Multidimensional scaling (MDS) is a family of statistical methods that represents the similarities or differences among objects as distances between points in a low-dimensional map. Objects that are more similar appear closer together, while more dissimilar objects appear farther apart. The map helps researchers explore patterns that are difficult to see in a distance matrix.

Introduction

Researchers often need to compare objects for which the most meaningful information is not a set of ordinary variables but a collection of pairwise relationships.

Consumers may judge how similar different brands are. Ecologists may calculate dissimilarities among biological communities. Psychologists may compare stimuli, emotions or concepts. Geneticists may estimate distances among populations. In each case, the dataset can be expressed as a matrix showing how similar or different every pair of objects is.

Multidimensional scaling converts that matrix into coordinates. The coordinates can then be displayed as a one-, two- or three-dimensional configuration in which geometric distance approximates the original dissimilarity.

This guide explains the main forms of MDS, the underlying mathematics, the complete analytical process, appropriate diagnostic methods, interpretation rules, software options and common mistakes.

Key takeaways

• MDS begins with pairwise similarities, dissimilarities or distances.
• The principal output is a spatial configuration, not a set of original-variable loadings.
• Classical MDS uses eigendecomposition, while metric and non-metric MDS commonly minimise stress iteratively.
• A lower stress value generally indicates a better representation, but stress must be interpreted with other diagnostics.
• Distances among points usually matter more than the direction or sign of the axes.
• MDS is an exploratory mapping method and does not by itself prove that clusters or dimensions are statistically significant.

What Is Multidimensional Scaling?

Multidimensional scaling is a method for placing objects in a geometric space so that the distances among their plotted positions reproduce a supplied set of pairwise similarities or dissimilarities as closely as possible.

Suppose researchers compare six universities. Instead of measuring each university with predetermined variables, participants rate the similarity of every university pair. MDS can translate those ratings into a map. Universities perceived as similar will occupy nearby positions, while those perceived as different will be separated.

The term multidimensional does not mean that the final chart must contain many visible dimensions. It refers to the possibility that the relationships among the objects reflect several underlying characteristics. A two-dimensional solution is often selected because it can be displayed on a page, even though the original relational structure may require more dimensions for an accurate representation.

MDS belongs to the broader families of ordination, dimensionality reduction and proximity analysis. Its intellectual foundations include the work of Torgerson, Shepard and Kruskal (Borg & Groenen, 2005; Kruskal, 1964a, 1964b; Shepard, 1962a, 1962b).

The Basic Idea Behind MDS

Imagine a table of road distances among cities. The table contains distances but no latitude or longitude. MDS can use those pairwise distances to reconstruct an approximate geographic map.

The same principle applies when the distances are not geographical. A “distance” might represent:

• Differences in consumer perception.
• Ecological dissimilarity between sampling sites.
• Genetic distance between populations.
• Semantic distance between words.
• Correlation-based dissimilarity between variables.
• Differences between political attitudes.
• Confusion rates among visual or auditory stimuli.

The resulting coordinates are constructed to represent the supplied relationships. They are not necessarily measurements of physical location.

Similarity, Dissimilarity and Distance

These terms are related but should not be treated as interchangeable.

Similarity

A similarity value increases as two objects become more alike. A correlation coefficient is one example, although its use requires careful consideration of negative values and substantive meaning.

Dissimilarity

A dissimilarity value increases as two objects become less alike. Dissimilarities do not always satisfy all mathematical requirements of a true distance metric.

Distance

A mathematical distance usually satisfies four properties:

1. Non-negativity.
2. Zero distance from an object to itself.
3. Symmetry.
4. The triangle inequality.

Some MDS procedures can analyse dissimilarities that do not fully satisfy these properties. However, non-Euclidean input may produce negative eigenvalues in classical MDS or increased distortion in a low-dimensional solution.

Converting similarity into dissimilarity

A common conversion for similarities bounded between 0 and 1 is:

[
\delta_{ij}=1-s_{ij}
]

where (s_{ij}) is the similarity between objects (i) and (j), and (\delta_{ij}) is their dissimilarity.

This conversion is not universally appropriate. The transformation should reflect the scale, range and theoretical meaning of the similarity measure. For correlations, researchers might consider (1-r), (1-|r|), or another transformation, but these alternatives represent different concepts and can produce different maps.

What Data Does MDS Require?

The usual input is an (n \times n) proximity matrix, where (n) is the number of objects.

A valid complete dissimilarity matrix normally has:

• The same objects in its rows and columns.
• Zeros along the diagonal.
• Non-negative off-diagonal values.
• Symmetry when the relationship is assumed to be symmetric.
• A documented method for calculating or collecting each proximity.

For five objects, the number of unique pairwise relationships is:

[
\frac{n(n-1)}{2}=\frac{5(4)}{2}=10
]

As the number of objects increases, the number of pairwise relationships increases quadratically. This affects data collection, storage and computation.

MDS may also begin with a conventional object-by-variable dataset. Software can first calculate a distance matrix from the variables and then fit the MDS model. In this situation, feature scaling and distance selection become essential analytical decisions.

Main Types of Multidimensional Scaling

Terminology varies slightly across textbooks and software packages. For practical purposes, three major forms should be distinguished.

Type	What it attempts to preserve	Main computational idea	Most suitable when
Classical MDS	Distances through a scalar-product representation	Double-centering and eigendecomposition	Input distances are Euclidean or approximately Euclidean
Metric MDS	Numerical magnitudes of dissimilarities or a specified metric transformation	Iterative stress minimisation	Dissimilarity values contain meaningful interval- or ratio-level information
Non-metric MDS	Rank order of dissimilarities	Monotonic transformation plus stress minimisation	Only the ordering of dissimilarities is trustworthy

Classical multidimensional scaling

Classical MDS is also called principal coordinates analysis, PCoA, Torgerson scaling or Torgerson–Gower scaling.

It converts squared distances into a centred inner-product matrix and obtains coordinates through eigendecomposition. When the input consists of Euclidean distances calculated from centred quantitative observations, classical MDS is closely related to principal component analysis.

Classical MDS has an analytical solution and is computationally convenient. However, large negative eigenvalues warn that the supplied dissimilarities cannot be represented exactly in Euclidean space.

Metric multidimensional scaling

Metric MDS attempts to make fitted distances numerically close to the input dissimilarities or to a prespecified transformation of them.

For an absolute metric model, a dissimilarity of 8 should be represented by a fitted distance close to 8, subject to scaling conventions and model constraints. Other metric formulations may permit ratio, interval, polynomial or spline transformations.

Metric MDS is appropriate when differences in the magnitudes of the proximities are meaningful, not merely their ordering.

Non-metric multidimensional scaling

Non-metric MDS, usually abbreviated NMDS, aims to preserve rank order rather than exact magnitudes.

If object A is more similar to B than to C, the fitted map should place A closer to B than to C. The exact ratio between the two distances is not the primary concern.

NMDS estimates a monotonic relationship between observed dissimilarities and fitted distances, often through isotonic regression. This makes it useful for ordinal judgments and for dissimilarities whose absolute scale should not be interpreted literally.

Because NMDS uses iterative optimisation, researchers should examine convergence and repeat the analysis from multiple starting configurations.

The Mathematics of Classical MDS

Let (\Delta) be an (n \times n) matrix of dissimilarities and let (\Delta^{(2)}) contain the squared dissimilarities.

Define the centring matrix:

[
J=I-\frac{1}{n}\mathbf{1}\mathbf{1}^{T}
]

Classical MDS constructs the double-centred matrix:

[
B=-\frac{1}{2}J\Delta^{(2)}J
]

The eigendecomposition of (B) is:

[
B=V\Lambda V^{T}
]

For a (p)-dimensional solution, the coordinates are obtained from the largest positive eigenvalues and their corresponding eigenvectors:

[
X_p=V_p\Lambda_p^{1/2}
]

Each row of (X_p) provides the coordinates of one object.

What positive and negative eigenvalues mean

Positive eigenvalues describe dimensions that can be represented in Euclidean space.

Small negative eigenvalues may arise from rounding or mild departures from Euclidean geometry. Large negative eigenvalues indicate that the input dissimilarities have a substantial non-Euclidean component. In that case, researchers should:

• Reconsider the distance measure.
• Examine correction options supported by the software.
• Compare classical MDS with a stress-based method.
• Report the negative eigenvalues rather than ignoring them.

Stress in Metric and Non-Metric MDS

Iterative MDS procedures estimate coordinates that minimise a loss function known as stress.

Let:

• (\delta_{ij}) be the observed dissimilarity.
• (\hat d_{ij}=f(\delta_{ij})) be its modelled disparity.
• (d_{ij}(X)) be the distance between points (i) and (j) in the fitted configuration.
• (w_{ij}) be an optional pair weight.

A common normalised stress formulation is:

[
\text{Stress-1}

\sqrt{
\frac{
\sum_{i<j}w_{ij}\left[\hat d_{ij}-d_{ij}(X)\right]^2
}{
\sum_{i<j}w_{ij}d_{ij}(X)^2
}
}
]

In metric MDS, the disparity may equal the observed dissimilarity. In NMDS, the disparity is a monotonic transformation of the observed dissimilarity.

Raw stress vs normalised stress

Software packages do not always report the same stress definition.

• Raw stress is an unnormalised sum of squared errors.
• Stress-1 is normalised and more suitable for comparison across some solutions.
• S-stress uses squared distances and squared dissimilarities.
• Other programs may report stress per point, residual sums, dispersion accounted for or an (R^2)-type measure.

Researchers must identify exactly which fit statistic their software reports before interpreting or comparing values.

How Multidimensional Scaling Works: Step by Step

Step 1: Define the objects and research purpose

Determine what will be represented by the points: people, brands, countries, stimuli, samples, documents or another set of entities.

State whether the goal is exploratory visualisation, theory evaluation, perceptual mapping, group comparison or dimensionality reduction.

Step 2: Obtain or calculate proximities

Proximities may come from:

• Direct pairwise ratings.
• Sorting or grouping tasks.
• Confusion frequencies.
• Correlations.
• Geographic or network distances.
• Raw-variable distance calculations.
• Biological, genetic or ecological indices.

Step 3: Inspect the proximity matrix

Check the diagonal, symmetry, missing values, impossible entries, duplicated objects, zero distances between different objects and extreme dissimilarities.

Step 4: Choose a distance or dissimilarity measure

The measure must match the type of data and the scientific question. Different measures can create substantially different configurations.

Step 5: Select the MDS model

Use classical MDS for Euclidean or near-Euclidean distances, metric MDS when numerical magnitudes are meaningful, and NMDS when rank order is more defensible than exact magnitude.

Step 6: Fit several dimensional solutions

Fit one-, two-, three- and possibly higher-dimensional models. Do not assume that two dimensions are adequate merely because a two-dimensional plot is convenient.

Step 7: Evaluate fit and stability

Examine:

• Stress or strain.
• Stress-versus-dimension plots.
• Shepard diagrams.
• Pairwise residuals.
• Stress per point.
• Eigenvalues for classical MDS.
• Convergence.
• Agreement across random starts.
• Stability under resampling or minor data changes.

Step 8: Interpret the configuration

Focus first on distances, neighbourhoods, gradients, isolated objects and regions of the map. Give axes substantive labels only when external evidence supports those interpretations.

Step 9: Validate interpretations

Relate the configuration to variables that were not used to construct it. Possible procedures include correlation with external variables, property fitting, regression, permutation tests, Procrustes comparison or confirmatory follow-up studies.

Step 10: Report the complete procedure

Report the objects, proximity source, transformation, distance measure, model, dimensions, algorithm, starting strategy, software, fit statistics, diagnostics and interpretation limitations.

Worked Example: A Smartphone Perceptual Map

Suppose consumers rate the dissimilarity among five smartphone brands:

	Alpha	Beta	Gamma	Delta	Epsilon
Alpha	0	2	7	6	4
Beta	2	0	6	5	3
Gamma	7	6	0	2	5
Delta	6	5	2	0	4
Epsilon	4	3	5	4	0

The matrix suggests that:

• Alpha and Beta are perceived as similar.
• Gamma and Delta are perceived as similar.
• Alpha and Gamma are perceived as very different.
• Epsilon occupies an intermediate position.

A two-dimensional MDS solution might place Alpha and Beta on one side of the map, Gamma and Delta on the other, and Epsilon between the groups.

The spatial arrangement does not automatically reveal why the brands differ. Researchers might hypothesise that one direction represents price and another represents innovation, but these labels require external brand-attribute data or additional consumer research.

The example demonstrates the central interpretive rule: MDS reveals the structure of relationships among objects, but the substantive meaning of the dimensions must be investigated rather than assumed.

Choosing an Appropriate Distance Measure

Euclidean distance

Euclidean distance is the straight-line distance between observations. It is suitable for continuous quantitative variables when their scales are comparable or appropriately standardised.

Manhattan distance

Manhattan distance sums absolute coordinate differences. It may be useful when additive differences across variables are more meaningful than straight-line geometry.

Jaccard dissimilarity

Jaccard dissimilarity is commonly used for binary presence–absence data when joint absences should not be treated as evidence of similarity.

Bray–Curtis dissimilarity

Bray–Curtis dissimilarity is widely used for abundance or compositional community data. Its interpretation and preprocessing requirements should be considered in relation to the field of study.

Correlation-based dissimilarity

A correlation-derived measure can compare profiles rather than absolute levels. However, choices such as (1-r) and (1-|r|) answer different questions.

Gower dissimilarity

Gower dissimilarity can combine continuous, ordinal, nominal and binary variables. Researchers must still examine variable weighting and how missing values are treated.

Why standardisation matters

If Euclidean distance is calculated from variables measured on very different scales, a high-variance variable may dominate the result. Standardisation can reduce this problem, but it also changes the scientific meaning of distance. It should therefore be justified, not applied automatically.

How Many Dimensions Should Be Retained?

Retain the smallest number of dimensions that provides acceptable fit, stable results and a substantively interpretable representation.

No single criterion is sufficient.

Stress-versus-dimension plot

Fit models with increasing dimensionality and plot stress against the number of dimensions. Look for an elbow after which additional dimensions produce only modest improvements.

Eigenvalue plot

For classical MDS, inspect the positive eigenvalues. Large early eigenvalues followed by a marked decline may support a low-dimensional representation.

Shepard diagram

A Shepard diagram compares observed dissimilarities with fitted distances or disparities. A close relationship indicates that the configuration reproduces the proximity structure reasonably well.

For NMDS, the relationship should be monotonic rather than necessarily linear.

Interpretability

A statistically better four-dimensional solution may be difficult to explain or display. Conversely, an easily plotted two-dimensional solution should not be retained when it severely distorts important relationships.

Stability

A credible dimensional solution should be reasonably reproducible across random starts, resamples, subsets or plausible preprocessing alternatives.

How Should Stress Be Interpreted?

Lower stress generally means that fitted distances reproduce the target dissimilarities more accurately. However, stress is affected by the number of objects, the number of dimensions, ties, the data structure and the formula used by the software.

Frequently cited informal categories describe values below approximately .05 as very good, values around .05–.10 as good, values around .10–.20 as increasingly distorted, and values above .20 as potentially problematic.

These categories should not be treated as universal pass–fail rules. Researchers should also examine:

• Stress reduction when another dimension is added.
• The Shepard diagram.
• Object-level and pair-level residuals.
• Convergence and repeated starts.
• Stability of neighbourhoods.
• Comparison with randomised or null data when appropriate.

Dexter et al. (2018) show why stress evaluation can require more context than a single conventional cutoff.

How to Interpret an MDS Plot

Interpret distances before axes

Points close together represent relatively similar objects. Points far apart represent relatively dissimilar objects.

The exact (x)- and (y)-coordinates are usually less important than the pairwise relationships.

Do not attach meaning to axis direction automatically

An MDS configuration can often be translated, rotated or reflected without changing its interpoint distances. A map and its mirror image may therefore be equivalent solutions.

“MDS1” and “MDS2” are coordinate dimensions. Their positive and negative directions do not automatically carry substantive meanings.

Look for several kinds of structure

An MDS map may reveal:

• Tight neighbourhoods.
• Broad group separation.
• Continuous gradients.
• Central and peripheral objects.
• Isolated or unusual objects.
• Curved or circular patterns.

These patterns are exploratory. A visible group is not automatically a statistically verified cluster.

Use external variables cautiously

External variables can be correlated or regressed onto the configuration to help explain it. This is sometimes called property fitting or environmental fitting.

Because axes can rotate, interpretation should consider the full configuration rather than only separate correlations with the first and second coordinates.

Check local distortions

A map with acceptable overall stress can still represent some pairs badly. Pairwise residuals and stress-per-point diagnostics identify objects or relationships that contribute disproportionately to the error.

MDS Compared With Related Methods

Method	Typical input	Primary objective	Interpretation
MDS	Distance or dissimilarity matrix	Preserve pairwise relationships	Relative locations of objects
PCA	Object-by-variable matrix	Preserve maximum variance through linear components	Components described by variable loadings
Factor analysis	Correlation or covariance structure	Model latent factors underlying observed variables	Latent constructs and factor loadings
Cluster analysis	Variables or distances	Form discrete groups	Cluster membership
t-SNE	Feature or distance information	Preserve local probabilistic neighbourhoods	Local grouping; global distances require caution
UMAP	Feature or distance information	Preserve selected local/topological structure	Local and some broader structure, parameter-dependent
Isomap	Feature data converted to neighbourhood-graph distances	Preserve approximate geodesic distances	Unfolded manifold coordinates

MDS vs PCA

Classical MDS applied to Euclidean distances from centred quantitative observations is closely related to PCA. The conceptual emphasis differs:

• PCA begins with variables and identifies directions of high variance.
• MDS begins conceptually with relationships among objects and seeks coordinates that reproduce those relationships.

PCA provides variable loadings directly. Ordinary MDS does not automatically explain its dimensions through original-variable loadings.

MDS vs factor analysis

Factor analysis models associations among observed variables using latent factors and an explicit error structure. MDS represents proximities among objects geometrically.

The methods can address related substantive questions, but they use different inputs and models.

MDS vs cluster analysis

MDS produces continuous coordinates. Cluster analysis assigns or estimates discrete groups.

Researchers sometimes display cluster labels on an MDS map, but agreement between the two methods should be evaluated rather than assumed.

MDS vs t-SNE and UMAP

Traditional MDS gives explicit importance to reproducing pairwise distances or their ordering. t-SNE concentrates strongly on local neighbourhood probabilities, while UMAP constructs and embeds a neighbourhood graph.

Consequently, distances between widely separated clusters in a t-SNE or UMAP plot should not automatically be interpreted in the same way as distances in a well-fitting global MDS model.

Applications of Multidimensional Scaling

Psychology and psychophysics

MDS can represent perceived similarities among colours, sounds, faces, emotions, concepts or symptoms. It can help researchers investigate whether psychological judgments correspond to interpretable dimensions.

Marketing and consumer research

Perceptual maps display how consumers position brands or products relative to one another. Researchers may add preference information or external attributes to investigate market positioning.

Ecology

NMDS is widely used to display compositional differences among ecological communities, often with measures such as Bray–Curtis or Jaccard dissimilarity.

Genetics and bioinformatics

MDS or PCoA can represent genetic distances among individuals, samples or populations. It can reveal broad relatedness patterns, potential outliers and population structure.

Neuroscience

Representational similarity and distance matrices can be embedded to compare neural activity patterns associated with stimuli or cognitive conditions.

Political and social research

Parties, countries, policies or respondents can be mapped from voting patterns, attitude similarities, trade relations or other relational data.

Text and language research

Researchers can map semantic dissimilarities among words, documents or language models. The validity of the map depends on how semantic distance is constructed.

Network and spatial analysis

MDS can support network visualisation, sensor localisation and reconstruction of spatial arrangements from pairwise distances.

Multidimensional Scaling in Modern Research

Modern MDS extends beyond a single two-dimensional perceptual map.

Important extensions include:

• Weighted MDS, in which some pairwise relationships receive more influence.
• Individual-differences scaling, which models variation among judges or groups.
• Unfolding models, which jointly position people and preference objects.
• Constrained MDS, which restricts coordinates using external variables.
• Asymmetric MDS, for directional relationships.
• Spherical or non-Euclidean MDS, for specialised geometric spaces.
• Robust MDS, designed to reduce the influence of contaminated distances.
• Bayesian MDS, which represents uncertainty probabilistically.
• Multi-view MDS, which combines several dissimilarity matrices.
• Landmark and stochastic methods, which reduce the computational burden for large datasets.

These methods require more specialised assumptions and should not be substituted for ordinary MDS merely because they are newer.

Software for Multidimensional Scaling

Software	Relevant procedures	Typical use
R	stats::cmdscale, MASS::isoMDS, vegan::metaMDS, smacof	Classical MDS, NMDS, ecological ordination and advanced stress models
Python	sklearn.manifold.MDS, sklearn.manifold.ClassicalMDS	Metric, non-metric and classical embeddings
IBM SPSS Statistics	ALSCAL and current proximity-mapping procedures	Menu-driven social-science and market-research analysis
MATLAB	cmdscale, mdscale	Classical and nonclassical MDS
XLSTAT	Metric and non-metric MDS models	Spreadsheet-oriented analysis
JMP and similar packages	MDS reports and Shepard diagrams	Interactive analysis and diagnostics

Basic classical MDS in R

# D is a complete symmetric dissimilarity matrix
fit <- cmdscale(
  as.dist(D),
  k = 2,
  eig = TRUE
)

coordinates <- fit$points
eigenvalues <- fit$eig
plot(coordinates, type = "n", asp = 1)
text(coordinates, labels = rownames(D))

Basic NMDS in R with vegan

library(vegan)

set.seed(42)

fit <- metaMDS(
  as.dist(D),
  k = 2,
  trymax = 100,
  autotransform = FALSE
)

fit$stress
stressplot(fit)
plot(fit)

Basic metric MDS in current scikit-learn

import numpy as np
from sklearn.manifold import MDS

# D is a square precomputed dissimilarity matrix
D = np.asarray(D, dtype=float)

model = MDS(
    n_components=2,
    metric_mds=True,
    metric="precomputed",
    normalized_stress="auto",
    random_state=42
)

coordinates = model.fit_transform(D)
print(model.stress_)

Scikit-learn parameter names have changed across versions. Older versions use metric=True and dissimilarity="precomputed". Researchers should check the documentation for the installed version rather than copying code without verification.

Artificial Intelligence and MDS Workflows

Generative AI can assist with:

• Explaining output terminology.
• Drafting initial R or Python code.
• Converting data into a required matrix format.
• Suggesting diagnostic plots.
• Improving figure captions.
• Checking whether a written interpretation distinguishes evidence from speculation.

AI should not be trusted to choose a dissimilarity measure, declare stress acceptable, label axes or diagnose substantive patterns without reference to the actual data and research design.

Researchers should independently verify:

• Package names and parameter syntax.
• The exact stress formula.
• Matrix orientation.
• Treatment of missing values and ties.
• Software-version changes.
• Citations and reported numerical results.

The analysis should remain reproducible without relying on an undocumented conversation with an AI system.

Advantages of Multidimensional Scaling

Flexible input

MDS can work directly with pairwise relationships, including distances that cannot easily be expressed as ordinary measured variables.

Intuitive visualisation

A proximity matrix containing hundreds of values can be summarised in a spatial map.

Broad applicability

The method is useful in behavioural science, marketing, ecology, genetics, neuroscience, linguistics, geography and machine learning.

Metric and ordinal options

Researchers can choose a model that preserves exact values or only rank order.

Compatibility with external validation

Configurations can be related to external variables, group labels or theoretical predictions.

Limitations of Multidimensional Scaling

Information is lost

A two-dimensional map cannot generally preserve all relationships from a complex proximity matrix.

Results depend on the input dissimilarities

Poorly chosen or poorly measured proximities produce a misleading configuration, even when the algorithm converges.

Axes may be difficult to name

Unlike PCA, ordinary MDS does not automatically supply variable loadings that explain each axis.

Iterative solutions may reach local minima

Metric and non-metric stress optimisation can produce different configurations from different starting points.

Outliers can distort the configuration

A small number of extreme objects or unusual pairwise relationships may influence many positions.

Computational cost increases rapidly

A complete analysis of (n) objects involves (n(n-1)/2) unique pairwise relationships and ordinarily requires storage proportional to (n^2).

Visual patterns can be overinterpreted

Humans readily perceive clusters and dimensions even in noisy maps. Diagnostics and external validation are therefore essential.

Common Mistakes

Choosing a distance measure only because it is the software default

The distance measure defines what “similar” means. It must be justified scientifically.

Treating metric and non-metric MDS as interchangeable

Metric MDS attempts to preserve magnitudes. NMDS primarily preserves order.

Interpreting MDS1 as automatically more important

In iterative MDS, axis orientation is arbitrary. Classical MDS dimensions can be ordered by eigenvalues, but their substantive meaning still requires evidence.

Using a single random start

A single solution may be a local minimum. Repeat iterative MDS from multiple starts.

Declaring a solution valid from stress alone

Stress should be combined with Shepard plots, residuals, stability and interpretability.

Hiding a poor two-dimensional fit

A two-dimensional map is convenient, not automatically adequate. Report when three or more dimensions are required.

Treating visible groups as confirmed clusters

MDS is not a hypothesis test or clustering algorithm.

Comparing unaligned configurations coordinate by coordinate

Equivalent configurations may differ by rotation, translation or reflection. Use Procrustes alignment before comparing their coordinates.

Failing to report preprocessing

Standardisation, transformations, missing-data handling and similarity-to-dissimilarity conversion can materially alter the result.

How to Report Multidimensional Scaling in a Research Paper

A reproducible report should include:

1. The objects and analytical sample.
2. How proximities were collected or calculated.
3. Any transformations or standardisation.
4. The selected distance measure.
5. The type of MDS.
6. The requested dimensionality.
7. The estimation algorithm and number of starts.
8. Software and package versions.
9. Stress, strain or eigenvalue information.
10. Shepard-plot or residual findings.
11. Stability checks.
12. The basis for substantive interpretation.
13. Important limitations.

Example methods statement

A two-dimensional non-metric multidimensional scaling solution was fitted to the Bray–Curtis dissimilarities among 64 sampling sites. The analysis was conducted in R using the metaMDS procedure with 100 random starts. Model adequacy was evaluated using normalised stress, a Shepard diagram, pairwise residuals and agreement among repeated solutions. Because the orientation of an NMDS configuration is arbitrary, interpretation focused on interpoint distances and externally fitted environmental variables rather than the signs of the coordinate axes.

Example results statement

The two-dimensional solution produced a normalised stress of .11. The Shepard diagram showed an approximately monotonic relationship, although several pairs involving Site 18 had comparatively large residuals. Three-dimensional MDS reduced stress to .07 but did not materially change the principal neighbourhood relationships. The two-dimensional configuration was retained for presentation, with its remaining distortion noted as a limitation.

These statements are templates. Numerical values and conclusions must be replaced with the actual analysis.

Conclusion

Multidimensional scaling converts pairwise similarities or dissimilarities into a geometric configuration that can reveal neighbourhoods, gradients, groups and unusual objects. A defensible analysis requires more than producing a two-dimensional chart. Researchers must justify the proximity measure, select the correct MDS model, assess fit and stability, interpret distances rather than arbitrary axis directions, and report uncertainty and distortion transparently.

Frequently Asked Questions

What is multidimensional scaling in simple terms?

Multidimensional scaling turns a table of pairwise differences into a map. Similar objects are positioned close together, while dissimilar objects are placed farther apart. The purpose is to make a complex relationship matrix easier to visualise and interpret.

What is multidimensional scaling used for?

MDS is used for perceptual mapping, ecological ordination, genetic-distance analysis, psychological similarity research, market positioning, semantic mapping, network visualisation and other problems in which pairwise relationships are more meaningful than ordinary variables.

What is the difference between metric and non-metric MDS?

Metric MDS attempts to reproduce the numerical magnitudes of dissimilarities. Non-metric MDS attempts primarily to preserve their rank order through a monotonic transformation. NMDS is therefore appropriate when ordering is more trustworthy than exact numerical differences.

Is classical MDS the same as PCoA?

Yes. Classical multidimensional scaling is commonly called principal coordinates analysis, or PCoA. It obtains coordinates from an eigendecomposition of a double-centred squared-distance matrix.

Is MDS the same as PCA?

No. PCA begins with an object-by-variable dataset and identifies linear directions that preserve variance. MDS begins conceptually with pairwise distances or dissimilarities and finds coordinates that reproduce them. Classical MDS on Euclidean distances is mathematically closely related to PCA.

What does stress mean in MDS?

Stress measures disagreement between the distances in the fitted map and the target dissimilarities or disparities. Lower stress generally represents less distortion, but the formula, number of objects, dimensionality and data structure must be considered.

What do MDS1 and MDS2 mean?

MDS1 and MDS2 are the first two coordinate dimensions of the configuration. They do not automatically represent named variables or constructs. Their interpretation should be supported with theory, external variables or additional evidence.

How many dimensions should an MDS solution use?

Use the smallest dimensionality that provides adequate fit, stable relationships and useful interpretation. Consider stress reduction, eigenvalues, Shepard diagrams, residuals, reproducibility and substantive meaning rather than relying on a single threshold.

Can MDS analyse categorical data?

MDS can analyse dissimilarities calculated from categorical or mixed data when the selected measure is appropriate. Examples include Jaccard dissimilarity for certain binary data and Gower dissimilarity for mixed variable types.

Is MDS a clustering method?

No. MDS produces continuous spatial coordinates. Clusters may appear visually, but they are not automatically statistically validated. A separate clustering or inferential procedure is required when group identification is the research objective.