PCoA Demystified: A Practical and Comprehensive Guide to Principal Coordinates Analysis

Principal Coordinates Analysis, commonly abbreviated as PCoA, is a versatile ordination technique used across ecology, microbiology, anthropology and beyond. Unlike classic PCA, which operates on raw data, PCoA starts from a distance or dissimilarity matrix and seeks a low‑dimensional representation that preserves as much of the original relationships as possible. This article explains what PCoA is, why it matters, how to perform it, and how to interpret the outputs in real-world research. Whether you are analysing microbial communities, ecological surveys or phenotypic profiles, understanding PCoA will help you reveal patterns that might otherwise remain hidden in complex datasets.

What is PCoA and why it matters?

PCoA, short for Principal Coordinates Analysis, is a distance‑based ordination method. It converts a matrix of pairwise dissimilarities into a set of coordinates in a few dimensions, typically two or three, that can be visualised. The method is particularly powerful when the data are not well described by linear relationships or when the researcher wants to work directly with a chosen distance metric—such as Bray–Curtis, UniFrac or Jaccard—that captures ecological or compositional differences among samples.

In practice, PCoA provides a map where samples that are similar according to the distance metric cluster together, while dissimilar samples occupy distant regions of the plot. Because PCoA is based on a distance matrix, it can accommodate a wide range of data types and distance definitions. This flexibility makes PCoA a staple in microbiome analyses, community ecology and any field requiring comparative ordination of samples.

Core concepts behind PCoA

Distance matrices and their role

The starting point for PCoA is a distance (or dissimilarity) matrix. Each entry reflects how different two samples are, according to a chosen metric. Common choices include Bray–Curtis for abundance data, Jaccard for presence–absence data, and UniFrac for phylogenetically informed comparisons in microbial studies. The choice of distance metric shapes the interpretation of the resulting ordination, so selecting a metric aligned with your hypotheses and data structure is essential.

Distance matrices must be symmetric with zero diagonal values. The ensuing PCoA then seeks to place samples into a low‑dimensional coordinate system that preserves those pairwise distances as closely as possible.

Double-centering and eigen decomposition

PCoA proceeds by transforming the distance matrix into a centred matrix through double‑centering. This mathematical step prepares the data for eigen decomposition, a process that extracts eigenvalues and eigenvectors. The eigenvectors define the axes (the principal coordinates) along which samples are distributed, and the corresponding eigenvalues quantify how much variation each axis captures.

In short, the largest eigenvalues correspond to axes that explain the most variation in the distance structure. Plotting the first two or three axes often reveals the most interpretable patterns, such as clusters of similar samples or gradients corresponding to environmental or experimental factors.

Axes, variance and interpretation

The percentage of total variation explained by each axis is a useful guide for interpretation. If the first two axes account for a substantial portion of the variability, a two‑dimensional plot will be a faithful representation of the underlying relationships. If not, a three‑axis plot or alternative ordination techniques may be more informative. Remember that PCoA visualisations reflect the chosen distance metric; a different metric may reveal alternative structure, so it is often beneficial to explore multiple distances.

Visual interpretation and caveats

Interpreting PCoA plots requires caution. Proximity on the plot suggests similarity under the selected distance, but it does not prove a direct causal or mechanistic relationship. Outliers can distort the visualization, and the scale of axes may be non‑linear in ways that obscure subtle patterns. Supplementary analyses, such as PERMANOVA or analysis of group dispersions, can help test whether observed groupings are statistically meaningful.

When to use PCoA

PCoA is well suited to questions like: Which samples share similar community composition? Do samples group by environment, treatment or host factors? How do dramatic changes in a factor influence overall structure? Because it works with a distance matrix, PCoA is flexible enough to incorporate complex data types, from species abundance counts to presence–absence matrices, and even phylogenetically informed distances.

It is particularly popular in microbiome research, where distances such as UniFrac integrate phylogenetic information and where Bray–Curtis highlights differences in species abundances between samples. PCoA can also be paired with ecological metadata to explore associations between ordination patterns and environmental variables.

Distance metrics for PCoA: what to choose

Bray–Curtis dissimilarity

Bray–Curtis is widely used for community composition data because it is sensitive to the abundances of shared taxa while down‑weighting shared absences. It is intuitive for ecological data and often reveals clear separation of sample groups based on community structure. However, it does not account for phylogeny, so interpretive depth may be limited when evolutionary relationships are important.

UniFrac distances

UniFrac distances incorporate phylogenetic information. They come in unweighted and weighted variants. Unweighted UniFrac emphasises presence/absence and is sensitive to rare lineages, while weighted UniFrac weights taxa by relative abundance, highlighting the impact of dominant lineages on overall community structure. For microbial ecology, UniFrac often yields biologically informative patterns that align with environmental gradients or host factors.

Jaccard distance

Jaccard focuses on shared presence–absence. It is especially useful when data are sparse or when the interest is in membership rather than abundance. While it can be less sensitive to abundance shifts, Jaccard remains a robust option for binary data or early exploratory analyses.

Euclidean and other distances

Euclidean distance is appropriate when data are subset‑summed and standardised, and when the analysis aims to connect PCoA with PCA concepts. Other metrics, including Bray–Curtis‑like or Chi‑squared distances, can be applied depending on data type and study design. The key is to align the distance metric with the scientific question and the data’s characteristics.

A practical workflow for performing PCoA

Step 1: Prepare your data

Clean, normalise and, if relevant, transform your data. In microbiome studies, this may involve rarefying, normalising counts, or using compositional data techniques. Decide how to handle zeros, sparsity and batch effects, as these can influence distance calculations and the resulting ordination.

Step 2: Choose a distance metric

Select a distance that aligns with your hypotheses and data structure. For community data with abundances, Bray–Curtis or UniFrac are common choices. For presence–absence data, Jaccard may be more appropriate. Consider running PCoA with multiple distances to assess the robustness of detected patterns.

Step 3: Compute the distance matrix

Use your preferred software to compute the pairwise distance matrix. Ensure the matrix is square, symmetric and contains non‑negative values. Some tools provide built‑in checks for distance matrix validity, which can help catch anomalies early.

Step 4: Run PCoA

Apply the PCoA algorithm to the distance matrix to obtain coordinates for each sample along principal axes. Extract eigenvalues to gauge how much variation each axis explains. In many software packages, you can also obtain a scree plot showing cumulative explained variance.

Step 5: Interpret and visualise

Plot the first two or three axes and colour or shape points by metadata (e.g., environment, treatment, host) to identify groupings and gradients. Add 95% confidence ellipses where appropriate, though note that ellipses assume underlying normality and homogeneity of dispersion. Consider overlaying environmental vectors or scores to help interpret axes in relation to metadata.

Step 6: Validate with statistical tests

Beyond visual inspection, use statistical tests such as PERMANOVA to assess whether group centroids differ significantly in the chosen distance space. Tests of dispersion can reveal whether groups differ more in variability than in centroids, which can influence interpretation of the ordination.

PCoA in practice: software options

Using R: vegan and beyond

In R, the vegan and ape packages are common choices for PCoA workflows. A typical sequence involves computing a distance matrix with dist or a specialised distance function, followed by cmdscale or eigen decomposition via the function pcoa from the ape package. Some researchers also use the vegan function capscale as a constrained alternative for ordination with environmental variables. Here is a simplified outline:

• Compute distance: dist(data, method = “bray”) or a specialised distance function for phylogenetic data.
• Run PCoA: library(ape); pcoa(dist_matrix)
• Visualise: plot(pcoa_object$vectors[,1], pcoa_object$vectors[,2], col = group_color)
• Assess variance: pcoa_object$values[1:2] give the explained variances.

The exact commands depend on the distance metric chosen and the structure of your data. Always consult the documentation for the specific packages you use to ensure correct interpretation of the outputs.

Using Python: scikit-bio and scikit-learn

Python users often rely on scikit-bio for PCoA, which supports a range of distance metrics and provides convenient plotting interfaces. A typical workflow includes computing a distance matrix with a function like skbio.diversity.distance_matrix, applying skbio.stats.ordination.pcoa to obtain coordinates, and then plotting the first two axes with seaborn or matplotlib. For large datasets, consider efficient data handling and memory‑friendly plotting strategies.

Interpreting axes and variance explained

When you run PCoA, you’ll receive a set of coordinates for each sample along axes, plus eigenvalues indicating the amount of variation captured by each axis. The first axis explains the largest portion of the distance‑structure in the data, the second axis the second largest, and so on. A scree plot helps you decide how many axes to interpret. If the first two axes capture a meaningful share of the variation, a two‑dimensional plot will reveal the major patterns. If not, a three‑axis plot or a two‑stage interpretation—focusing on both the first two and the third axis—might be more informative.

Interpretation should link to metadata. For example, samples from a particular environment may cluster together, or a gradient in PCoA space might align with pH, temperature or geographic origin. This interpretive step is where PCoA translates mathematical structure into biological or ecological insight.

Limitations and common pitfalls

PCoA is a powerful tool, but it has limitations. Here are several points to keep in mind:

• Distance choice matters. Different metrics emphasise different aspects of the data, potentially leading to different ordination patterns. Always justify the chosen distance in light of your research question.
• Non‑Euclidean distances can produce negative eigenvalues. In such cases, the PCoA plot represents a best‑fit approximation, and interpretation should be cautious. Some software provides corrections or alternative representations when negative eigenvalues arise.
• Effect of rare features and zeros. Sparse data or many zeros can distort distances. Consider data transformations or alternative abundance handling when appropriate.
• Batch effects and sampling depth. Technical variation can drive separation in PCoA plots. Apply normalization, randomisation or batch correction techniques to mitigate such effects.
• Over‑interpretation. The visualization is exploratory. Confirm patterns with appropriate statistical testing and biological reasoning.

Comparing PCoA with PCA and other ordination methods

PCoA is distinct from PCA in that it operates on a distance matrix and is therefore not limited to linear relationships in the raw data. While PCA assumes Euclidean geometry and continuous, normally distributed variables, PCoA accommodates a variety of distance metrics that capture ecological or compositional structure. When your data are count data with a meaningful distance definition, PCoA often provides more faithful representations of relationships than PCA.

Other ordination techniques, such as non‑metric multidimensional scaling (NMDS) or canonical correspondence analysis (CCA), offer complementary perspectives. NMDS emphasises rank order of distances and is less sensitive to the scale of distances, whereas CCA integrates external explanatory variables directly into the ordination. Choosing among these methods depends on the data, hypotheses and desired interpretation.

Practical case study: microbiome sample analysis

Imagine a study comparing gut microbiome samples from three dietary groups: plant‑based, omnivorous, and high‑fat diets. After sequencing 16S rRNA gene amplicons and processing the data, you compute a Bray–Curtis distance matrix. Running PCoA reveals clear clustering by diet, with the first axis separating plant‑based communities from the others and the second axis differentiating omnivorous from high‑fat groups. A PERMANOVA confirms that diet explains a substantial portion of the variation, while dispersion tests show similar within‑group variability. Overlaying metadata such as age and sex indicates no systematic bias, strengthening the diet‑driven interpretation. This example highlights how PCoA can translate complex microbial composition into actionable ecological insight.

Best practices for reporting PCoA results

When presenting PCoA results in a report or publication, clarity and transparency are key. Consider including:

• The distance metric used (e.g., Bray–Curtis, UniFrac) and rationale for its choice.
• Details of data preprocessing, including any normalisation or transformation steps.
• The axes shown in plots and the corresponding variance explained (e.g., 1st axis explains 35% of the variation).
• Statistical tests supporting observed group differences (e.g., PERMANOVA) and dispersion analyses.
• Supplementary plots, such as scree plots of eigenvalues and distance heatmaps, to provide a fuller picture of the data structure.

Advanced topics and future directions

As datasets grow more complex, researchers explore constrained PCoA and phylogenetically informed distances to better reflect ecological relationships. Partial PCoA can be used to control for specific covariates while examining residual structure. Integrating PCoA with machine learning approaches or combining it with other ordination methods can yield richer insights into community structure and environmental drivers.

Frequently asked questions about PCoA

What is PCoA used for?

PCoA is used to visualise and explore patterns in sample similarity or dissimilarity, based on a chosen distance metric. It helps researchers identify clusters, gradients and outliers in ecological, genomic or microbiome data.

How many axes should I plot?

Typically, the first two axes are shown for readability, but always check the explained variance. If the first two axes capture a modest portion of the total variation, consider including the third axis or citing the scree plot to justify the visual representation.

Can PCoA handle missing data?

Most implementations require complete distance matrices. Handle missing data during preprocessing by imputation, exclusion or using distance metrics robust to missing values, and document your approach clearly.

Is PCoA the same as PCA?

Not exactly. PCA is a feature‑based method operating on the original data with Euclidean distances, while PCoA is distance‑based and can use non‑Euclidean distances that capture ecological or phylogenetic relationships more effectively.

How can I test whether groups differ in PCoA space?

PERMANOVA (permutational multivariate analysis of variance) is a common choice to test for differences in centroids between groups in the multivariate space defined by the distance matrix. Pairwise tests and dispersion assessments can accompany the analysis for a robust interpretation.

Closing thoughts

PCoA remains a cornerstone of modern ecological and microbiological data analysis, offering a flexible, intuitive way to translate complex relational data into digestible visual patterns. By carefully selecting the distance metric, ensuring proper data preparation, and validating observations with appropriate statistics, researchers can leverage PCoA to uncover meaningful structure within their datasets. With thoughtful implementation and clear reporting, the PCoA approach will continue to illuminate the relationships that shape biological communities and environmental systems.