PCA vs PCoA vs NMDS: Key Differences, Applications, and When to Use Each

In contemporary biological research, the handling of high-dimensional data has emerged as a significant challenge. For instance, in the analysis of microbial communities and gene expression data, samples often encompass thousands of features (such as Operational Taxonomic Units (OTUs) or gene expression values). Direct analysis of these high-dimensional datasets not only involves substantial computational effort but also renders it difficult to intuitively display similarities and differences among samples. Consequently, dimensional reduction analysis has become a vital tool for projecting high-dimensional data into lower-dimensional spaces, all while preserving the primary structural information. This simplification enhances both analytical ease and visualization quality.

The fundamental premise of dimensional reduction analysis is to reduce data complexity by decreasing the number of dimensions, endeavoring to retain essential information as effectively as possible. Common dimensional reduction techniques include Principal Component Analysis (PCA), Principal Coordinate Analysis (PCoA), and Non-Metric Multidimensional Scaling (NMDS). These methods have been widely adopted in fields such as microbiomics and gene expression analysis.

This article focuses on comparing the core differences, applicable scenarios, and selection criteria for PCA, PCoA, and NMDS ordinal analysis methods. PCA, which is based on linear models, is well-suited to situations with a limited number of species and minimal variability in environmental factors and species abundance. PCoA, on the other hand, relies on a distance matrix and is appropriate for studying the similarities or dissimilarities in sample community composition. NMDS, being a nonlinear dimensional reduction method, is suitable for ordinal analysis while preserving the original relationships between samples.

By contrasting the characteristics and application contexts of these three methods, this article aims to provide researchers with a systematic perspective, assisting them in choosing the most appropriate dimensionality reduction method for their specific studies.

Overview of Dimensionality Reduction Techniques

Dimensionality reduction is a crucial process in data analysis, especially when handling large and complex datasets. Among the most commonly utilized techniques are PCA, PCoA, and NMDS. Each of these methods offers unique definitions and applications pertinent to different research needs.

PCA vs PCoA vs NMDS. (Kaysar, Md Salahuddin, et al., 2022; Dang, Chenyuan, et al., 2020; Brehaut, et al., 2022)

A detailed overview of these techniques is provided below:

1. PCA

Definition:

PCA is a dimensionality reduction technique fundamentally based on linear modeling. It functions by identifying principal components through variance maximization, thereby reducing the dimensionality of high-dimensional datasets. The process involves the orthogonal transformation of the original dataset into a set of new, linearly independent variables termed principal components, arranged in order of decreasing variance.

Applicable Scenarios:

• Linear Data: PCA is particularly effective for data exhibiting linear relationships, facilitating the extraction of primary feature components.
• Data Preprocessing and Feature Extraction: Extensively used in preprocessing to remove redundant features, diminish computational demands, and improve model training efficacy. Its applications span across fields including image processing, gene expression analysis, and financial modeling.
• High-Dimensional Data Reduction: By prioritizing principal components with maximal variance, PCA minimizes dimensions while preserving substantial information content.

Advantages:

• Computational efficiency.
• Simplifies interpretation by omitting the need for hyperparameter adjustments.
• Effectively retains variance from the original dataset.

Disadvantages:

• Assumes linearity, thus limiting performance with nonlinear data distributions.
• Computationally intensive for extensive datasets, with a complexity approximated by O(n^2d) for n samples and d dimensions.

2. PCoA

Definition:

PCoA is a technique that reduces dimensionality via a distance matrix, portraying the similarities or dissimilarities between samples. The central aim of PCoA is to project this matrix into a lower-dimensional space while maintaining relative distances as accurately as possible.

Applicable Scenarios:

• Distance Metric-Based Analysis: Suited for scenarios where sample similarity is assessed through distance measures, such as species composition in ecology and microbiome studies.
• Ecology and Microbiome Analysis: Commonly deployed to evaluate species diversity and community structures using distance matrices like Bray-Curtis or Jaccard.

Advantages:

• Effective for distance-based dimensionality reduction analyses.
• Capable of managing datasets with limited samples yet numerous features.

Disadvantages:

• Although it projects data linearly, its efficacy hinges on the distance matrix’s choice and data distribution traits.
• Exhibits high computational demands when applied to large datasets.

3. NMDS

Definition:

NMDS is a rank-based dimensionality reduction technique prioritizing qualitative over quantitative distances. This method transforms high-dimensional data into lower dimensions while preserving the rank-order of relative distances among samples.

Applicable Scenarios:

• Complex Datasets: Particularly beneficial for complex or high-dimensional datasets with multiple samples or species, where data structure interpretation proves challenging.
• Ecology and Microbiome Analysis: Widely applied for uncovering the relative relations between samples, characterizing species diversity and community structure.

Advantages:

• Well-suited for nonlinear data analyses.
• Provides strong interpretive utility for complex data configurations.
• Adapts to a variety of data types thanks to its emphasis on rank-order information.

Disadvantages:

• Requires intensive computation.
• Outcomes may vary based on initial settings, necessitating iterative refinement.

CD Genomics offers custom bioinformatics pipeline development, enabling automated distance matrix computation and PCoA visualization. This service helps researchers streamline their microbiome diversity analyses with standardized, reproducible workflows.

4. Conclusion:

Each method-PCA, PCoA, and NMDS-offers unique advantages and potential shortcomings, making them applicable to particular data types and analytical needs:

• PCA is optimal for linear datasets and dimensionally reducing extensive datasets, assisting in preprocessing and elaborating features.
• PCoA excels in analyses reliant on distance matrices, often employed in ecological and microbiomic contexts.
• NMDS is adept at handling complex datasets, well-suited for unveiling relative relationships in nonlinear data.

Choosing the most appropriate dimensionality reduction technique requires careful attention to the data’s specific characteristics and the objectives of the analysis.

To assist researchers in simplifying these analyses, CD Genomics offers comprehensive next-generation sequencing (NGS) services, including metagenomic sequencing and amplicon sequencing. These services provide extensive datasets that can be transformed using techniques like PCA, PCoA, and NMDS, facilitating advanced insights into microbial community structures and gene expression patterns.

For instance, metagenomic sequencing is ideal for obtaining a holistic view of microbial community genomes and their functions within a sample, paving the way for comprehensive ecological and functional analyses. Following sequencing, PCA or PCoA can be used to reduce the complexity of these high-dimensional datasets, enabling researchers to efficiently visualize differences in community composition and function across various samples or environmental conditions.

Similarly, amplicon sequencing, which focuses on specific gene regions (e.g., 16S rRNA) to profile microbial community composition, can benefit from these dimensional reduction techniques. When researchers employ PCoA, differences in β-diversity among samples can be effectively visualized, contributing to a better understanding of microbial diversity and ecological relationships.

Key Differences in PCA, PCoA, and NMDS

PCA, PCoA, and NMDS are integral methodologies for dimensionality reduction and ordination analysis. They exhibit significant differences in terms of input data, distance measures, and suitable applications. Below, the core differences of these techniques are delineated:

1. Input Data:

• PCA: Utilizes the original feature matrix (e.g., species abundance) for analysis. It is best suited for data with linear structures and is commonly employed for feature extraction and dimensional reduction.
• PCoA: Operates on a distance matrix (such as Bray-Curtis, Jaccard, or UniFrac distances). This approach is suitable for scenarios where sample similarity is measured through distance metrics, widely used in ecological and microbiome research.
• NMDS: Also based on a distance matrix but focuses on preserving the relative order of samples rather than exact distance values. It is well-suited for complex datasets, particularly in analyses involving multiple samples or species.

2. Distance Measures:

• PCA: Relies on the covariance matrix or correlation matrix for dimensionality reduction, inherently assuming data conformity to Euclidean space structures.
• PCoA: Accommodates various distance measures, including Bray-Curtis, Jaccard, and both weighted and unweighted UniFrac. These measures can capture different interspecies relationship nuances, such as evolutionary connections or abundance variations.
• NMDS: While capable of using any distance matrix, it ultimately retains only the rank-order information rather than absolute distances. Common distance measures include Bray-Curtis and Jaccard.

3. Suitable Applications:

• PCA: Ideal for datasets with linear structures, frequently applied for feature extraction and dimensionality reduction. It is effectively used in gene expression analysis, financial data examination, and image processing, offering dimension reduction while preserving maximal information.
• PCoA: Appropriate for data analysis grounded in distance matrices, adept at exploring sample similarities and divergences. It is prevalent in ecological and microbiomic studies for visualizing community structures and analyzing inter-sample relationships.
• NMDS: Tailored for complex datasets, particularly when distributions are irregular or when linear methods fall short. NMDS iteratively optimizes the rank-order of samples to preserve their relative order in a lower-dimensional space, making it valuable in ecology and community structure studies.

Comparison Table of Dimensionality Reduction Methods: PCA, PCoA, and NMDS

Conclusion

Each method-PCA, PCoA, and NMDS-exhibits unique characteristics and applications:

• PCA is suited for linear data and serves well for feature extraction and dimensionality reduction.
• PCoA is ideal for data interpretations based on distance matrices, excelling in visualizing inter-sample relationships.
• NMDS excels with complex datasets, particularly in the realm of nonlinear data analysis and rank-order retention.

The choice of method largely depends on the specific research objectives and data characteristics. For instance, in microbial community analysis, if the data clearly boast a linear structure, PCA may be the most appropriate. Conversely, in ecological studies, employing PCoA or NMDS may better accommodate complex relationships among samples.

Applications of PCA, PCoA, and NMDS in Various Fields

These sophisticated methodologies are indispensable for unveiling complex patterns within biological and ecological data, enriching our understanding of diverse scientific phenomena.

The application of PCA, PCoA, and NMDS across diverse research areas highlights their robust capabilities in data analysis. PCA is well-suited for dimensionality reduction and feature extraction in high-dimensional datasets. In contrast, PCoA excels in β-diversity analysis using distance matrices, and NMDS effectively handles nonlinear dimensionality reduction and differential analysis in complex data scenarios. These methodologies have found extensive application in fields such as gene expression, image processing, and microbial community studies.

Applications of PCA:

1. Gene Expression Data Analysis: PCA is frequently employed in the dimensionality reduction of gene expression data, facilitating the identification of key components to reveal similarities and differences between samples.
2. Image Processing and Pattern Recognition: Widely used in image processing, PCA plays a crucial role in dimensionality reduction and feature extraction. By decreasing the dimensionality of image data, PCA effectively reduces computational complexity while preserving core image features, thereby enabling efficient pattern recognition.

Applications of PCoA:

1. β-Diversity Analysis of Microbial Communities:
   PCoA is a dimensionality reduction method based on distance matrices, often applied in the analysis of β-diversity within microbial communities. By projecting high-dimensional data into a lower-dimensional space, PCoA allows researchers to observe differences in microbial composition across samples.
2. Species Distribution Studies in Ecology:
   In ecological research, PCoA is employed to investigate species distribution variability. By analyzing species composition under diverse environmental conditions, PCoA aids in uncovering the relationships between species distribution and environmental factors.

Applications of NMDS:

1. Time-Series Expression Profile Analysis: As a nonlinear dimensionality reduction method based on distance matrices, NMDS is adept at handling high-dimensional data. In time-series analysis, NMDS is employed to compare expression profile differences at various time points.
2. Differential Analysis of Microbial Communities in Metagenomics: NMDS is widely applied in metagenomics for differential analysis of microbial communities. By projecting metagenomic data into lower-dimensional spaces, NMDS helps identify differences in microbial composition across samples.

Graph Interpretation:

1. PCA Graph:

PCA biplot depicting the relationship between the estimated variables and the rice cultivars. (Kaysar, Md Salahuddin, et al., 2022)

• Axes: Represent the first and second principal components, with percentages indicating the contribution of each component to total data variance.
• Sample Points: Closer distances among samples of the same group suggest greater reproducibility; larger distances among different groups indicate more pronounced differences.

2. PCoA Graph:

PCoA plots show the composition differences (Bray-Curtis distances) of (a) sOTUs and (b) ARGs subtypes between water and sediment in different seasons. (Dang, Chenyuan, et al., 2020)

• Axes: Represent the first and second principal coordinates, reflecting the distribution of distances between samples.
• Sample Points: Greater proximity among samples indicates higher similarity.

3. NMDS Graph:

NMDS ordination (k = 3, stress = 0.152) of treeline site plant community functional groups and environmental conditions grouped by burned (red) and unburned (blue) locations. (Brehaut, et al., 2022)

• Stress Value: Evaluates the consistency of the model with original data; generally, a Stress value <0.1 suggests an acceptable model fit.
• Sample Points: Greater proximity among samples indicates more consistent rank-order relationships.

These sophisticated methodologies are indispensable for unveiling complex patterns within biological and ecological data, enriching our understanding of diverse scientific phenomena.

How to Choose the Right Technique

Selecting the right analytical technique is critical in data analysis, as different methods are tailored for various data types, analytical goals, and distance measurement needs. This section focuses on guiding researchers in choosing suitable dimensionality reduction and analytical methods based on the characteristics of their data and research objectives. By understanding the core differences and application scenarios of these techniques, researchers can better apply them to solve real-world problems.

1. Data Type: Linear vs. Nonlinear vs. Complex Datasets

• Linear Data: For datasets with linear relationships, linear models such as linear regression and logistic regression are appropriate. These models are applicable when data are regularly distributed with straightforward inter-variable relationships.
• Nonlinear Data: Nonlinear datasets require models like decision trees, support vector machines, or neural networks, which are adept at capturing intricate relationships among variables.
• Complex Datasets: Complex datasets often encompass various types of data (such as images, text, and genetic sequences) and typically necessitate approaches like deep learning or other pattern recognition algorithms for processing.

2. Analytical Objectives: Feature Extraction vs. Visualization of Sample Relationships vs. Rank-Order Analysis

• Feature Extraction: The goal of feature extraction is to reduce data dimensionality while retaining essential information. Techniques such as PCA and Linear Discriminant Analysis (LDA) are commonly employed to lower computational complexity or enhance model performance.
• Visualization of Sample Relationships: Visualization techniques such as t-SNE and UMAP are instrumental in uncovering latent structures and patterns within data. These methods are particularly effective for the dimensionality reduction and visualization of high-dimensional data.
• Rank-Order Analysis: This approach is used for handling ordinal data or situations where analysis based on relative positioning is necessary. Non-parametric methods like Spearman’s rank correlation and Kendall’s tau-b are frequently used to assess monotonic relationships and consistency in ordering between variables.

3. Distance Measurement Needs: Euclidean vs. Custom Distances vs. Rank-Order

• Euclidean Distance: A prevalent measurement method suitable for continuous numerical data. It is simple and efficient but may require standardization due to sensitivity to variable scales.
• Custom Distances: In specific cases, Euclidean distance may not accurately capture the true relationships within data. For instance, in time series analysis, dynamic time warping (DTW) distance is more adept at managing nonlinear variations.
• Rank-Order Relationships: Applicable to ordinal data or scenarios requiring analysis based on relative positioning. Methods like Spearman’s rank correlation and Kendall’s tau-b effectively evaluate monotonic relationships and order consistency.

4. Recommendations Based on Sample and Species Numbers

• Many Samples, Few Species: Prioritize PCA, known for its high computational efficiency and effective handling of large-scale data, suitable for linear data dimensionality reduction.
• Few Samples, Many Species: Opt for PCoA, which captures sample similarities and differences through distance matrices.
• Nonlinear, Complex Data with Important Sample Rank-Order Differences: NMDS is preferred, as it effectively reflects rank-order relationships among samples, suitable for nonlinear data and complex ordinal analyses.

Adopting these strategies ensures that researchers choose the most appropriate technique for their specific challenges and data characteristics, thereby enhancing the accuracy and efficacy of their analytical outcomes.

Conclusion

PCA, PCoA, and NMDS are fundamental techniques in data dimensionality reduction and visualization, each playing a crucial role in different analytical contexts. PCA is best suited for handling linear data, primarily facilitating feature extraction by projecting high-dimensional data into a lower-dimensional space for more accessible analysis. PCoA focuses on analyzing relationships among samples based on distance matrices, making it ideal for β-diversity studies in microbiome research and ecology. NMDS, as a nonlinear dimensionality reduction method, excels in managing complex datasets and rank-order relationships, particularly when the ordering of differences among samples is critical.

When selecting the appropriate analytical method, it is essential to consider the characteristics of the data and the objectives of the analysis. PCA is preferable for linear datasets, PCoA is an excellent choice for visualizing inter-sample relationships, and NMDS is undoubtedly more suitable for nonlinear and complex datasets. Understanding the core differences and applicable scenarios of these methods will enhance the accuracy and efficiency of data analysis.

For those seeking further learning, both R and Python offer extensive tools and libraries that support the implementation of these techniques. In R, the ‘prcomp()’ function and the ‘vegan’ package are useful for performing PCA and PCoA analyses, while the ‘metaMDS()’ function is appropriate for NMDS. In Python, PCA can be executed using ‘scikit-learn’, PCoA is supported by the ‘skbio’ package, and NMDS can be implemented via the ‘MDS’ class in ‘scipy’ and ‘sklearn.manifold’. These resources provide invaluable support for mastering data analysis techniques.

References

1. Kaysar, Md Salahuddin, et al. "Dissecting the relationship between root morphological traits and yield attributes in diverse rice cultivars under subtropical condition." Life 12.10 (2022): 1519. https://doi.org/10.3390/life12101519
2. Dang, Chenyuan, et al. "Metagenomic insights into the profile of antibiotic resistomes in a large drinking water reservoir." Environment International 136 (2020): 105449. https://doi.org/10.1016/j.envint.2019.105449
3. Brehaut, L., Brown, C.D. Wildfires did not ignite boreal forest range expansion into tundra ecosystems in subarctic Yukon, Canada. Plant Ecol 223, 829–847 (2022). https://doi.org/10.1007/s11258-022-01242-9