original variance; hierarchical clustering (Ward linkage, consolidated with a k-means pass) is then performed on those components instead of the raw variables. This reduces noise from correlated variables and can sharpen cluster separation. The results in the HCPC tab are entirely separate from the Analysis Results tab above, they come from a different clustering run on the principal-component space, not the cutree-based clusters from Section 12.
When to use HCPC?
- When you have many continuous, correlated clustering variables and want to reduce dimensionality before clustering.
- When you want an integrated PCA + hierarchical clustering + k-means refinement pipeline instead of a single method.
- When you want richer visualisations, factor maps and chord diagrams, for presenting the grouping.
Difference between HCPC and classical HCA
- HCPC clusters the uncorrelated principal components; classical HCA (Section 12) clusters the raw variables directly.
- HCPC combines PCA, hierarchical clustering, and a k-means consolidation step; classical HCA only performs hierarchical clustering.
- HCPC produces a factor map and chord diagrams in addition to a dendrogram; classical HCA’s main output is the dendrogram and the tables in Section 12.
After uploading your data, selecting your labels and clustering variables, and clicking Run Analysis, open the HCPC tab. Six tables appear first, in order: PCA eigenvalues & variance explained, variable contributions, variable correlations with components, the Clustering Summary (standardized cluster means from the PCA input data), the top variables per cluster, and the cluster quality indicators, followed by five plot buttons in the order HCPC Dendrogram, Factor Map, HCPC 3D Dendrogram, Chord Diagram, and Chord (Means).
Eigen Values PCA
The eigenvalue is the variance captured by each principal component; components with an eigenvalue greater than 1 are typically the meaningful ones. In our worked example, component 1 alone captures 62.01% of the total variance (eigenvalue 2.48), and component 2 adds a further 24.74%, so the first two components together already explain 86.75% of the variance, comp 3 and comp 4 explain the remaining 8.91% and 4.34%.
Cluster-wise Contribution
This table shows how much each clustering variable contributes to each principal-component dimension. In our run, UrbanPop dominates Dim.2 (76.18% contribution) while Murder, Assault, and Rape all load heavily on Dim.1 (28–34% each), confirming that Dim.1 is essentially an overall “crime intensity” axis and Dim.2 an “urbanisation” axis.
Correlation Table
This table reports each clustering variable’s correlation with each component, from −1 to 1. Assault (0.92) and Rape (0.86) correlate most strongly with Dim.1, while UrbanPop (0.87) correlates most strongly with Dim.2, the same pattern as the contributions table above, expressed as a signed correlation rather than a percentage.
Clustering Summary
The standardized (z-score) mean of each clustering variable within each HCPC cluster. Cluster 1 sits below the overall average on every variable (Murder −0.67, Assault −0.68, UrbanPop −0.13, Rape −0.56), while cluster 2 sits above average on all four (Murder 1.00, Assault 1.01, UrbanPop 0.20, Rape 0.85), a similar high-crime versus low-crime split to the classical HCA results in Section 12, though HCPC clusters on the principal components rather than the raw variables.
For each cluster, the variables with the largest absolute v.test are shown, alongside the mean and standard deviation within the cluster versus overall, and a p-value for the variable’s contribution. Larger absolute v.test and smaller p-value mean that variable more strongly characterises that particular cluster. In our run, Assault is the strongest characterising variable for both clusters (v.test of −5.85 for cluster 1, +5.85 for cluster 2), followed closely by Murder and Rape, all with p-values of 0.00.
These four indicators are computed in the PCA-coordinate space, not on the raw or scaled variables directly. Total_SS is the total variability across all observations in that PCA space, representing overall data dispersion. Within_SS is the variation of points around their own cluster’s centroid, a compactness measure. Between_SS is the variation between cluster centroids, how distinct the clusters are. The Between/Total ratio is a separation indicator, higher values mean the clusters are better separated from one another. In our worked example, Total_SS = 200.00, Within_SS = 104.96, Between_SS = 95.04, giving a Between/Total ratio of 0.48, close to half the total variance is explained by the two-cluster split.
From Figure 32 to Figure 36, you can see the five HCPC visualisations available in RAISINS. Each is visually illustrated and accompanied by a clear description below — hover over any plot thumbnail to read it.
HCPC Dendrogram — Hierarchical clustering performed in the principal-component space rather than on the raw variables, showing how observations merge based on their PCA coordinates.
Factor Map — Plots every state in the space of the first two principal components, coloured and enclosed by cluster. Well-separated, non-overlapping regions indicate a clean grouping, while heavy overlap suggests the clusters are not clearly distinct.
HCPC 3D Dendrogram — The dendrogram projected onto the PCA factor map, letting you see the cluster structure and the underlying component space together.
Chord Diagram — Visualises the relationship between your original variables and the HCPC clusters. Wider ribbons mean a variable is more characteristic of that cluster, weighted by its v.test statistic and normalised within each cluster.
Chord (Means) — The same variable-to-cluster relationship as the Chord Diagram, but weighted by cluster mean values instead of v.test.
The Factor Map (Figure 33) plots every state in the space of the first two principal components, coloured and enclosed by cluster, well-separated, non-overlapping regions indicate a clean grouping, while heavy overlap suggests the clusters are not clearly distinct on the variables you selected. The Chord Diagram (Figure 35) visualises the relationship between your original variables and the HCPC clusters. The outer ring shows two types of segments, your dataset’s variables and the clusters, and the ribbons connecting them represent the strength of association: wider ribbons mean a variable is more characteristic of that cluster, weighted by its v.test statistic and normalised within each cluster so you can compare relative importance. In our worked example, Assault and Murder form the widest ribbons into both clusters, consistent with them being the top variables by v.test above. Chord (Means) presents the same relationship, but weighted by cluster mean values instead of v.test.
17 Cluster Comparisons: Tanglegram
Every choice in Section 4 and Section 3 changes the resulting tree, so it helps to compare two runs side by side. The Cluster Comparisons tab lets you pick a Distance measure and Linkage method independently for Method 1 and Method 2 (the defaults compare average linkage against ward.D2, both on euclidean distance), plus a Number of clusters (k) to highlight slider (2 to 20, defaulting to 6), then draws a tanglegram, two dendrograms facing each other with lines connecting the same state on each side.
Above the plot, RAISINS writes an automatic plain-language summary of the comparison. Straight, uncrossed connecting lines mean the two linkage/distance choices grouped the states almost identically; heavy crossing means the two methods disagree about which states belong together. This is quantified by the entanglement score, from 0 (the two dendrograms align perfectly) to 1 (maximal tangling), shown beneath the plot. For the default average vs ward.D2 comparison on USArrests, both at k = 6, the entanglement score is 0.0754, close to 0, so the two linkage methods agree closely on this dataset even though their formulas differ.
If two very different linkage methods (say, single vs. Ward.D2) produce a low-entanglement tanglegram, your grouping is robust to that choice, a reassuring sign. A high entanglement score means the linkage method genuinely matters for your data, and it is worth reading Section 4 again before deciding which one to report.
Below the tanglegram, the Linkage Method Correlation (Cophenetic) heatmap compares all six standard linkage methods (single, complete, average, centroid, median, Ward) at once, on a distance metric you choose, using the correlation between their cophenetic distances. This gives a quick overview of which linkage choices tend to agree with each other, complementing the two-method tanglegram above. On USArrests with euclidean distance, complete and Ward agree almost perfectly (correlation 0.94), while median linkage stands apart from every other method (correlations of just 0.26–0.33), a sign that median linkage is producing a noticeably different tree structure on this dataset.
18 Chat with your data using RA-One
RA-One is the built-in conversational assistant for the Cluster Analysis module, available from the RA-One tab or the chat bubble. You ask questions in plain language and it answers using your own results.
Within a single conversation, RA-One can:
- Interpret your results - explain what the distance matrix, cluster means, intra-/inter-cluster statistics, cophenetic correlation, or the HCPC tables mean for your specific run.
- Generate any of the eight dendrogram plot types from Section 13 on request, colour coded, rectangular, vertical, horizontal, base, Ward’s, circular, or the coloured-rectangle cluster dendrogram, styled and downloaded directly in the chat.
- Draw the tanglegram or the linkage correlation heatmap from Section 17 on request, and refine an existing plot (palette, labels, branch colour, and more) by simply asking for changes.
- Prepare your data - build a correctly formatted dataset template (Section 10.4) or fetch a model dataset (Section 10.3) so you can try the module immediately.
- Answer format and replication questions - how many columns you need, how to lay out labels versus variables, and similar general questions about preparing a file for this module.
Within a single conversation, RA-One can interpret your clustering results, produce any of the eight dendrogram styles or the tanglegram, build a data template, and fetch a model dataset, so a routine HCA session can largely be conducted without leaving the chat window.
19 FAQs
The module includes a dedicated FAQs tab that clarifies common doubts, linkage and distance choices, scaling, reading a dendrogram, and more, with detailed answers and tips for a smooth experience.
20 View data
View Data is the primary diagnostic tool for confirming data integrity before analysis. When you upload your dataset, RAISINS performs an automated health check: it confirms your chosen label column contains unique, non-numeric values with no duplicates, that all selected clustering variables are numeric, and that there are no missing values, since distances cannot be computed to a missing cell. Use this tab any time the Analysis tab reports a problem with your file.
21 Wrapping up
Hierarchical Cluster Analysis rests on one simple mechanism: start with every observation in its own group, repeatedly merge the two most similar groups, and record every merge as a dendrogram. Everything else in this tutorial, scaling, distance metrics, linkage methods, the optimal-k plots, HCPC, the tanglegram, exists to help you make the judgment calls the algorithm itself cannot: how to make your variables comparable, which linkage and distance genuinely suit your data, and where to cut the tree.
If your data instead form clean, evenly sized, roughly spherical groups and you already know how many groups you expect, RAISINS’s K-Means Cluster Analysis module is a companion tool built for exactly that case, and it uses the same scaling options described in Section 2. If you get stuck at any point, RA-One is available 24 × 7, or write to us at [email protected].