Consensus MDS: Aggregating Ideal Points

• CoMDS is a method that aggregates diverse ideal point estimates into a single spatial representation reflecting shared ideological positions.
• It uses diagonal rescaling and SMACOF-based optimization to align multiple distance matrices and handle missing data.
• The approach isolates stable, cross-source signals by mitigating endogeneity and source-specific noise in political measurement.

Consensus Multidimensional Scaling (CoMDS) is a distance-based aggregation method that takes in multiple, heterogeneous estimates of political actors’ “ideal points” and returns a single, common embedding that preserves the cross-source structure those estimates share. It is designed for the way applied scholars actually use ideal points in practice: as general indicators of ideological position or extremity, irrespective of the particular data source or behavioral model that generated them. In that sense, CoMDS captures the shared, stable associations of a set of underlying ideal point estimates and can be interpreted as their common spatial representation. In a broader multi-view setting, closely related work formulates the same general problem as multidimensional scaling on multiple input distance matrices, seeking a single consensus embedding from several dissimilarity views (Meisels et al., 8 Jan 2026, Bai et al., 2016).

1. Problem setting and motivation

CoMDS arises from a setting in which researchers have many estimates of candidates’ positions derived from different data sources and models, including roll-call votes, campaign finance, speeches, platforms, social media, endorsements, and surveys. These sources often exhibit weak within-party relationships and strong cross-party separation. Each source mixes the candidate’s general positioning, which is the shared component, with domain-specific idiosyncrasies, which are source-specific components. Different measures are nevertheless used interchangeably in most substantive analyses, even though weak relationships raise questions about the extent to which they capture a shared quantity rather than idiosyncratic, domain-specific factors (Meisels et al., 8 Jan 2026).

A central motivation is endogeneity. Using an ideal point and an outcome measured from the same domain creates endogeneity concerns through mechanical correlations and measurement co-production. CoMDS addresses this by extracting only the structure shared across sources and delivering a common spatial representation of candidates. In practice, it recovers a consensus embedding that is invariant to each source’s rotation, scale, and shift and tolerant to missingness; it enables relating consensus ideal points to source-domain variables without relying on any single domain’s idiosyncrasies; and it isolates source-specific features via a projection residual, helping clarify what conclusions hinge on the domain rather than the underlying ideology (Meisels et al., 8 Jan 2026).

The same general problem can be stated outside political measurement. In the multi-view MDS formulation, the inputs are multiple distance or dissimilarity matrices over the same set of objects, and the target is a single consensus embedding shared across all views. This broader framing treats each input distance matrix as one view and asks how to do multidimensional scaling on multiple input distance matrices in a consensus sense (Bai et al., 2016).

2. Formal model and objective functions

Let $C$ be the set of $n$ candidates, and let $S$ denote the number of sources, indexed by $s=1,\dots,S$. For each source $s$, one may observe either a one-dimensional estimate $\theta_i^{(s)}$ for candidate $i$ or a higher-dimensional embedding $z_i^{(s)} \in \mathbb{R}^{r_s}$. From each source, one constructs a pairwise dissimilarity matrix $D^{(s)}$ with entries $d_{ij}^{(s)} \ge 0$. If $n$0 is one-dimensional, then

$n$1

If $n$2 is multi-dimensional, then

$n$3

after source-specific standardization (Meisels et al., 8 Jan 2026).

The CoMDS target is a consensus embedding $n$4, with $n$5 or $n$6 in most applications, that preserves these source-specific pairwise dissimilarities up to source-specific diagonal rescalings. Let $n$7 be a missingness or availability indicator for pair $n$8 in source $n$9, and let $S$0 be a source-level weight. The metric CoMDS stress objective is

$S$1

Here $S$2 is a $S$3 diagonal matrix allowing each source to rescale the consensus axes, so sources can have different effective units across dimensions (Meisels et al., 8 Jan 2026).

A non-metric CoMDS variant replaces metric matching with source-specific monotone transformations $S$4:

$S$5

where $S$6 is a source-provided proximity or dissimilarity and $S$7 is estimated via isotonic regression mapping consensus distances to source dissimilarities. This variant preserves rank orders rather than metric scales within sources (Meisels et al., 8 Jan 2026).

Construction of $S$8 is part of the model definition. For one-dimensional $S$9, the source is centered and rescaled to unit variance or unit interquartile range before taking absolute differences. For multi-dimensional $s=1,\dots,S$0, each dimension is standardized within source to unit variance, and Euclidean distances are then used. If one begins from similarities, they may be converted to dissimilarities by $s=1,\dots,S$1 or another monotone transformation, followed by non-metric CoMDS (Meisels et al., 8 Jan 2026).

Identifiability follows standard MDS invariances. MDS-type embeddings are invariant to translation, rotation, and global scale. CoMDS enforces identifiability by centering the columns of $s=1,\dots,S$2, fixing average squared distance to $s=1,\dots,S$3, and orienting axes for interpretability. For $s=1,\dots,S$4, the sign may be set so larger values denote greater conservatism; for $s=1,\dots,S$5, Procrustes alignment to a reference or anchoring with known ideologues may be used. Sign flips or rotations across runs with different initializations do not affect distances (Meisels et al., 8 Jan 2026).

3. Estimation, optimization, and computational structure

CoMDS can be estimated by an extension of SMACOF, an iterative majorization algorithm for MDS that guarantees monotone stress decrease and linear convergence. A practical scheme is alternating minimization. In Step A, one updates $s=1,\dots,S$6 given the current $s=1,\dots,S$7 using a SMACOF-style update. In Step B, one updates each diagonal $s=1,\dots,S$8 given $s=1,\dots,S$9 via a weighted nonnegative least squares fit on squared distances. The alternation typically converges in tens of iterations (Meisels et al., 8 Jan 2026).

For the $s$0-update, define the current fitted distances

$s$1

Then construct per-source majorization matrices $s$2 with entries

$s$3

Here $s$4 is a small constant to avoid division by zero; if $s$5, one sets $s$6. With

$s$7

where

$s$8

the SMACOF update is

$s$9

After the update, one re-centers the columns of $\theta_i^{(s)}$0 to mean zero and rescales to fix the global scale; if using $\theta_i^{(s)}$1, one may optionally orthogonalize columns to improve numerical stability (Meisels et al., 8 Jan 2026).

For the diagonal scaling update, write $\theta_i^{(s)}$2. The squared fitted distance is

$\theta_i^{(s)}$3

Defining $\theta_i^{(s)}$4 and $\theta_i^{(s)}$5, one fits $\theta_i^{(s)}$6 by weighted nonnegative least squares:

$\theta_i^{(s)}$7

Then $\theta_i^{(s)}$8. This subproblem is convex and can be solved efficiently, including by projected gradient or NNLS (Meisels et al., 8 Jan 2026).

Initialization and computational details are explicit. One may initialize $\theta_i^{(s)}$9 using classical MDS on the average distance matrix $i$0 with

$i$1

One may initialize $i$2 to identity or solve the NNLS once given $i$3. A standard stopping rule is relative stress decrease less than $i$4 or reaching the maximum number of iterations. Complexity per iteration is $i$5 for $i$6 and $i$7 for NNLS, often faster in practice. Missingness is handled through $i$8, so candidates missing from a source are still embedded via other sources (Meisels et al., 8 Jan 2026).

4. Diagnostics, validation, and treatment of endogeneity

CoMDS is accompanied by a suite of diagnostic tools intended to aid practical usage. A basic diagnostic is per-source fit. Define the source-specific sum of squared errors

$i$9

and

$z_i^{(s)} \in \mathbb{R}^{r_s}$0

with $z_i^{(s)} \in \mathbb{R}^{r_s}$1 the mean of the observed $z_i^{(s)} \in \mathbb{R}^{r_s}$2. Then the per-source $z_i^{(s)} \in \mathbb{R}^{r_s}$3-type fit is

$z_i^{(s)} \in \mathbb{R}^{r_s}$4

A second diagnostic is the relative error contribution

$z_i^{(s)} \in \mathbb{R}^{r_s}$5

which lies in $z_i^{(s)} \in \mathbb{R}^{r_s}$6 and sums to $z_i^{(s)} \in \mathbb{R}^{r_s}$7 across sources; larger values indicate more source-specific idiosyncrasy and less shared structure (Meisels et al., 8 Jan 2026).

Leave-one-source-out stability assesses how much the consensus depends on any one source. One estimates CoMDS with all sources to obtain $z_i^{(s)} \in \mathbb{R}^{r_s}$8, then re-estimates after leaving out source $z_i^{(s)} \in \mathbb{R}^{r_s}$9 to obtain $D^{(s)}$0. For $D^{(s)}$1, the similarity metric is

$D^{(s)}$2

For $D^{(s)}$3, subspace correlation is defined by

$D^{(s)}$4

where the $D^{(s)}$5 are singular values of $D^{(s)}$6. Values near $D^{(s)}$7 indicate stability. Dimension selection proceeds by plotting stress versus $D^{(s)}$8 or total $D^{(s)}$9 versus $d_{ij}^{(s)} \ge 0$0 and looking for elbows, or by cross-validating on held-out pairs or sources and measuring predictive $d_{ij}^{(s)} \ge 0$1. In political applications, $d_{ij}^{(s)} \ge 0$2 or $d_{ij}^{(s)} \ge 0$3 usually suffices (Meisels et al., 8 Jan 2026).

Candidate influence and uncertainty can also be examined. Influence may be assessed by recomputing stress after deleting candidate $d_{ij}^{(s)} \ge 0$4 or by computing the leave-one-candidate-out correlation between full $d_{ij}^{(s)} \ge 0$5 and $d_{ij}^{(s)} \ge 0$6. Uncertainty may be assessed by bootstrapping pairs within sources or bootstrapping sources, then reporting standard errors or confidence bands for $d_{ij}^{(s)} \ge 0$7. Residual maps based on

$d_{ij}^{(s)} \ge 0$8

can reveal systematic residual patterns suggestive of source-specific biases, including agenda or measurement artifacts (Meisels et al., 8 Jan 2026).

Endogeneity is a principal use-case rather than a secondary consideration. CoMDS mitigates endogeneity by purging source-specific co-production. If one uses campaign-finance-based ideal points and relates them to campaign-finance outcomes, estimates may be mechanically inflated. The consensus embedding is trained to preserve what multiple sources agree on, and source-specific distortions are downweighted through the multi-source stress. Recommended practice includes split-source analyses, in which one re-estimates CoMDS excluding the outcome’s source or assigns that source a small weight, and projection decomposition, in which outcomes are regressed on the shared component only (Meisels et al., 8 Jan 2026).

Projection decomposition makes the shared-versus-idiosyncratic distinction explicit. After obtaining $d_{ij}^{(s)} \ge 0$9, define

$n$00

the projection onto the consensus subspace. For source $n$01, the shared component of the original coordinates $n$02 is $n$03, and the idiosyncratic residual is

$n$04

By construction, $n$05 is orthogonal to $n$06. This tool quantifies how much a source’s measure deviates from the consensus and can be related to external domain variables to understand source-specific biases (Meisels et al., 8 Jan 2026).

5. Empirical applications in congressional ideal-point estimation

In the U.S. House application for 2016–2024, the method combines NOMINATE, campaign finance scores, and platform positions. The sources are described as NOMINATE, which is roll-call based, one- to two-dimensional, and available for incumbents only; campaign finance scores, including static, dynamic, and DW-DIME variants; and platform positions, based on Wordfish-scaled campaign website issue positions. The analysis includes all candidates with at least two of the three sources, comprising approximately $n$07 candidate-year observations. A one-dimensional consensus embedding displays strong partisan bimodality, and within-party correlations with the consensus are stronger than cross-source correlations alone (Meisels et al., 8 Jan 2026).

The paper reports three substantive findings. First, on roll-call partisan disloyalty, consensus points reveal that moderates defect more, whereas NOMINATE-only analysis exaggerates magnitudes because of common-source endogeneity. Comparing source-specific measures yields conflicting signs or magnitudes, while the consensus resolves the relationship. Second, on fundraising base, measured as the number of unique donors, the consensus indicates that Democrats’ extremism and Republicans’ moderation have distinct effects; campaign finance scores alone suggest opposite conclusions and much larger magnitudes. Third, on lexical diversity, measured by platform CTTR, the consensus reveals that moderation associates with greater rhetorical sophistication within party; platform-based ideal points alone overstate magnitudes, and other sources sometimes disagree in sign (Meisels et al., 8 Jan 2026).

These applications define the concrete scope of CoMDS in the political science setting. The method is intended for cases in which ideal points are used as domain-agnostic measures of ideology or extremity across heterogeneous sources, especially when the researcher plans to study outcomes measured in any one domain, including votes, donors, speech, or platforms. A plausible implication is that the method is particularly useful when the substantive target is a stable ideological component rather than a domain-bound behavioral signature, because the projection residual can then be used to study agenda control, rhetoric, donor targeting, or other source-specific phenomena orthogonal to the underlying ideology (Meisels et al., 8 Jan 2026).

6. Relation to multi-view MDS, alternative methods, limitations, and extensions

A broader consensus-MDS formulation is given by Multi-View Multidimensional Scaling (MVMDS), which takes $n$08 distance matrices $n$09 over the same set of objects and learns a single consensus embedding $n$10 together with view weights $n$11. Its objective is

$n$12

subject to

$n$13

where $n$14 is a weight controller. Given $n$15, the weights admit the closed-form update

$n$16

with

$n$17

The $n$18-step uses a SMACOF-style majorization update $n$19, and with no missing entries the update simplifies to

$n$20

This formulation makes consensus and complementarity explicit: views with lower per-view stress receive higher weight, but $n$21 prevents trivial collapse to a single view (Bai et al., 2016).

The political-science CoMDS and the multi-view MVMDS formulation share a common structural idea: a single embedding is estimated so that multiple dissimilarity structures are jointly respected. CoMDS differs by emphasizing source-specific diagonal rescalings, tolerance to missingness, diagnostic decompositions, and endogeneity mitigation in applications involving heterogeneous ideal point estimators. MVMDS, by contrast, is strictly metric MDS in the paper’s formulation and does not use non-metric monotone transformations. This suggests that CoMDS occupies a more application-specific position within a broader family of consensus MDS methods (Meisels et al., 8 Jan 2026, Bai et al., 2016).

Alternative methods have distinct assumptions and failure modes.

Method	Pros	Cons
Bayesian IRT joint scaling	coherent probabilistic inference	misspecification risk; heavy computation; stringent functional-form assumptions
MD2S	explicit decomposition	assumes normality and linearity; sensitive to rotations/scales
PCA on stacked measures	simple and fast	sensitive to source rotations/scales; overweights sources with more dimensions
Joint or bridge alignments	useful with a gold standard	requires anchors or overlap and a trusted reference
JIVE and related consensus methods	shares CoMDS’ spirit	typically assumes linear decompositions

CoMDS has three stated advantages. It is invariant to source rotations, scales, and shifts because it operates on distances and learns source-specific diagonal scales. It is flexible with missingness and balances across sources rather than over-weighting high-dimensional ones. It does not impose a parametric behavioral model and therefore respects each source’s domain-specific generative assumptions. Its stated limitations are equally clear: it uses only pairwise dissimilarities rather than raw data likelihoods and has no built-in probabilistic uncertainty; the choice of distance and weights matters and can become problematic if a source is very noisy or adversarial; and the algorithm converges to local minima, so initialization matters, even though SMACOF tends to be stable in practice (Meisels et al., 8 Jan 2026).

Several extensions are proposed. Dynamic CoMDS estimates time-indexed consensus embeddings $n$22 with temporal smoothness penalties, for example

$n$23

Hierarchical or multi-level CoMDS introduces group-level embeddings and penalties for nested actors. Incorporating uncertainty can be done by weighting pair contributions by inverse variance or adjusting source weights to reflect reliability. Probabilistic formulations treat observed distances as noisy observations of $n$24 with additive Gaussian noise and estimate by maximum likelihood or Bayesian posterior sampling while retaining SMACOF-like majorization for MAP. Theoretical guarantees are local rather than global: the SMACOF majorization step guarantees non-increasing stress and linear convergence to a local minimum, and the NNLS updates for $n$25 are convex and converge to a global minimum for each fixed $n$26 (Meisels et al., 8 Jan 2026).