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REVIEW 3 major objections 5 minor 34 references

One theorem makes ratio variables safe for data envelopment analysis: per-student ratios under VRS are exactly equivalent to volume ratios under CRS.

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2026-08-01 18:44 UTC pith:OKGO2IUL

load-bearing objection Theorem is correct, but the paper's own economic-effort input violates its common-denominator condition—fixable, but currently load-bearing. the 3 major comments →

arxiv 2607.17193 v1 pith:OKGO2IUL submitted 2026-07-19 stat.ME math.OC

On the suitability of ratio variables in data envelopment analysis: An application to education with methodological extensions

Vicente J. Bol\'os , V\'ictor J. Espa\~na , Rafael Ben\'itez , Vicente Coll-Serrano This is my paper
classification stat.ME math.OC MSC 90C0590C90
keywords data envelopment analysisratio variablesvariable returns to scaleconstant returns to scaleequivalence theoremnon-controllable variablesefficiency measurementeducational performance
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a general equivalence theorem that removes a long-standing obstacle to using ratio data in data envelopment analysis (DEA). It proves that if every input and output is a ratio sharing the same denominator—here, the number of students—then any reasonable DEA model under variable returns to scale applied to those ratios yields exactly the same efficiency scores as the same model under constant returns to scale applied to the corresponding volume variables, with the denominator treated as a non-controllable variable. This justifies the common practice of running VRS models on per-student test scores, teaching time, and spending. The paper then applies the framework to 32 countries in the 2022 international student assessment, adds a socio-economic index as an input, and proposes data-driven directions of improvement. A sympathetic reader should care because the result puts ratio-based efficiency analysis on a theoretical footing that many practitioners assumed was unavailable.

Core claim

Central claim: Theorem 3.1, an equivalence theorem for ratio variables. When every input and output is a ratio with a common denominator p_j (here, student count), any CRS-invariant DEA model under constant returns to scale applied to volumes x̃_ij = x_ij p_j, ỹ_rj = y_rj p_j, with p_j non-controllable, gives exactly the same efficiency scores as the same model under variable returns to scale applied to the ratios alone. The proof is a change of variables (λ_j = p_j λ̃_j / p_o) that turns the non-controllable constraint into the VRS convexity constraint. The application to 32 countries in the 2022 assessment shows that adding a translated socio-economic index as an input raises the efficien

What carries the argument

Theorem 3.1 (the equivalence theorem for ratio variables): the change of variables λ_j = p_j λ̃_j / p_o converts the non-controllable-variable constraint of a CRS model on volumes into the convexity constraint of the corresponding VRS model on ratios. It is the hinge that turns the known inconsistency of ratio data in DEA into an exact equivalence for any CRS-invariant model (radial, directional, slacks-based).

Load-bearing premise

The load-bearing premise is that every included variable—instruction time per student, economic effort, and translated ESCS—is a true ratio with the same denominator, the number of students; economic effort actually divides by GDP per capita as well, and ESCS is a translated composite index, so the applied analysis stands or falls on this premise.

What would settle it

Reconstruct the volume variables by multiplying each per-student variable by the country's actual student count (and, for the socio-economic index, recomputing the country mean from individual values) and run the CRS model with student count as non-controllable; if the efficiency scores or country rankings differ from those of the VRS ratio model for any country, the common-denominator premise fails and Theorem 3.1 does not apply to that variable.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Analysts can run standard VRS radial and directional DEA on per-student ratios without needing the specialized ratio-technology models previously required; the convex combinations are automatically feasible.
  • The equivalence is two-way: a CRS volume problem with a non-controllable size variable can be re-expressed as a VRS ratio problem, so any tool that needs VRS (such as archetypal analysis) becomes applicable.
  • Including the socio-economic index as an input meaningfully changes the efficiency map: the scores of Chile, Colombia, and Mexico rise by roughly 0.05–0.09, so fairness comparisons depend on controlling for student background.
  • Directional models with data-driven orientation coefficients produce different targets than radial models, so recommended score improvements are sensitive to the chosen improvement strategy.
  • Regularizing the frontier with adaptive constrained enveloping splines pulls several near-frontier countries slightly below efficiency, indicating that classical DEA's frontier may overstate their standing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the theorem is really a duality between 'extensive' and 'intensive' representations of a production technology; the same reweighting should transfer ratio-VRS/volume-CRS equivalence to any setting where per-capita or per-unit ratios are studied, such as public health, firm productivity, or energy intensity.
  • Editorial inference: because the application's economic-effort input is expenditure per student divided by GDP per capita, its denominator is not literally the student count; if one insists on the theorem's premise, the input set should be redefined (e.g., using expenditure per student alone or including GDP per student separately) or the empirical rankings should be treated as conditional on that
  • Editorial inference: a natural stress test is to vary the ESCS translation parameter across the fitted range and check whether the five efficient countries and the top ranking gains persist; robustness over the translation would make the framework actionable.
  • Editorial inference: the equivalence suggests a direct falsification protocol—reconstruct true volumes for all variables and compare CRS-with-non-controllable results against VRS ratio results; disagreements locate exactly which variable violates the common-denominator assumption.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a theoretical equivalence (Theorem 3.1) between (i) a CRS-invariant DEA model applied to volume variables together with a non-controllable denominator variable and (ii) the same model under VRS applied to ratio variables obtained by dividing all inputs and outputs by the denominator. The proof is carried out for radial/directional models using the change of variables λ_j = (p_j/p_o)λ~_j, and a sketch is given for other CRS-invariant models. The authors apply the result to justify output-oriented VRS DEA on 32 OECD countries using PISA 2022 mean scores as outputs and instruction time per student, economic effort per student, and a translated ESCS index as inputs. Further contributions include directional orientation coefficients estimated from annual PISA changes or percentile spreads, an ACES-based robustness analysis, chance-constrained and Kao–Liu fuzzy extensions, and publicly available replication code.

Significance. The equivalence in Theorem 3.1 is a clean and potentially useful formal result: it connects the long-standing ratio-variables debate to a simple CRS/VRS duality and is proved with an explicit construction. The manuscript supplies reproducible data/scripts, and the theorem is not circular in the target-efficiency sense. If the applied variables satisfy the common-denominator hypothesis, the framework offers practitioners a principled way to use VRS with per-student PISA ratios and to alternate between ratio-VRS and volume-CRS formulations. However, the empirical illustration as currently specified uses an economic-effort input that does not satisfy that hypothesis, so the application is not fully justified by the theorem; this is a load-bearing but repairable gap.

major comments (3)
  1. [§2.1, §2.2, §3.2] The 'economic effort' input is not a common-denominator ratio in the sense required for the VRS PPS to contain only realizable activities. With E_j total expenditure, G_j GDP, N_j population, and p_j students, e_j = (E_j/p_j)/(G_j/N_j) = E_j N_j/(p_j G_j). The convex-combination identity (2) holds algebraically for any positive values, but the quantity p_j e_j = E_j N_j/G_j does not aggregate linearly across DMUs, so the 'correct average' interpretation fails. For example, E=(100,200), p=(10,20), G=(1000,4000), N=(100,200), λ=(0.5,0.5) gives Σλ e = 0.75, while the actual economic effort of the equal-weight combined entity is 0.60 (and the tildeλ-weighted entity from (3) gives 0.67). Hence the VRS PPS built with this input contains non-realizable activities, and the paper's claim in §2.2 that the PPS 'accurately represents the set of all feasible activities' does not hold for the illustra
  2. [§3.2, Theorem 3.1] The theorem is stated for any CRS-invariant model, but the proof is given only for the directional program (6)–(8). The sentence after (8) asserting that the same change of variables applies to SBM and other CRS-invariant models is not a proof: one must verify that every constraint and the objective of each model transform correctly under (7), including possible slack variables. Since the abstract and conclusions advertise a 'general result', the authors should either restrict the statement to the class of envelopment models with a linear objective of the form used in (6) (which covers radial, directional, and standard SBM formulations) or supply the full general argument. This does not invalidate the radial/directional application, but it matters for the theorem's advertised scope.
  3. [§2.3, §5] The ESCS translation parameter τ is estimated from the same PISA data via regression, with R² ≈ 0.58 for the mean slope; the chosen value τ = 2.55 is one of several plausible values (2.45–2.70, with a 95% upper value around 3.33). Since the translated ESCS is an input in the central empirical analysis and the results in Tables A.4–A.12 depend on it, the absence of a sensitivity analysis is a substantive gap: a different τ changes the input values and can alter efficiency scores and targets. The theorem does not depend on τ, so this is not a correctness issue for the theoretical result, but it is load-bearing for the paper's methodological claim that ESCS can be incorporated as an input in this way.
minor comments (5)
  1. [§4, Tables A.2, A.4–A.9] Equation cross-references are inconsistent: Section 4 defines orientation coefficients in equations (16)–(18), but the tables and Section 5.1 refer to directions '(21)', '(22)', '(23)'. Please renumber or correct all citations.
  2. [§2.2, eq. (3)] The derivation that the tildeλ defined in (3) sum to 1 is omitted; it is a short algebraic step and should be shown explicitly, especially because the feasibility of the convex combination depends on it.
  3. [§5.3] The chance-constrained models classify all DMUs as stochastically efficient even at α = 0.5, a null result the authors acknowledge. Consider presenting this more tersely as a limitation or moving the detailed table to supplementary material, since it currently adds little to the empirical section.
  4. [§3.3, §5.2, Table 1] The final ACES configuration selected by LOOCV sets monotonicity = No and concavity = No, while Section 3.3 describes these as core constraints imposed through (10). Clarify that these shape restrictions are optional in the implemented version and discuss the economic interpretation of the resulting frontier, which is no longer guaranteed to be concave.
  5. [§4.1, §4.2] The minima defining m_eval are written with subscripts 'r=1,...,s; j∈eval' but without a visible double subscript; the notation should make clear that the minimum is over both outputs and DMUs in eval.

Circularity Check

0 steps flagged

No load-bearing circularity: Theorem 3.1 is a self-contained algebraic equivalence; the self-citations are auxiliary and the fitted ESCS translation is preprocessing, not a predicted efficiency.

full rationale

The central derivation is self-contained. Theorem 3.1 proves the VRS-ratio / CRS-volume equivalence by an explicit change of variables (λ_j = p_j/p_o · \tilde λ_j) that converts program (6) into program (8); the convexity condition Σλ_j = 1 is obtained algebraically from the non-controllable constraint Σ\tilde λ_j p_j = p_o. No empirical quantity is fitted to efficiency scores and then presented as a prediction. The ESCS translation parameter τ is fitted by regression, but it is an input-preprocessing constant and does not enter the objective function of the DEA programs; it is not used to derive the efficiency scores it later explains. The orientation coefficients in Section 4 are also user-chosen inputs for directional models, not outputs of the analysis. The self-citations (ACES from España et al. 2024/2025a, the Farrell-oriented-score paper Bolós et al. 2026, and the deaR/SdeaR software) support auxiliary or robustness components and are not load-bearing for Theorem 3.1. There is a real correctness caveat: the applied common-denominator assumption is questionable for the 'economic effort' input, since that variable is expenditure per student divided by GDP per capita and does not satisfy the same convex-combination interpretation as a genuine per-student average; the paper itself warns that mixed-nature ratio variables may generate PPS inconsistencies. That is an assumption-violation concern, not circularity: the theorem is still valid under its stated hypotheses, and the derivation does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central theorem itself is self-contained and uses only standard DEA axioms and a change of variables. The applied framework introduces two kinds of free choices: the ESCS translation tau fitted from regression data, and the orientation coefficients/heuristics chosen by the researcher. No new physical or conceptual entities are invented.

free parameters (3)
  • ESCS translation parameter tau = 2.55 (alternatives 2.70 and 3.19-3.59 at 95% confidence)
    Chosen in Section 2.3 as the ESCS value where the regression line of the PISA ESCS gradient crosses zero (R^2=0.58); used to shift ESCS to strictly positive values before using it as a DEA input.
  • Orientation threshold delta = 1% for Eq. (16), 5% for Eq. (18)
    Researcher-selected thresholds in Section 4 that translate raw relative rates of change or percentile differences into positive direction coefficients; directly changes targets in directional models.
  • ACES hyperparameters = qmax=3/2, xi(q)=0.05/0, gamma=1, psi=0.01/0.025, LOF=MAE, monotonicity/concavity off
    Selected by a frontier-oriented LOOCV in Section 5.2 that keeps only countries with eta*_j < 1.025 and uses median RMSE; controls the ACES frontier estimate and hence the efficiency scores.
axioms (6)
  • domain assumption Banker-Charnes-Cooper Axioms 1-3 (feasibility, free disposability, convexity) define the VRS production possibility set.
    Invoked in Sections 2 and 2.2 through Eq. (1); free disposability is argued to hold 'for practical purposes' even though instruction time and expenditure are bounded.
  • domain assumption All inputs and outputs are ratio variables sharing a common denominator p_j (number of students).
    Central premise of Theorem 3.1 and Section 2.2. Clear for PISA mean scores and instruction time per student, but weaker for economic effort and the translated ESCS index.
  • standard math The DEA model is CRS-invariant in the sense of Definition 3.2.
    Needed for Theorem 3.1; holds for radial and directional models and is asserted for SBM-type models when unit-invariant. The proof demonstrates the directional case explicitly.
  • domain assumption The number of students p_j is a non-controllable variable in the CRS volume model.
    Section 3.2 imposes the constraint sum tilde-lambda_j p_j = p_o; the paper justifies this by exogeneity of student counts and the need to compare scaled activities.
  • ad hoc to paper The ESCS gradient regression identifies a meaningful zero point for the translated ESCS input.
    Section 2.3 selects tau=2.55 from a linear regression of ESCS slope on ESCS values (R^2=0.58); different tau would change relative input levels and efficiency scores.
  • domain assumption Strong disposability of inputs and outputs is assumed.
    Section 2 states strong disposability, which ensures the weakly efficient frontier coincides with the boundary of the PPS.

pith-pipeline@v1.3.0-alltime-deepseek · 48816 in / 12960 out tokens · 116506 ms · 2026-08-01T18:44:05.083730+00:00 · methodology

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read the original abstract

It is well known that the use of ratio variables is inconsistent with the fundamental assumptions of convex Data Envelopment Analysis (DEA). However, in this paper, we establish a general result demonstrating the equivalence between DEA models with ratio variables under variable returns to scale and DEA models with volume (non-ratio) variables under constant returns to scale, provided that all variables are ratio variables sharing a common denominator. This significant result enables the development of a framework for evaluating the efficiency of the Organisation for Economic Co-operation and Development (OECD) countries based on the results of the Programme for International Student Assessment (PISA) report, using mean performance scores as outputs. In this framework, we give some methodological innovations, such as the incorporation of the index of economic social and cultural status (ESCS) as an input, thereby enabling fairer comparisons with countries with a lower socio-economic level. Furthermore, we introduce different methods for estimating directions of improvement and calculating targets appropriate to the difficulty of improving each performance score. Finally, we review and introduce several novel contributions to emerging methodologies that can complement classical radial and directional models, such as efficient frontier estimation with adaptive constrained enveloping splines (ACES), stochastic chance-constrained models, and fuzzy models. All these methodologies can be used to analyse data from other PISA or similar reports, allowing non-specialists to implement DEA appropriately.

Figures

Figures reproduced from arXiv: 2607.17193 by Rafael Ben\'itez, Vicente Coll-Serrano, Vicente J. Bol\'os, V\'ictor J. Espa\~na.

Figure 1
Figure 1. Figure 1: ESCS vs (a) slope in mathematics, (b) slope in reading and (c) slope in science per￾formances, and (d) the mean slope of all the performances, for a sample of 80 countries (OECD members and partners). Regression lines are computed in order to find the translation parameter τ for transforming the ESCS index into a strictly positive one. The grey zone corresponds to the 95% confidence bands. 1.3679 for Türki… view at source ↗
Figure 2
Figure 2. Figure 2: Representation of two ways in which a DMU can achieve efficiency if its target is not efficient. We consider an output oriented model with one input and one output under VRS. The PPS P is the grey area. The efficient DMUs are A and B, which define the efficient frontier ∂ SP represented by a black dashed line. The rest of the weakly efficient frontier ∂WP is represented by grey dashed lines. Since the targ… view at source ↗
Figure 3
Figure 3. Figure 3: Construction of a fuzzy number from percentiles. For the first scenario, the resulting fuzzy efficiency scores ρ ∗ are presented in Table A.11 and Figure A.1. The membership functions of the fuzzy efficiencies for Ireland, Poland, and Türkiye degenerate to a crisp value equal to 1. Furthermore, the associated inefficiency slacks are zero, con￾firming that they are therefore efficient countries. In other wo… view at source ↗

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