PCA weighting reshuffles sustainability rankings in 11 of 13 models

A principal component analysis-based weighting method changed decision-relevant rankings in 11 of 13 multicriteria decision-making models for European countries, according to a computational study of sustainability indicators. Rather than simply averaging data-derived objective weights with subjective preferences, the method uses more subjective weight when criteria behave independently and shifts emphasis toward objective weights when criteria are strongly correlated. The aim is to limit overrepresentation: when several indicators measure nearly the same underlying feature, an alternative can receive disproportionate influence simply because that feature appears more than once in the model.

The researchers tested the approach in three previously published sustainability case studies containing 13 models. These covered circular-economy and agricultural-production indicators for 2012 and 2020; Sustainable Development Goal and sustainable-city indicators for 2015 through 2019, plus a DARIA-TOPSIS model; and circular-economy indicators for 2016, 2018 and 2020. European Union countries were the alternatives in every model, and the case studies used 33 distinct indicators as criteria. To create comparable inputs, the researchers supplied missing objective weights with Shannon entropy and missing subjective weights with a simulated analytic hierarchy process organized around economic, social and environmental sustainability pillars. These simulations isolated the integration procedure; they did not represent preferences expressed by actual stakeholders in a live decision.

Technically, the procedure begins with historical Eurostat data aggregated for each country and indicator. Because the indicators have different units and levels of variability, principal component analysis is performed on a correlation matrix rather than the raw historical values. Eigenvectors from the first two components are treated as loadings, and the angle between two criteria’s loadings is used to estimate their relationship. An angle of 90 degrees represents independence, while angles of zero or 180 degrees represent perfect correlation in either direction. The resulting degree of independence is zero for perfectly correlated criteria and one for uncorrelated criteria. Mathematical checks confirmed the intended handling of perfect positive correlation, independence, inverted correlation and reordering of the criteria.

The degree of independence controls a pairwise blending of objective and subjective weights. For a strongly correlated pair, the method places more emphasis on the objective component, reducing the possibility that stakeholder preferences amplify information already represented by another criterion. For an uncorrelated pair, the subjective component receives more influence because the criterion appears to contribute distinct information. The method solves separate pairwise systems for the correlated and uncorrelated components, combines them using the estimated independence, and normalizes the final criterion weights so that they sum to one. A coefficient of 0.5 divides the influence equally between the correlated, objective side and the uncorrelated, subjective side, although this balance can be adjusted from zero to one to favor either source of weight.

The first application generated 25 rank changes in each circular-economy model, 16 in agricultural production for 2012 and 12 in agricultural production for 2020. It also produced moves of five or more places and changes in the membership of the leading or trailing five countries. The Sustainable Development Goal models showed decision-relevant changes in every model, including a move for Czechia from seventh to second in 2017 and for Slovakia from 11th to fourth in the DARIA-TOPSIS model. Some final weights changed substantially. Across the Sustainable Development Goal models, the integrated weight of the share of buses and trains in total passenger transport increased by 23.3% to 38.4%, while the integrated weight of the overcrowding-rate indicator fell by 27.3% to 30.1%. In the separate circular-economy case, the consumption-footprint integrated weight rose by 196% to 383%, although it remained among the lowest-weighted criteria.

Not every model responded in the same way. The third case study, which ranked countries with circular-economy indicators, experienced 10 changes in 2016, nine in 2018 and 15 in 2020, but none met the study’s decision-relevant criteria. Only three indicators changed their positions in the weight ordering, and those changes were insufficient to alter the important parts of the country rankings. Across all 13 models, Kendall’s tau remained high, indicating that most overall orderings were similar, while median numbers of rank changes ranged from zero to two. Some countries nevertheless crossed important boundaries or moved at least five positions. The study therefore distinguished numerical stability from practical significance rather than treating every reordered country as evidence of a wholly different result.

Sensitivity analyses exposed several constraints. Bootstrapping the principal-component eigenvectors at 95% and 99% confidence levels produced changes in multiple rankings, including swaps at the boundary between fifth and sixth place, showing that the procedure is sensitive to variability in historical observations. Replacing the PCA-based blending with a simple arithmetic mean or a weighted mean also changed rankings. In one circular-economy model, Denmark moved from fifth to 24th under weighted-mean aggregation. Altering the balance between objective and subjective influences by only 0.125 could introduce a decision-relevant change, so the coefficient should be accompanied by explicit sensitivity testing rather than treated as a neutral technical default. Threshold effects also appeared when countries with close scores exchanged places around the fifth-place or bottom-five boundaries.

Other tests supported several design choices while exposing a central weakness. Using individual years instead of aggregated historical observations did not add new decision-relevant changes, and replacing historical aggregates with the immediate decision matrix was an adequate alternative only for the separate circular-economy case. Gender-differentiated data for three Sustainable Development Goal indicators produced no substantive effect beyond one threshold swap. By contrast, retaining enough principal components to explain at least 75% of the variance changed rankings considerably. The original two-component approximation captured 85.6% of variance for agricultural production, 74.3% for the circular-economy theme, 63.8% for the separate circular-economy case and only 57.0% for the Sustainable Development Goal case. The last value falls below the cited 60%–90% range used to judge whether correlations are adequately represented, and comparisons with direct correlation coefficients showed that approximation errors could be substantial for particular indicator pairs.

The findings support the method’s capacity to redistribute weight and change consequential rankings, but they do not establish that the resulting decisions are better. The analysis compared computational rankings; it did not evaluate decision quality, stakeholder satisfaction or whether correlated criteria actually distorted real choices. The authors also note that no real stakeholder engagement was conducted, some source indicators had been discontinued, and several models were sensitive to rounded published scores. The approach requires historical data beyond the decision matrix, statistical expertise for PCA and careful interpretation when observed relationships conflict with expected ones. In the circular-economy data, for example, countries with higher resource productivity or circular material use also showed higher consumption footprints or material dependency, a pattern that may deserve investigation rather than automatic down-weighting. Further work must test the method with actual stakeholders, complete objective and subjective weight sets, non-European data and more realistic correlation structures, including possible nonlinear relationships among criteria.

Subject of Research: PCA-based integration of objective and subjective weights for correlated sustainability criteria in European multicriteria decision-making models.

Article Title: A novel integrated objective-subjective weighting method using principal component analysis to correct overrepresentation of correlated criteria in sustainability decision-making

Article References: Anglehart-Nunes, J., Glaus, M., & Villot, J. (2026). A novel integrated objective-subjective weighting method using principal component analysis to correct overrepresentation of correlated criteria in sustainability decision-making. Environmental and Sustainability Indicators, 32, Article 101573. https://doi.org/10.1016/j.indic.2026.101573

Image Credits: AI Generated

DOI: 10.1016/j.indic.2026.101573

Keywords: principal component analysis, multicriteria decision-making, sustainability indicators, criterion weighting, correlated criteria, objective-subjective integration, circular economy, agricultural production, Sustainable Development Goals, European countries, sensitivity analysis, decision rankings