Modeling Ranking Concordance, Dispersion, and Tail Extremes with a Joint Copula Framework

1. Introduction

1.1. The Problem of Ranking Analysis

Ranking analysis underpins consequential decisions across domains [1,2]. In academia, peer review allocates resources and recognizes contributions [3,4]; in sports, rankings drive championships and seedings [5,6]; in medicine, ordered risk stratification informs diagnosis and treatment [7,8]; and in business, rankings shape performance evaluation and strategy [9,10].

Despite this ubiquity, methodological practice remains fragmented [11,12]. Analysts routinely summarize concordance (agreement among raters), dispersion (concentration of rank usage), and extremeness (prevalence of unusually high/low ranks) as if these were independent. In real systems, however, these properties co–move: patterns that boost agreement can also compress dispersion and alter the frequency of extremes, and vice versa. Treating them in isolation risks mischaracterizing the integrity of the ranking process.

1.2. Limitations of Current Approaches

Classical measures—Kendall’s W for multi-rater concordance [13] and rank correlations in the Spearman family [14]—are informative but siloed. When multiple characteristics are considered, they are typically tested separately under implicit independence. This creates three well–known pitfalls.

First, summary indices ignore constraints among ranking properties: high agreement mechanically restricts allowable dispersion and alters the chance of extreme scores [15]. Second, separate testing inflates false positives when dependence is present [17,18]. Third, independence-based reporting obscures whether “agreement” reflects genuine consensus or shared structure in how ranks are distributed and how extremes co-occur across raters.

Copula ideas have appeared around ranking problems, but prior uses generally address pairwise association or specialized tasks rather than a system-level diagnosis of how agreement, dispersion, and extremeness interact. To our knowledge, no widely adopted framework operationalizes these interactions to audit ranking reliability and surface dependence-driven artifacts.

1.3. The Copula Perspective

Copula theory separates marginal behavior from dependence [19], enabling flexible multivariate modeling with appropriate margins [20,21]. It has proved effective in finance [22,23], environmental extremes [24,25], and risk settings where tail co-movements matter [26,27]. For ranking systems, this perspective is compelling because the events with greatest practical weight are often joint extremes (e.g., multiple raters pushing the same items to the top or bottom).

In the present work, copulas are not used to posit a monolithic trivariate latent model of “concordance–dispersion–extremeness.” Instead, we take a pragmatic, data-driven dependence view: estimate upper-tail association pairwise between raters using Gumbel copulas on mid-rank pseudo-observations, summarize tail dependence with both pairwise parameters and an aggregate scaled measure, and read those in concert with global concordance and information-theoretic screening.

1.4. Research Contribution and Novelty

We introduce the Concordance–Dispersion–Extremity Framework (CDEF), a pipeline that reinterprets ranking reliability through the lens of dependence. Its novelty is not a new copula family; it is a ranking-specific diagnostic that (i) detects the appropriate ranking regime, (ii) quantifies tail-dependent agreement between raters, (iii) selects a dispersion mechanism consistent with the data, and (iv) integrates these ingredients to separate genuine consensus from dependence-induced artifacts:

• Regime detection. Starting from long-format data, CDEF pivots to wide form and auto-detects forced vs. non-forced rankings (ranking regime). For forced permutations with dependence, it scores Mallows structure; for forced independence, it compares to the uniform-permutation baseline; for non-forced rankings, it distinguishes independent (multinomial) from dependent regimes via ${\chi }^2$ screening.
• Tail-dependence estimation. Using mid-rank pseudo-observations and the Kendall link $\theta =1/(1-\tau )$ where $\tau$ is Kendall’s $\tau$, CDEF fits bivariate Gumbel copulas to every rater pair and aggregates these into an extremeness summary attuned to upper-tail co-movement [19,20,21,23,26,27].
• Dispersion mechanism. A ${\chi }^2$ test distinguishes independent rank allocation (multinomial) from without-replacement structure (hypergeometric), aligning inference with the data-generating process [40].
• Screening and reporting. Global concordance (W), mutual information across representative pairs, and likelihood summaries provide complementary views; decision rules then diagnose genuine vs. phantom agreement—phantom when high apparent concordance is largely attributable to tail dependence and shared dispersion patterns rather than stable consensus.

This system-level reading of dependence addresses the core failure mode of traditional practice: high W by itself can be an echo of coordinated extremes or constrained rank usage. By design, CDEF shows when that apparent agreement dissolves once tail co-movement and dispersion mechanics are made explicit—delivering a practically actionable notion of phantom concordance that classical summaries miss, while remaining compatible with established copula methodology [22,23,24,25,26,27] and long-standing rank theory [13,14,15].

2. Background and Related Work

2.1. Ranking Analysis Fundamentals

Ranking analysis concerns statistical evaluation of ordered preferences or performance assessments across multiple entities [30,32]. Work in this area spans pairwise and multivariate formulations, from classic correlation measures to dependence models that handle complex rater systems.

Three characteristics organize the behavior of ranking systems. Concordance captures overall agreement among m raters ranking n entities, commonly summarized by Kendall’s W [13]:

$$W={\displaystyle \frac{12{\sum }_{i=1}^n{(R_i-\overline{R})}^2}{m^2(n^3-n)}},$$

(1)

where $R_i$ is the sum of ranks for entity i and $\overline{R}=m(n+1)/2$ is the expected sum under random assignment.

Dispersion (or concentration) refers to how rank assignments spread across the available scale, independent of inter–rater agreement [30,33]. Depending on the sampling mechanism, dispersion can reflect independent draws (naturally modeled by a multinomial process) or without–replacement structure (aligned with multivariate hypergeometric mechanisms) [40]. Because the choice has inferential consequences, we treat it as a data-driven decision rather than fix a single summary index in the background discussion.

Extremeness describes the prevalence and influence of unusually high or low ranks. Beyond marginal “extreme response styles” [34], the phenomena that most affect downstream decisions often involve co–occurring extremes across raters. This motivates a tail-dependence view rooted in extreme value ideas [41,42,43,44,45] and copula modeling [26,27], where extremeness is operationalized through the strength of upper-tail association rather than only marginal outliers.

2.2. Traditional Ranking Evaluation Methods

2.2.1. Concordance Measures

Kendall’s W is the standard for multi-rater concordance [13]. Related agreement coefficients (e.g., Fleiss’ $\kappa$[35] and Krippendorff’s $\alpha$ [36]) assess agreement beyond chance, but can be challenging to interpret in rank-permutation settings and may have non-Gaussian asymptotics that complicate inference in discrete data analysis [37,38].

2.2.2. Dispersion Analysis

Dispersion in ranking data can arise from fundamentally different stochastic mechanisms. When rank assignments are independent, category counts follow a multinomial law [39]:

$$P(X_1=x_1,\dots ,X_k=x_k)={\displaystyle \frac{N!}{x_1!\dots x_k!}}{p}_1^{x_1}\dots p_k^{x_k}.$$

(2)

When assignments are dependent through sampling without replacement, a multivariate hypergeometric model is more appropriate [40]:

$$P(X_1=x_1,\dots ,X_k=x_k)={\displaystyle \frac{{\prod }_{i=1}^k\left(\genfrac{}{}{0pt}{}{N_i}{x_i}\right)}{\left(\genfrac{}{}{0pt}{}{N}{n}\right)}},$$

(3)

with $N={\sum }_i{N}_i$ and $n={\sum }_i{x}_i$. Foundational work notes that misspecifying this mechanism biases dispersion conclusions [30]; accordingly, empirical pipelines benefit from a data-driven choice between regimes (see Materials and Methods).

When rankings are forced permutations, dispersion is structured not by sampling frequencies but by proximity to a modal ranking. In this setting, the Mallows model [31] provides a principled baseline:

$$P\left(\pi \right)={\displaystyle \frac{1}{Z\left(\varphi \right)}}exp\left[-\varphi d(\pi ,{\pi }_0)\right],$$

(4)

where $\pi$ is a ranking, ${\pi }_0$ is the consensus ordering, $d(\pi ,{\pi }_0)$ is a permutation distance (typically Kendall’s tau), and $\varphi$ controls concentration. As $\varphi \to 0$, the model approaches a uniform distribution over permutations; as $\varphi$ increases, rankings cluster tightly around ${\pi }_0$. Misspecifying this structured dependence as multinomial or hypergeometric can lead to inflated or deflated dispersion estimates, masking the underlying mechanism.

2.2.3. Extremeness Detection

Extreme value theory provides limiting models for maxima and minima [41,42,43,44,45]. While marginal Generalized Extreme Value (GEV) families are informative, ranking systems often hinge on joint extremes—e.g., distinct raters simultaneously producing very high (or very low) ranks. This motivates tail-dependent multivariate models [26,27] in which extremeness is assessed through the probability of co-occurring extremes rather than only the marginal tail thickness.

2.3. Copula Theory in Dependence Modeling

2.3.1. Theoretical Foundations

Sklar’s theorem decomposes any multivariate distribution into its marginals and a copula that encodes dependence [19,46,47]:

$$F(x_1,\dots ,x_d)=C\left(F_1\left(x_1\right),\dots ,F_d\left(x_d\right)\right).$$

(5)

This separation is crucial when marginal scales are discrete (ranks) but dependence—especially in the tails—is the quantity of interest [26,27]. In ranking contexts, each observation is assigned a rank $r\in \{1,\dots ,n\}$, where r denotes the rank of the observation and n is the total number of ranked items. These ranks are then mapped to pseudo-observations on the unit interval $[0,1]$ via the mid-rank transformation

$$‘U={\displaystyle \frac{r-0.5}{n}},$$

(6)

which centers the ranks within their respective intervals. This transformation is consistent with empirical copula practice and order-statistic theory [61,63,66].

2.3.2. Copula Families for Ranking Applications

Different copula families capture distinct dependence shapes. Elliptical (Gaussian, t) copulas model symmetric dependence [48], while Archimedean families can express asymmetric tail features. The Gumbel copula emphasizes upper tail dependence and is therefore well-suited to settings where unusually high (or concordantly extreme) ranks dominate decisions [20,21,23]:

$$C(u_1,\dots ,u_d)=exp\left(-{\left(\sum _{i=1}^d{(-ln{u}_i)}^{\theta }\right)}^{1/\theta }\right),\theta \ge 1.$$

(7)

In practice, system-level evaluation can proceed by estimating pairwise bivariate Gumbel copulas for rater pairs using pseudo-observations, linking Kendall’s $\tau$ to $\theta$ via $\theta =1/(1-\tau )$, and summarizing extremeness by aggregating pairwise parameters. To reflect both local (pairwise) and global (all-raters) agreement, an auxiliary summary that scales the Gumbel parameter by $(1+W)$ can be tracked alongside pairwise estimates [20,21].

2.3.3. Recent Advances in Copula Applications

Copulas have proved effective in finance and risk [22,23], environmental extremes [24,25], and biostatistics [28,29]. High-dimensional extensions leverage vines [49,50] and factors [51], while goodness-of-fit procedures continue to mature [52,53,55]. Despite this breadth, direct applications to system-level ranking evaluation remain rare; most work targets pairwise rank association or theoretical properties of permutations [60] rather than full diagnostics of ranking systems.

2.4. Comparison with Recent Multivariate Ranking and Copula-Based Methods

To situate the method explicated here (CDEF), we provide Table 1 which contrasts our aim—joint evaluation of concordance, dispersion, and extremeness in ranking systems—with recent approaches that employ copulas or address multivariate ranking from other angles [56,57,58,59,60]. CDEF’s distinctive contribution is not a new copula family per se, but a ranking-specific diagnostic that makes dependence structure the primary object of inference.

2.5. Gaps in Current Literature and CDEF’s Unique Contribution

Although ranking analysis and copula methodology are both mature, we found no prior framework that jointly models concordance, dispersion mechanism, and tail-dependent extremeness for system-level evaluation. Traditional reliability metrics [37,38] often treat high inter-rater correlation as genuine consensus, even when shared dispersion patterns or tail behaviors drive much of the apparent agreement. Existing copula work in ranking contexts emphasizes estimation procedures [61], software and algorithms [54,62,64], asymptotics [63], and special domains [56,57,58,59], but not the overarching diagnostic question of whether “consensus” is genuine or dependence-induced.

CDEF fills this gap with a theoretically grounded yet tractable pipeline: mid-rank pseudo-observations [61,63,66], pairwise Gumbel copulas for upper-tail dependence with the $\tau \to \theta$ link [20,21,23], a data-driven dispersion regime choice (multinomial vs. without-replacement) [30,40], and model comparison using likelihood summaries and dependence-aware diagnostics [26,27]. This combination provides practical tools for identifying phantom reliability—apparent agreement explained by shared dependence—versus genuine consensus, while remaining compatible with advances in high-dimensional dependence (vines [49,50]) and goodness-of-fit testing [52,53,55].

3. The Concordance–Dispersion–Extremeness Framework (CDEF)

3.1. Conceptual Foundation

CDEF tackles a core limitation in ranking analysis: treating concordance, dispersion, and extremeness as if they were independent features. In many applied settings, dependence among these components is structural, not incidental [30,32]. Ignoring it can inflate evidence of “agreement” and misstate uncertainty [29].

CDEF therefore (i) quantifies global agreement via Kendall’s W[13], (ii) diagnoses cross-rater information sharing and (in)dependence via mutual information and ${\chi }^2$ tests [17,18], and (iii) models tail comovement of raters using a copula layer that is sensitive to upper-tail clustering (Gumbel) [19,20,21,22,23]. The framework then chooses an appropriate dispersion model and baseline likelihood according to the ranking regime (forced vs. non-forced) and dependence diagnostics. Rather than reporting pseudo–conditional probabilities across heterogeneous scales, CDEF summarizes the three components as normalized contributions that reveal where the dependence “mass” resides [20,23].

Figure 1 visualizes the CDEF joint surface for a Gumbel dependence regime with $\theta =5.00$, evaluated at observed concordance $W=0.85$ and concurrence $\mathrm{MI}=1.30$. The rising ridge toward the upper–right shows probability mass concentrating where high W co-occurs with high MI—consistent with the “GENUINE: Natural agreement” diagnosis. The surface height (joint density weighted by $\theta$) reflects upper-tail dependence; as $\theta$ increases, the surface steepens and curvature intensifies, emphasizing extreme co-movements. Reproducible, interactive code that generates this plot and allows $\theta$, W, and MI to be varied is available at https://github.com/dustoff06/CDEF.

3.2. Operational Formulation

3.2.1. Data Layout and Regime Detection

Rankings arrive in long format with columns Rater, Ratee, Ranking; we pivot to a wide $N\times m$ matrix (items × raters) and drop rows with missing ranks [30,32]. For each rater j, if the column is a permutation of $\{1,\dots ,N\}$ (no ties), the dataset is labeled forced; otherwise non-forced. This branching determines the dispersion baseline and the likelihood used downstream.

3.2.2. Concordance

Global agreement is measured by Kendall’s W[13,30]. Let $R_i$ be the sum of ranks for item i, $\overline{R}$ their mean, and $S={\sum }_{i=1}^N{(R_i-\overline{R})}^2$. Then Kendall’s W is

$$W={\displaystyle \frac{12S}{m^2(N^3-N)}}\in [0,1].$$

3.2.3. Concurrence and (In)dependence Diagnostics

For each pair of raters $(j,k)$ we construct a joint contingency of their rank columns and extract two dependence measures. When rankings are forced (strict permutations with no ties), we form an exact rank–by–rank crosstab (no binning). When rankings are non-forced (ties allowed), we build a binned 2D histogram using an adaptive rule (e.g., $b=max\{5,\lceil \sqrt{n}\rceil \}$), which stabilizes sparse cells for moderate n. To avoid $log\left(0\right)$, a small Laplace smoothing constant $\alpha >0$ is added to every cell before normalization. From each contingency we compute (a) mutual information (MI) in natural units,

$$I(j;k)=H\left(X_j\right)+H\left(X_k\right)-H(X_j,X_k),$$

where $H(·)$ denotes Shannon entropy in natural units (nats) [16], and (b) a Pearson ${\chi }^2$ test of independence on the unsmoothed counts [17,18]. We summarize MI at the panel level by averaging across all $\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)$ rater pairs (and optionally reporting range or IQR). For a panel-level hypothesis test of independence, we aggregate the pairwise p-values using Fisher’s method: $X^2=-2{\sum }_{j<k}log{p}_{jk}\sim {\chi }_{2\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)}^2$ under the global null. Binning rule, smoothing $\alpha$, and MI units (nats vs. bits) are fixed and reported for reproducibility; in sensitivity checks, conclusions were invariant to using alternative binning heuristics (e.g., Sturges or Freedman–Diaconis).

3.2.4. Extremeness via Copulas (Upper-Tail Co-Movement)

Ranks are converted to pseudo-uniforms by the empirical CDF as previously explicated which centers each order statistic in its probability interval and aligns with empirical-copula practice and order-statistic theory [61,63,66]. For each rater pair we fit a Gumbel copula $C_{\theta }$ capturing upper-tail dependence [19,20,21]. The Gumbel parameter relates to Kendall’s $\tau$ via

$$\tau =1-{\displaystyle \frac{1}{\theta }}\Rightarrow {\widehat{\theta }}_{\tau }={\displaystyle \frac{1}{1-\widehat{\tau }}},$$

and we also compute the average copula log-likelihood per observation as a fit diagnostic [20,23]. To reflect interaction between global concordance and tail dependence, we report

$${\theta }_{\mathrm{scaled}}={\widehat{\theta }}_{\tau }·(1+W),$$

a scalar summary that rises when both pairwise tail comovement and system-wide agreement are high [26,27].

3.2.5. Dispersion Baselines and Model Selection

CDEF selects the dispersion/likelihood branch by combining the regime label (forced/non-forced) with the ${\chi }^2$/MI diagnostics:

• Forced rankings and dependence detected: fit a Mallows model

  $$P\left(\sigma \right)\propto exp\{-\varphi d_K(\sigma ,{\sigma }_0)\},$$

  where $d_K$ is Kendall distance and ${\sigma }_0$ is a Borda-based consensus. We report an approximate Mallows log-likelihood and average distance to consensus [30].
• Forced rankings + no dependence: baseline is the uniform distribution over $S_N$ with log-likelihood $-log(N!)$ (Stirling approximation for large N) [30,65].
• Non-forced rankings + no dependence: baseline is a multinomial likelihood over observed $(\mathrm{team},\mathrm{rank})$ cells (MLE via empirical frequencies) [17,18].
• Non-forced rankings + dependence: dispersion is left nonparametric; we characterize dependence structure via the copula layer and report diagnostics (pairwise $\widehat{\theta }$, average copula log-likelihood). For concentrated draws, we discuss links to multivariate hypergeometric sampling [40].

3.2.6. Copula Selection and Tail Dependence

While several copula families can model dependence structures in multivariate distributions, the Gumbel copula was selected for this analysis based on its theoretical properties and alignment with the characteristics of extreme software dependencies. The Gumbel copula exhibits upper tail dependence, capturing the probability that multiple variables jointly take extreme high values, which is particularly relevant for dependency analysis where the primary concern is simultaneous occurrence of extreme behavior across multiple components—such as high coupling, complexity, or instability—posing the greatest risk to system maintainability and reliability. In contrast, the Clayton copula exhibits lower tail dependence, focusing on joint occurrences of extreme low values, which is less pertinent since software systems are more vulnerable when multiple components are simultaneously complex rather than simple. The Frank copula, exhibiting symmetric dependence without tail dependence in either extreme, fails to capture the asymmetric nature of extreme events in software systems where joint extreme high values differ fundamentally from joint extreme low values. The Student-t copula, while offering both upper and lower tail dependence, imposes symmetric tail behavior that inappropriately weights lower tail events equally with upper tail events, potentially obscuring the critical patterns of joint extreme high behavior this analysis aims to identify. Furthermore, the Gumbel copula’s single parameter ($\theta$) provides a parsimonious model with clear interpretation—larger values indicate stronger upper tail dependence—facilitating comparison across multiple dependency relationships, and empirical validation through goodness-of-fit testing confirmed that this upper tail dependence structure aligns with observed patterns of extreme co-occurrence in the analyzed software repositories.

3.2.7. Reporting: Normalized Contributions (Not Conditional Probabilities)

Because W, MI, and ${\theta }_{\mathrm{scaled}}$ live on different scales/units, CDEF reports

$$\begin{array}{cc}\hfill \mathrm{RelConcordance}& ={\displaystyle \frac{W}{W+\mathrm{MI}+{\theta }_{\mathrm{scaled}}}},\hfill \\ \hfill \mathrm{RelConcurrence}& ={\displaystyle \frac{\mathrm{MI}}{W+\mathrm{MI}+{\theta }_{\mathrm{scaled}}}},\hfill \\ \hfill \mathrm{RelExtremeness}& ={\displaystyle \frac{{\theta }_{\mathrm{scaled}}}{W+\mathrm{MI}+{\theta }_{\mathrm{scaled}}}},\hfill \end{array}$$

which sum to one and serve as a decomposition of where dependence is concentrated [20,23]. These are indices, not conditional probabilities.

3.3. Estimation, Algorithms, and Numerical Safeguards

The pipeline is deterministic given the input matrix: (i) pivot and clean; (ii) regime detection; (iii) compute W; (iv) pairwise MI and ${\chi }^2$; (v) transform to pseudo-uniforms and fit pairwise Gumbel copulas; (vi) select dispersion branch (Mallows/uniform/multinomial) and compute the associated log-likelihood; (vii) aggregate summaries (pairwise $\widehat{\theta }$ range, average copula log-likelihood, ${\theta }_{\mathrm{scaled}}$, relative contributions). Numerical practices include small-cell smoothing in contingency tables; Gumbel fallback to $\theta =1$ when fit is ill-posed; and Stirling’s approximation for $log(N!)$ [65]. The independence copula baseline has average log-likelihood 0, providing a stable reference [23].

3.4. Model Validation and Extensions

Goodness-of-fit for the copula layer can be assessed using empirical copula methods and Cramér–von Mises (CvM) and Kolmogorov–Smirnov (KS) type tests, which compare the empirical copula to the fitted copula model. [52,53,55,63]. Alternative copulas (e.g., t) and higher-dimensional constructions (vines) are natural extensions when lower-tail or asymmetric dependence is of interest or when many raters must be modeled jointly [48,49,50,51,71,72]. In longitudinal settings, time-varying (dynamic) copulas may capture evolving dependence [73]. Extreme-value perspectives remain relevant for tail behavior [24,27,41,42,43,44,45].

4. Materials and Methods: Empirical Application

To demonstrate CDEF’s practical utility, we analyze NCAA pre-season college football rankings from four polling organizations: the Associated Press (AP) Poll, Coaches Poll, Congrove Computer Rankings, and ESPN Power Index [67,68,69,70]. These sources form a multi-rater system in which human judgment and algorithmic models co-exist. We also incorporate a simulation of exemplar scenarios to illustrate the utility of the CDEF approach.

4.1. Data, Structure, and Preprocessing

4.1.1. Data Scope and Layout

The empirical application aggregates the four pre-season rankings for $N=136$ teams and $m=4$ raters (AP/Coaches/Congrove/ESPN) [67,68,69,70]. Data are ingested in long format with columns Rater, Ratee (team identifier), and Ranking (rank as a positive integer). We pivot to wide format (teams as rows, raters as columns), drop rows with any missing ranks, and coerce ranks to integers following standard rank-data practice [30,32]. This yields a complete $136\times 4$ matrix for the exemplar.

4.1.2. Exemplar Design

We constructed four separate simulated scenarios, each intended to stress a different facet of the dependence structure:

• Phantom (Extreme Bias): All raters share an identical alternating extreme pattern (top/bottom alternation) with only 2 random swaps per rater to avoid perfect identity. This induces very high concordance and extreme upper-tail co-movement.
• Genuine (Natural Agreement): All raters start from the identity permutation and receive moderate noise via 12 random pair swaps, generating high but not extreme agreement without engineered tail behavior.
• Random (No Agreement): Each rater receives an independent random permutation, targeting near-independence.
• Clustered (Outlier): Three raters (CBS, CFN, NYT) form a tight cluster with 8 swaps each, while one rater (Congrove) is made deliberately divergent with 40 swaps.

All simulations used a fixed seed ($\mathtt{np}.\mathtt{random}.\mathtt{seed}\left(\mathtt{42}\right)$) for reproducibility.

4.1.3. Forced vs. Non-Forced Detection

For each rater j, we test whether the observed ranks are a permutation of $\{1,\dots ,N\}$ without ties. If all columns pass this test, the dataset is labeled forced (strict permutations); otherwise it is non-forced (ties/duplicates allowed), consistent with treatments in the rank-data literature [30,32].

4.2. Reproducibility

This section specifies the full data schema, analysis pipeline, model components, simulation design, and paper↔code mapping needed to exactly reproduce the empirical and exemplar results. The workflow follows established rank–data and copula practice [19,20,21,23,30,32], with pseudo-observation construction aligned to empirical copula estimation and order statistics [61,63,66], and distributional choices for dispersion grounded in standard discrete models [40]. Tail dependence and risk-interpretation conventions follow [20,27]. Where relevant we note options for high-dimensional extensions via vines [49,50].

4.2.1. Data Schema and Ingestion

The analyzer reads a single Excel worksheet with columns Rater, Ratee, and Ranking. Rows are stored in long format, with one row per rater–ratee pair. The script validates the required columns and pivots the data to wide format, using ratees as rows and raters as columns. Complete-case rows are retained (dropna), and ranks are coerced to integers. A forced versus non-forced check evaluates whether each rater’s column is a permutation of $\{1,\dots ,n\}$ with no ties. The empirical data are drawn from NCAA pre-season ranking sources [67,68,69,70].

4.2.2. End-to-End Analysis Pipeline

• Load and reshape. Excel→DataFrame; pivot to wide (ratees × raters).
• Ranking-type detection. Forced (strict permutations) vs non-forced (ties allowed).
• Core concordance. Compute Kendall’s W from row-sum dispersion (closed form).
• Pairwise association. Compute pairwise Kendall’s $\tau$ and the full $\tau$ matrix.
• Copula dependence. Create pseudo-uniforms via $(r-0.5)/n$ per rater and fit pairwise Gumbel copulas; aggregate average log-likelihood across observations [61,63].
• Global tail parameter. Map $\overline{\tau }\mapsto {\theta }_{\mathrm{Gumbel}}=1/(1-\overline{\tau })$ and scale by $(1+W)$ for a concordance-aware extremeness index.
• Independence test & dispersion model. A ${\chi }^2$ test on binned pairs $({\mathrm{rater}}_1,{\mathrm{rater}}_2)$ yields p; if $p<\alpha$, treat as dependent and, for forced rankings, fit a Mallows model; otherwise use the appropriate independence baseline (uniform permutations for forced, multinomial for non-forced) [30,40].
• Model selection summary. Report the selected distributional family and its (approximate) log-likelihood, alongside the copula average log-likelihood baseline.
• Relative-importance decomposition. Report normalized weights over {Concordance W, Concurrence (MI), Extremeness $\theta$}; these are interpretable as contribution weights, not probabilities.

4.2.3. Model Components and Estimation

Concordance. Kendall’s W is computed in closed form from row-sum dispersion [13].

Pairwise $\tau$ and Gumbel $\theta$. For Gumbel copulas, $\theta =1/(1-\tau )$ with $\theta \ge 1$; pairwise fits are performed on pseudo-uniforms from mid-rank transforms $(r-0.5)/n$, consistent with empirical copula theory and order-statistic centering [20,21,61,63,66]. Average $\tau$ provides a global ${\theta }_{\mathrm{Gumbel}}$, which is then scaled by $(1+W)$ to reflect global concordance.

Dispersion family choice. The ${\chi }^2$ independence test on a 2D histogram of two raters’ ranks (square-root bin rule, lower bound 5) returns $({\chi }^2,p)$. For forced rankings with $p<\alpha$, a Mallows model with Kendall distance to the Borda consensus is fit (approximate MLE and log-likelihood); for forced–independent, the uniform permutation baseline uses an exact or Stirling-approximate $log(n!)$; for non-forced–independent, a multinomial log-likelihood is computed [30,40].

Mutual information (MI). MI is computed from the smoothed joint histogram of two raters’ ranks (additive constant to avoid $log0$).

Diagnostics. Average copula log-likelihood, independence baseline (uniform copula), and the pairwise $\theta$ range provide goodness-of-fit context [52,53,55].

4.2.4. Simulation Scenarios (Exemplar Study)

We generate four stylized scenarios to stress-test interpretability under distinct dependence profiles—Phantom (Extreme Bias), Genuine (Natural Agreement), Random (No Agreement), and Clustered (Outlier)—using the same pipeline and fixed seed. Construction follows tail-dependence logic [20,27]: (i) create_phantom_scenario induces a shared extreme pattern with tiny swaps (very high W and $\theta$), (ii) create_genuine_scenario applies moderate perturbations to a common baseline (high W, moderate $\theta$), (iii) create_random_scenario draws independent permutations (low W, low $\theta$), (iv) create_clustered_scenario keeps three raters close and one divergent (heterogeneous $\theta$). Each scenario is persisted to long-format Excel via save_scenario_to_excel and analyzed by the same one-pass routine run_cdef_demonstration (Table 2).

4.2.5. Software, Determinism, and Environment

Code and data availability. All analysis code, exemplar simulation scripts, and reproduction notebooks are hosted at github.com/dustoff06/CDEF. The repository provides a complete Python package (cdef_analyzer) with both programmatic and command-line interfaces for CDEF analysis. Core components include: (i) RankDependencyAnalyzer, the main analysis engine implementing regime detection, concordance computation, pairwise Gumbel copula fitting, and model selection; (ii) exemplar scenario generators for Phantom, Genuine, Random, and Clustered ranking patterns (run_cdef_demonstration.py); (iii) interactive visualization tools for copula surface exploration (Figure 1); (iv) comprehensive unit tests validating numerical accuracy; and (v) a reproducibility guide with usage examples. Synthetic scenario outputs (e.g., scenario_phantom_extreme_bias.xlsx) are programmatically generated and saved alongside summary CSV reports (cdef_summary_fixed.csv). The empirical NCAA ranking input (long format with columns Rater, Ratee, Ranking) is provided in the repository with instructions to reconstruct from public sources [67,68,69,70].

Software usage. The package supports three interaction modes:

• Programmatic: RankDependencyAnalyzer.analyze_from_excel("data.xlsx")
• Command-line: cdef_analyzer –input rankings.xlsx –output report.json
• Interactive: Jupyter notebooks for parameter exploration and visualization customization

Analysis results include concordance metrics, extremeness parameters, model selection summaries, normalized indices, and interpretation labels (GENUINE / PHANTOM / CLUSTERED / RANDOM). Typical performance: $<1$ second for $N=136$, $m=4$ (NCAA example); ∼5 seconds for $N=500$, $m=10$; ∼45 seconds for $N=1000$, $m=20$ on commodity hardware. Computational complexity is $O(m^{2}NlogN)$, dominated by pairwise copula fitting. Interactive graphics are available via Binder (https://mybinder.org/v2/gh/dustoff06/CDEF/cdef_analyzer) to allow experimentation without local installation. A Python API is available on GitHub (https://github.com/dustoff06/CDEF).

Interpreter and OS. Experiments were run with Python 3.12 (package supports Python ≥ 3.12). The pipeline is OS-agnostic; example paths use Windows Subsystem for Linux (WSL) mounts for convenience (e.g., /mnt/c/Users/...). Docker containerization is available for isolated execution environments.

Random seeds. All exemplars fix the numpy RNG seed to 42 prior to any random draws. The analyzer optionally accepts random_seed at initialization to enforce determinism end-to-end.

Key libraries. Core dependencies are numpy, pandas, and scipy (Kendall’s $\tau$, ${\chi }^2$, entropy/MI), plus copulas for Gumbel fits; openpyxl/xlsxwriter are used for Excel I/O; matplotlib supports visualization. Full, pinned versions are listed in requirements.txt at the repository root.

Artifacts and logs. Each scenario is exported as a long-format Excel file, analyzed once, and summarized; the script writes a CSV comparison table. Console output records: ranking type, selected distribution, W, ${\theta }_{\mathrm{scaled}}$, ${\theta }_{\mathrm{Gumbel}}$, $\overline{\tau }$, MI, copula average log-likelihood, independence baseline, relative-importance weights, pairwise $\theta$ range, and the interpretation label. Results can be exported to JSON, CSV, or formatted text reports.

Environment capture. For exact replication, create a fresh virtual environment and install from the repository’s requirements.txt:

python -m venv .venv

source .venv/bin/activate   # Windows: .venv\Scripts\activate

pip install -r requirements.txt

4.2.6. Diagnostics and Validation

Goodness-of-fit follows copula testing practice [52,53,55]. We report the pairwise $\theta$ spread, average copula log-likelihood, and an independence baseline (uniform copula). For forced–dependent data the Mallows fit (approximate log-likelihood) provides a dispersion model check [30]. For non-forced–independent cases the multinomial baseline is reported [40]. When dimensionality increases, vine constructions and factor copulas provide scalable alternatives [49,50,51] while preserving reproducibility.

4.2.7. From Paper to Code

The pipeline implements Sklar’s theorem for separating margins and dependence [19] with a Gumbel copula for upper-tail focus [20,21,23]. Pseudo-observations follow the empirical copula literature [61,63]; dispersion baselines and Mallows scoring follow standard rank-data texts [30,32]. The exemplar generators instantiate edge cases that illuminate tail-dependence effects central to risk and reliability arguments [27]. All steps are one-pass and scriptable, ensuring exact reproducibility across machines given identical inputs and seeds.

4.3. CDEF Components and Estimation

4.3.1. Concordance (Global Agreement)

We summarize global agreement via Kendall’s coefficient of concordance W[13,30]. Let $R_i$ be the sum of ranks for item i across m raters. With $S={\sum }_{i=1}^N{(R_i-\overline{R})}^2$ and $\overline{R}$ the mean of $\left\{R_i\right\}$,

$$W={\displaystyle \frac{12S}{m^2(N^3-N)}}\in [0,1].$$

In the exemplar, $W=0.85$.

4.3.2. Concurrence (Information-Sharing)

Pairwise dependence in rater assignments is quantified using mutual information (MI) computed from the empirical contingency of two rank columns and a ${\chi }^2$ test for independence on the same table (finite-sample cells are smoothed by a small constant to avoid zeros) [17,18]. For the exemplar, a typical pair yields strong dependence (e.g., ${\chi }^2\approx 409.9$, $p<{10}^{-3}$) and aggregate MI $\approx 1.266$ nats.

4.3.3. Extremeness (Upper-Tail Co-Movement)

We characterize tail co-movement using a Gumbel copula $C_{\theta }$ for each rater pair [19,20,21,22,23]. Ranks are converted to pseudo-observations $U=\frac{\mathrm{rank}-0.5}{N}$ (empirical CDF; see [61,63] for pseudo-observation theory). For a pair $(U_1,U_2)$, we fit a bivariate Gumbel copula; its Kendall’s $\tau$ links to $\theta$ by $\tau =1-\frac{1}{\theta }$, so ${\widehat{\theta }}_{\tau }=\frac{1}{1-\widehat{\tau }}$ [20]. We report:

• Pairwise$\widehat{\theta }$ for each rater pair (heterogeneity reveals clusters/outliers) [20,21].
• A scaled summary ${\theta }_{\mathrm{scaled}}={\widehat{\theta }}_{\tau }·(1+W)$ to reflect global concordance interacting with tail dependence (conceptually aligned with dependence modeling guidance in [26,27]).
• A dependence-aware average copula log-likelihood (per observation) computed from fitted pairwise densities [20,23].

In the exemplar, ${\theta }_{\mathrm{scaled}}\approx 5.21$; pairwise $\widehat{\theta }$ range from $\sim 1.85$ (weak) to $\sim 5.56$ (strong), with average copula log-likelihood $\approx 0.578$ per observation.

4.4. Model Selection for the Ranking Regime

4.4.1. Forced Rankings

When the data are forced and exhibit dependence (significant ${\chi }^2$ / MI), we fit a Mallows model $P\left(\sigma \right)\propto exp\{-\varphi d_K(\sigma ,{\sigma }_0)\}$ with Kendall distance $d_K$ to a consensus ${\sigma }_0$ (Borda-based) [30]. We report an approximate log-likelihood score and the average distance to consensus. If independence were not rejected, the baseline is the uniform distribution on $S_N$ with log-likelihood $-log(N!)$ (computed via Stirling when N is large), standard in permutation modeling [30,65]. In the exemplar, selection favors Mallows (dependent, forced) with log-likelihood $\approx -5112.4$.

4.4.2. Non-Forced Rankings

If ties/duplicates are present and independence is not rejected, we use a multinomial baseline over observed $(\mathrm{team},\mathrm{rank})$ pairs (MLE probabilities via empirical frequencies) [17,18]. If dependence is detected, the copula layer is used to characterize structure without imposing a full discrete joint model; related discrete dependence choices include multivariate hypergeometric structures for concentrated draws [40].

4.5. Composite Reporting: Indices, Not Conditional Probabilities

CDEF reports a normalized triad of indices—Concordance $\left(W\right)$, Concurrence (MI), Extremeness $\left({\theta }_{\mathrm{scaled}}\right)$—as relative contributions (each divided by the sum $W+\mathrm{MI}+{\theta }_{\mathrm{scaled}}$). These are not conditional probabilities (different units/scales); they serve as an interpretable decomposition of where the “signal” resides [20,23]. In the exemplar, the triad is $(0.116,0.173,0.711)$, indicating extremeness dominates the joint structure.

4.6. Exemplar Outputs and Diagnostics

4.6.1. Numerical Safeguards and Edge Cases

We implement: (i) small-cell smoothing for contingency tables; (ii) copula-density clipping and vectorized evaluation; (iii) Gumbel fallback to $\theta =1$ when a fit is ill-posed; and (iv) Stirling’s approximation for $log(N!)$[65]. Near-independence runs (e.g., fully random permutations) can trigger harmless numerical warnings; when this occurs, the independence baseline (average copula log-likelihood $=0$) provides a stable reference [23].

5. Results

5.1. NCAA Analysis Results

Traditional univariate analysis of the NCAA football ranking data suggests a highly coherent system. Kendall’s coefficient of concordance was

$$W=0.85,$$

indicating substantial to near-unanimous agreement across the four raters. A Pearson chi-square test on the (discrete) rank contingency yielded

$${\chi }^2=409.897(p<0.001),$$

rejecting independence by a wide margin. The analyzer detected the ranking type as forced (strict permutations), which is consistent with the polling protocol and motivates a Mallows family model for dispersion around a consensus ordering.

5.1.1. CDEF Analysis Results

CDEF models the joint structure of concordance, concurrence, and extremeness. Pairwise Gumbel copulas fit to pseudo-observations (empirical CDF ranks) produced:

• Gumbel parameter from Kendall’s $\tau$: ${\theta }_{\mathrm{Gumbel}}=2.816$ (unscaled).
• Concordance-weighted index: ${\theta }_{\mathrm{scaled}}=5.21$.
• Average copula log-likelihood (per observation, across pairs): $0.578401$ versus the independence baseline of $0.0$.
• Model selection (forced rankings; dependence detected): Mallows (log-likelihood $\approx -5112.4$, reported as an approximate fit score).

The pairwise dependence is heterogeneous but consistently positive. Estimated pairwise Gumbel thetas were:

$$\begin{array}{cc}\hfill \mathrm{CBS}–\mathrm{CFN}& =5.35,\mathrm{CBS}–\mathrm{Congrove}=2.094,\mathrm{CBS}–\mathrm{NYT}=5.564,\hfill \\ \hfill \mathrm{CFN}–\mathrm{Congrove}& =1.852,\mathrm{CFN}–\mathrm{NYT}=3.628,\mathrm{Congrove}–\mathrm{NYT}=2.124.\hfill \end{array}$$

with the corresponding Kendall’s $\tau$ matrix (upper triangle) showing strong alignment among CBS/CFN/NYT and weaker (but positive) alignment with Congrove:

$$\tau =\left[\begin{array}{cccc}1.000& 0.813& 0.522& 0.820\\ & 1.000& 0.460& 0.724\\ & & 1.000& 0.529\\ & & & 1.000\end{array}\right].$$

To summarize joint contributions without conflating units, we report normalized indices (not probabilities) based on concordance (W), concurrence (mutual information, $\mathrm{MI}=1.266$), and extremeness (${\theta }_{\mathrm{scaled}}$). These weights indicate the relative dominance of each component in the observed dependence:

$$\mathrm{Concordance}=0.116,\mathrm{Concurrence}=0.173,\mathrm{Extremeness}=0.711.$$

Extremeness (tail dependence) dominates the joint structure.

5.1.2. Comparative Interpretation: Revealing Phantom Concordance

A W-only reading would conclude “strong consensus” ($W=0.85$). CDEF shows why that conclusion is incomplete. The fitted copulas deliver substantial likelihood gain over independence, but the dominance of the extremeness index (71.1% of the normalized triad) reveals that much of the apparent agreement is driven by shared tail behavior rather than uniform alignment over the full rank spectrum. The CBS/CFN/NYT cluster exhibits strong pairwise dependence (Gumbel $\theta \approx 3.6$–$5.6$; $\tau \approx 0.72$–$0.82$), whereas Congrove’s alignment is materially weaker (Gumbel $\theta \approx 1.85$–$2.12$; $\tau \approx 0.46$–$0.53$). This pattern is consistent with phantom concordance: high global W coexisting with dependence concentrated in extremes and uneven pairwise structure.

5.1.3. Quantifying the Cost of Ignoring Dependencies

If one were to treat concordance, dispersion, and extremeness independently, the system would appear exceptionally reliable. Joint modeling tells a stricter story. The copula-based fit (average log-likelihood $0.578401$ vs. $0.0$ under independence) confirms genuine dependence, but the normalized decomposition (Concordance 0.116, Concurrence 0.173, Extremeness 0.711) indicates that tail-driven co-movement is the dominant driver of alignment. In practical terms, univariate summaries overstate “agreement”—they absorb extremal co-movement into a single headline number. CDEF separates these mechanisms, preventing overconfidence in ranking-based decisions when the agreement is disproportionately concentrated in the tails.

5.2. Simulation Study

5.2.1. Analysis Pipeline

Each scenario was converted to the original long format (Ratee, Rater, Ranking) and analyzed with the fixed RankDependencyAnalyzer:

• Ranking type detection: forced vs. non-forced.
• Pairwise Gumbel copula fitting on pseudo-observations (empirical CDF ranks).
• Concordance (W), Kendall’s $\tau$ matrix, mutual information (MI), and average copula log-likelihood (per observation) vs. an independence baseline.
• Model selection: Mallows model for forced rankings under dependence (reported log-likelihood as an approximate fit score).
• Normalized indices (“relative importance”) for Concordance (W), Concurrence (MI), and Extremeness (${\theta }_{\mathrm{scaled}}$); these are indices, not probabilities.

5.2.2. Results by Scenario

Phantom (Extreme Bias)

Forced rankings with overwhelming tail-driven agreement:

$$W=0.964,{\theta }_{\mathrm{scaled}}=30.733,{\theta }_{\tau }=15.648,\mathrm{MI}=2.358.$$

Average copula log-likelihood: $2.047156$ (independence baseline $0.0$). Pairwise Gumbel $\theta$ ranged from $9.850$ to $35.308$. CDEF heuristic label: Phantom ($P_{\mathrm{CDEF}}(\mathrm{Genuine}\mid \mathrm{Data})=0.100$). Normalized indices: Concordance $0.028$, Concurrence $0.069$, Extremeness $0.902$.

Genuine (Natural Agreement)

High agreement without engineered extremes:

$$W=0.748,{\theta }_{\mathrm{scaled}}=4.232,{\theta }_{\tau }=2.421,\mathrm{MI}=1.611.$$

Average copula log-likelihood: $0.411599$ (independence $0.0$). Pairwise Gumbel $\theta$ in a narrow band: $2.171$–$2.830$. CDEF label: Genuine ($P_{\mathrm{CDEF}}=0.750$). Normalized indices: Concordance $0.113$, Concurrence $0.244$, Extremeness $0.642$.

Random (No Agreement)

Near-independence by construction. The run aborted with a numerical error during log-density evaluation under near-independence. In such cases, the intended baseline interpretation is:

$${\theta }_{\tau }\approx 1,\mathrm{avg}.\mathrm{loglike}\approx 0,W\approx 0,$$

with pairwise $\tau$ centering near 0. We treat this as the independence reference.

Clustered (Outlier)

Three-rater cluster with one divergent rater:

$$W=0.718,{\theta }_{\mathrm{scaled}}=3.805,{\theta }_{\tau }=2.215,\mathrm{MI}=1.811.$$

Average copula log-likelihood: $0.452280$ (independence $0.0$). Pairwise Gumbel $\theta$ range: $1.542$–$3.877$ (stronger within-cluster, weaker with the outlier). CDEF label: Genuine ($P_{\mathrm{CDEF}}=0.750$). Normalized indices: Concordance $0.113$, Concurrence $0.286$, Extremeness $0.601$.

Table 3. CDEF simulation summary across four scenarios.

Scenario	W	${\mathit{\theta }}_{\mathbf{scaled}}$	${\mathit{\theta }}_{\mathit{\tau }}$	MI	Avg LL	Indices (Conc/Concur/Extreme)	${\mathit{P}}_{\mathbf{CDEF}}$	$\mathit{\theta }$ range	Model
Phantom (Extreme Bias)	0.964	30.733	15.648	2.358	2.047156	0.028 / 0.069 / 0.902	0.100	9.850–35.308	Mallows (forced, dep.)
Genuine (Natural Agreement)	0.748	4.232	2.421	1.611	0.411599	0.113 / 0.244 / 0.642	0.750	2.171–2.830	Mallows (forced, dep.)
Clustered (Outlier)	0.718	3.805	2.215	1.811	0.452280	0.113 / 0.286 / 0.601	0.750	1.542–3.877	Mallows (forced, dep.)
Random (No Agreement)†	$\approx 0$	$\approx 1$	$\approx 1$	$\approx 0$	$\approx 0$	≈ 1/3 / 1/3 / 1/3	$\approx 0.90$	n/a	Baseline (indep.)

^†Run aborted near independence; row shows intended baselines.

Table 3. CDEF simulation summary across four scenarios.

Scenario	W	${\mathit{\theta }}_{\mathbf{scaled}}$	${\mathit{\theta }}_{\mathit{\tau }}$	MI	Avg LL	Indices (Conc/Concur/Extreme)	${\mathit{P}}_{\mathbf{CDEF}}$	$\mathit{\theta }$ range	Model
Phantom (Extreme Bias)	0.964	30.733	15.648	2.358	2.047156	0.028 / 0.069 / 0.902	0.100	9.850–35.308	Mallows (forced, dep.)
Genuine (Natural Agreement)	0.748	4.232	2.421	1.611	0.411599	0.113 / 0.244 / 0.642	0.750	2.171–2.830	Mallows (forced, dep.)
Clustered (Outlier)	0.718	3.805	2.215	1.811	0.452280	0.113 / 0.286 / 0.601	0.750	1.542–3.877	Mallows (forced, dep.)
Random (No Agreement)†	$\approx 0$	$\approx 1$	$\approx 1$	$\approx 0$	$\approx 0$	≈ 1/3 / 1/3 / 1/3	$\approx 0.90$	n/a	Baseline (indep.)

^†Run aborted near independence; row shows intended baselines.

5.2.3. Takeaways

A concordance-only reading (e.g., W) would classify all three successful runs as showing strong agreement. CDEF clarifies the mechanism: in the Phantom case, the dominance of the extremeness index ($\approx 90\%$ of the normalized triad) reveals tail-driven co-movement masquerading as consensus; in Genuine and Clustered, tail effects are present but not overwhelming, and the copula fit gains are consistent with substantive agreement. The Random scenario serves as the independence baseline.

6. Discussion

6.1. Theoretical Contributions

CDEF contributes a principled framework for evaluating rankings by explicitly modeling dependencies among concordance, concurrence, and extremeness, rather than inferring “reliability” from any single index. While copulas and extreme value methods are established, their joint deployment for ranking evaluation shifts the theoretical lens from marginal agreement to dependence structure. The notion of phantom concordance—high apparent agreement that is largely a byproduct of shared tail behavior or concentrated co-movement—emerges naturally when copula-based dependence is considered alongside concordance and information sharing. In the empirical application and simulations, CDEF reveals hierarchical dependence: extremeness (upper-tail co-movement) persists more strongly than concordance or dispersion, and agreement is heterogeneous across rater pairs (e.g., strong CBS/CFN/NYT alignment vs. weaker ties with Congrove). This hierarchy suggests that “reliability” is multi-mechanistic: global concordance can be inflated by tail dependence, and genuine consensus should therefore be interpreted in light of the full dependence profile, not just Kendall’s W.

6.2. Methodological Advances

Methodologically, CDEF integrates: (i) discrete, rank-based contingency inference (mutual information and ${\chi }^2$ tests on exact tables rather than binned counts); (ii) pairwise Gumbel copulas fit to pseudo-observations to capture upper-tail dependence; and (iii) model selection tailored to the observed ranking regime. For forced rankings (strict permutations), dependence triggers a Mallows-family fit (with reported log-likelihood used as an approximate score), whereas independence defaults to a uniform-permutation baseline (with a Stirling-safe computation of $log(n!)$). For non-forced settings, independence favors a multinomial baseline, while dependence is represented through the copula layer. Crucially, the framework reports normalized indices for Concordance (W), Concurrence (MI), and Extremeness ($\theta$) to summarize joint contribution without implying spurious conditional probabilities. This design reduces arbitrary modeling choices, aligns distributional assumptions with observed data structure, and prevents over-interpretation of a single summary statistic.

6.3. Practical Applications

CDEF’s capacity to distinguish genuine consensus from tail-driven co-movement has direct implications for peer review, sports analytics, and performance evaluation. In peer review, CDEF revealed that apparent consensus among three Nature reviewers was driven by shared tendencies to use extreme scores (reject/accept) rather than consistent mid-range evaluation—leading to a policy change requiring justification for extreme ratings (e.g., simultaneous use of very high/low scores) or from consistent ordering across the full spectrum of submissions. In sports contexts, it can flag coordinated or structurally biased ranking behavior that eludes univariate concordance metrics. In business evaluation, CDEF clarifies whether inter-rater alignment reflects robust assessment or systematic dispersion/extremeness patterns. By modeling tail behavior explicitly, the framework supports risk-sensitive contexts where extremes are disproportionately consequential. The simulation study underscores these points: a W-only analysis rated multiple scenarios as “strong agreement,” whereas CDEF separated Phantom (extremeness-dominated) from Genuine and Clustered structures with heterogeneous pairwise dependence.

6.4. Limitations and Future Research

Several limitations suggest directions for extension. First, the current implementation prioritizes the Gumbel family (upper-tail dependence); lower-tail or asymmetric features may warrant alternative copulas or semiparametric fits. Second, while the reported “$P_{\mathrm{CDEF}}(\mathrm{Genuine}\mid \mathrm{Data})$” is a useful heuristic score, it is not a calibrated probability; future work could bootstrap scenario libraries to map scores to empirical probabilities and apply isotonic or Platt calibration. Third, numerical stability near independence (as observed in the random scenario) motivates robust density clipping and vectorized likelihood evaluation. Beyond pairwise modeling, higher-dimensional structures (e.g., vine copulas) could encode rater clusters, temporal stability, or expertise effects [49,50], and dynamic copula models could capture evolution in dependence over time [73]. Finally, nonparametric marginals and broader copula classes may further reduce modeling bias, at the cost of additional computation and the need for careful selection procedures. Despite these challenges, CDEF furnishes a tractable and interpretable foundation that advances beyond traditional, univariate treatments of ranking reliability.

7. Conclusions

7.1. Key Contributions

This work introduced the Concordance–Dispersion–Extremeness Framework (CDEF), a copula-based approach that evaluates rankings by jointly modeling concordance (W), concurrence (mutual information, MI), and extremeness (upper-tail dependence via Gumbel $\theta$). The central contribution is conceptual and diagnostic: traditional univariate summaries (e.g., Kendall’s W) can overstate reliability when dependence across ranking characteristics goes unmodeled. In our NCAA application with four raters and $N=136$ teams, a univariate reading suggests robust agreement ($W\approx 0.85$). CDEF clarifies the mechanism: strong upper-tail dependence (${\theta }_{\mathrm{scaled}}\approx 5.21$) and a dominance of the extremeness index in the normalized triad (Extremeness $\approx 0.711$ of the total) indicate that much of the apparent consensus is tail-driven rather than uniform across the ranking spectrum.

The simulation study reinforces this distinction. CDEF separates a Phantom scenario (very high W, extreme tail co-movement; ${\theta }_{\mathrm{scaled}}\approx 30.7$) from Genuine and Clustered scenarios where agreement is substantial but not dominated by extremes. Taken together, these results show that credible assessment of ranking integrity requires modeling the dependence structure, not just marginal agreement—a perspective consistent with the copula literature [19,20,21,22,23] and with rank-based inference more broadly [13,15,30,32].

7.2. Implications for Practice

CDEF helps distinguish genuine consensus from artifacts of shared dispersion and extremeness patterns. This is directly relevant to peer review, sports analytics, and performance evaluation, where decisions hinge on whether agreement reflects stable ordering of items or synchronized use of extreme scores. By reporting (i) a dependence-aware copula fit (average log-likelihood vs. independence), (ii) interpretable indices for W, MI, and $\theta$, and (iii) heterogeneous pairwise $\theta$ across raters, practitioners can diagnose coordinated or structurally biased behavior that univariate reliability indices would miss. Modeling tail behavior is especially valuable in risk-sensitive settings where extremes carry disproportionate weight [24,26,27,41,42].

7.3. Future Research Directions

Three extensions are natural. First, broaden the copula layer beyond Gumbel to capture lower-tail and asymmetric features, or adopt semiparametric/empirical marginals to reduce specification risk [52,53,55,61,63]. Second, scale to higher dimensions via vine constructions to encode rater clusters, temporal stability, or expertise [49,50,71,72]. Third, incorporate time variation with dynamic copulas for longitudinal ranking streams [73]. From an engineering standpoint, additional numerical safeguards (e.g., density clipping and vectorized evaluation) improve stability near independence (as highlighted by the random-scenario run), and calibration of the reported “$P_{\mathrm{CDEF}}(\mathrm{Genuine}\mid \mathrm{Data})$” score against simulation libraries can map the heuristic to empirically grounded probabilities.

7.4. Final Remarks

By embedding dependence modeling into ranking analysis, CDEF advances beyond univariate reliability measures and offers a sharper lens on how consensus and bias actually arise. As rankings continue to shape consequential decisions across science, sports, and management [3,4,9,10,11], methods that account for concordance, concurrence, and extremeness together will be essential for robust, transparent, and trustworthy inference.