The instrument at the center of the analysis is the Learning Management of University Students Scale, known as GAEU-1, a 19-item self-report questionnaire developed for undergraduates in Chile and Ecuador. Students rate statements about planning, motivation, and study techniques on a five-point agreement scale. Earlier validation work had retained three of the scale’s dimensions: a self-regulated learning factor covering study preparation and time management, a dimension of conscious motivational strategies, and a dimension of deep learning techniques. The new study, led by Xinglong Pang, Lin Zhou, and Jin-Ho Cho, asked whether these three retained dimensions produce measurement parameters that are statistically comparable across the two countries.

The question matters because score differences between groups can be deceptive. If an item functions differently in Chile than in Ecuador, an apparent gap in self-regulated learning might partly reflect group-specific item behavior rather than genuine psychological differences. Measurement invariance testing addresses this by progressively constraining model parameters to be equal across groups and checking whether the fit of the model deteriorates meaningfully. For Likert-type items, the technical challenge is sharper still: ordered responses arise from thresholds on underlying response propensities, so procedures designed for continuous indicators cannot simply be transferred. The team therefore used weighted least squares mean- and variance-adjusted estimation, WLSMV, with theta parameterization, treating all 15 retained indicators as ordered categorical variables.

The data came from an openly licensed Version-5 dataset containing 1,316 undergraduates, 631 in Chile and 685 in Ecuador, with a mean age of 20.45 years. All 19 items were complete and within range, and only nine participants, less than one percent, showed zero response variance across the full item set. Internal consistency was adequate for the three analyzed dimensions, with pooled coefficient alpha values of 0.782 for self-regulated learning, 0.793 for conscious motivational strategies, and 0.753 for deep learning techniques, and model-based omega estimates reaching 0.890 for the motivation dimension. A fourth dimension, perceived academic performance, was excluded from the invariance model because it had shown weaker reliability in the original validation.

Country-specific confirmatory factor analysis produced a mixed picture. The comparative fit index and standardized root mean square residual were relatively favorable in both countries, with CFI of 0.947 in Chile and 0.952 in Ecuador and SRMR of 0.059 in both. But the root mean square error of approximation told a different story, reaching 0.094 in Chile and 0.085 in Ecuador, above conventional close-fit values. The researchers did not gloss over this. Analytical power calculations showed the elevated RMSEA could not be attributed to low statistical sensitivity, meaning the baseline models carried genuine residual misspecification. Their conclusion was therefore deliberately conditional: the inherited three-dimensional representation was usable for invariance testing, but it was not a perfectly specified model.

The core of the study was a staged multigroup invariance sequence following the Wu-Estabrook identification framework for ordinal indicators. The sequence moved from a configural model, in which the same factor structure was imposed in both countries, to equality constraints on item thresholds, then on thresholds plus factor loadings, and finally on the intercept-related parameters required by the identified ordinal model. The decision guides were prespecified: a decrease in CFI no larger than 0.010 and an increase in RMSEA no larger than 0.015 at each stage. Every planned stage passed. Constraining thresholds changed CFI by only 0.001 while RMSEA actually improved. Adding loading equality improved both indices further, and the intercept-related constraints changed CFI by less than 0.001. No partial invariance was needed, and all 15 indicators retained their cross-group equality constraints through the strong, or scalar, level.

Complementary differential item functioning analyses added an item-level view. Using iterative hybrid ordinal logistic regression procedures with Chile as the reference group, the team found statistically detectable DIF signals for eight items after false discovery rate correction. Yet the largest total change in McFadden pseudo-R-squared was 0.0103, for item GA3, well below the prespecified 0.020 practical-effect criterion. No item therefore required release from the multigroup constraints. The researchers were careful to note that this conclusion depends on the chosen threshold: a sensitivity analysis showed GA3 would be flagged under a more permissive 0.010 rule. The finding supports small rather than absent item-level differences, a distinction the authors argue future work should address with formal equivalence testing rather than simply failing to detect effects.

Post hoc diagnostics sharpened the study’s limitations. The strongest residual dependence was localized to the pair GA7-GA8, both items involving motivational self-talk, in both samples, with residual correlations of 0.174 in Chile and 0.201 in Ecuador. A deliberately parsimonious one-factor model fit appreciably worse than the retained three-factor representation in both countries, with CFI drops of 0.059 and 0.069, arguing against simple unidimensionality. Discriminant validity evidence was mixed: polychoric HTMT values all fell below the conventional 0.85 heuristic, but Fornell-Larcker diagnostics were not satisfied for factor pairs involving the self-regulated learning factor, whose latent correlations with the other dimensions were substantial, reaching 0.836 in Chile. The factors are not empirically identical, but their separation is criterion-dependent.

After invariance was established, the team estimated latent sample contrasts as a secondary illustration. With effects coded Ecuador minus Chile, the latent contrast for self-regulated learning was small and non-significant, while conscious motivation and deep learning showed significant latent differences favoring Chile, with latent Cohen’s d of 0.225 and 0.284 respectively. The researchers stressed these values are descriptive of these two samples after measurement adjustment, not estimates of national or cultural effects, because the samples differed markedly in gender and university-sector composition. Women made up 46.9 percent of the Chilean sample but 65.7 percent of the Ecuadorian sample, and private-university students represented 57.5 percent in Chile versus 98.4 percent in Ecuador.

Two planned sensitivity analyses, one removing zero-variance respondents and one allowing residual covariances for the data-informed item pairs, preserved the main comparability result, as did a post hoc private-university-only analysis of 1,037 respondents. The overall conclusion is bounded but useful: the three retained GAEU-1 dimensions showed approximately stable thresholds, loadings, and intercept-related parameters across these Chilean and Ecuadorian university samples, conditional on the inherited scoring representation. For researchers using the scale in similar Latin American university settings, that is meaningful evidence for model-based cross-sample comparison, while broader structural, equivalence, and country-level claims remain outside what was actually tested.

Subject of Research: Ordinal measurement invariance testing of the GAEU-1 learning-management scale across Chilean and Ecuadorian university students

Article Title: Ordinal measurement invariance of three retained GAEU-1 dimensions in Chilean and Ecuadorian university samples

Article References: Pang, X., Zhou, L., & Cho, J.-H. (2026). Ordinal measurement invariance of three retained GAEU-1 dimensions in Chilean and Ecuadorian university samples. Frontiers in Psychology, 17, Article 1990515. https://doi.org/10.3389/fpsyg.2026.1990515

Image Credits: AI Generated

DOI: 10.3389/fpsyg.2026.1990515

Keywords: measurement invariance, self-regulated learning, GAEU-1, Chile, Ecuador, higher education, psychometrics, differential item functioning, Ordinal, measurement, invariance, three

The problem is more than academic pedantry. Standard tools in orbit determination rely on matrices—the state-transition matrix and the target-sensitivity matrix—that describe how small changes in an asteroid’s current position and velocity translate into changes at some future target time. Analysts often inspect the eigenvalues of these matrices to gauge how fast uncertainty grows. But as Tommei’s paper demonstrates, those raw eigenvalues depend on the coordinate system and the physical units chosen. Express positions in kilometers instead of astronomical units, or mix Cartesian coordinates with orbital elements, and the same physical situation produces different numbers. Worse, such eigenvalues conflate two distinct things: the intrinsic sensitivity of the dynamics to initial conditions, and the actual uncertainty that has been propagated from the observations.

The proposed solution is a diagnostic built from a comparison that anyone can state physically: take the covariance matrix of the orbit solution—the formal description of how the observational errors spread into state uncertainty—propagate it forward to the target quantity of interest, and compare it against a tolerance that the analyst explicitly declares. The scalar summary of that comparison is the largest generalized eigenvalue of the propagated covariance paired with the tolerance matrix. In plain terms, the diagnostic answers the question: relative to the accuracy I actually need, how badly has my knowledge of the target state degraded? Because both the covariance and the tolerance are stated in the same units and coordinates, their ratio is meaningful in a way that a bare eigenvalue never is.

The mathematical heart of the paper is an invariance theorem. Tommei proves that this scalar diagnostic, and the time at which it crosses its threshold, do not change under smooth reparametrizations of the state and the target quantity. Switch from Cartesian coordinates to Keplerian elements, from geocentric to heliocentric frames, from position to range and range-rate: within the local covariance model, the diagnostic reports the same number. This is the coordinate-invariance that the title advertises, and it elevates the diagnostic from a convenient heuristic to a genuine physical quantity, one that belongs to the encounter itself rather than to the analyst’s choice of bookkeeping.

The work also revisits a classical construct of impact monitoring: the Line of Variations, the curved path through orbit space that traces the spine of the uncertainty region and along which impact solutions are traditionally searched. Existing theory treats this line under assumptions of linear behavior and non-degenerate geometry. Tommei establishes a local theorem for the Line of Variations that incorporates the residual curvature of the least-squares objective—the second-order structure left over after the best-fit orbit is found—and the transverse non-degeneracy condition required when constrained differential corrections are applied. These refinements matter in exactly the situations that are hardest to handle: weakly determined orbits and encounters where the linear approximation strains at its limits.

Connecting the diagnostic to the physics of close encounters, the paper derives a lower bound on a finite post-encounter interval that links the target-uncertainty diagnostic to encounter stretching, the rapid separation of initially nearby trajectories under the planet’s gravitational perturbation. The bound combines coordinate-covariant state metrics with an explicit target-visibility condition—a criterion stating when the target quantity can actually be resolved from the state at all. When visibility fails, no amount of propagation refinement can recover the target information, and the diagnostic flags the breakdown. A controlled nonlinear map, deliberately constructed for the purpose, verifies all three failure modes the theory warns about: violations of invariance, failure of target visibility, and the collapse of first-order covariance propagation when nonlinearity dominates.

To show the diagnostic working on real data, Tommei turns to the most closely watched close approach in modern astronomy: the April 2029 encounter of asteroid 99942 Apophis with Earth. Apophis, roughly 340 meters across, will pass within about 31,000 kilometers of Earth’s surface—closer than many geostationary satellites—an event that has been tracked with extraordinary precision since its discovery in 2004. The benchmark uses published numerical target-uncertainty values from earlier work on Apophis’s non-zero Yarkovsky acceleration, the tiny thermal recoil force that subtly alters an asteroid’s orbit over years, together with data from NASA/JPL’s Small-Body Database Close-Approach API. No raw astrometry is reprocessed; the point is to demonstrate how the diagnostic responds as observations accumulate and as the analyst tightens or relaxes the stated tolerance.

The Apophis exercise yields a nuanced picture. As observational refinement shrinks the covariance, the diagnostic falls, quantifying exactly how much better the 2029 encounter is known than earlier predictions were. As the tolerance selection changes, the diagnostic rescales accordingly, making explicit a choice that is usually left implicit in uncertainty reporting. Crucially, Tommei is careful about what the diagnostic is not: it is a normalized summary of the target covariance, a single number that condenses a complicated situation, not a replacement for the b-plane analysis—the standard target-plane formalism used to compute impact probabilities—or for full nonlinear sampling of the uncertainty region. The diagnostic tells you when to worry and how much; it does not by itself tell you whether an impact will occur.

The paper closes with practical guidance that should make the method usable in operational settings. Threshold calibration is discussed, addressing how analysts should choose the tolerance that gives the diagnostic its meaning. The growth-bound constants appearing in the finite-interval lower bound are examined for practical evaluation, and invariant alternatives based on the trace and determinant of the covariance–tolerance pair are offered for situations where a single largest eigenvalue is too coarse a summary. Reproducibility is built in: the scripts that generate the paper’s figures and tables, along with all derived numerical values, are supplied as supplementary online material, and the synthetic data are produced by the supplied script itself.

The broader significance reaches beyond a single asteroid. Impact monitoring systems such as ESA’s Aegis services at the NEO Coordination Centre and NASA’s Sentry computations continuously propagate thousands of near-Earth objects and must decide, automatically, which encounters deserve human scrutiny. A coordinate-invariant, physically interpretable measure of target uncertainty offers such systems a principled ranking criterion—one that does not shift when software, coordinates, or units change, and that flags when linear covariance propagation can no longer be trusted. The work draws on deep connections to information geometry, optimal experimental design, and the theory of Lagrangian coherent structures in fluid dynamics, fields that have independently grappled with how to measure stretching and information loss in ways that survive changes of description. For a discipline whose highest-stakes judgments—impact probabilities, keyhole passages, resonant returns—rest on knowing exactly how well an orbit is known, a yardstick that measures the asteroid rather than the mathematics is a quietly powerful addition to the toolkit. The research was supported by the Italian Space Agency through the Monitoring Asteroids project, and the paper’s single-author provenance, from conception through final verification, underscores the individual mathematical labor behind what appears, on the surface, to be a modest scalar. Sometimes the most useful number is the one that stays the same no matter how you look at it.

Subject of Research: A coordinate-invariant measure of target uncertainty propagation for near-Earth asteroid close encounters

Article Title: Coordinate-invariant target-uncertainty diagnostic for near-earth object encounters

Article References: Tommei, G. (2026). Coordinate-invariant target-uncertainty diagnostic for near-earth object encounters. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 62. https://doi.org/10.1007/s10569-026-10335-w

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10335-w

Keywords: near-Earth objects, orbit determination, close encounters, uncertainty propagation, covariance, Line of Variations, Apophis, impact monitoring, generalized eigenvalues, celestial mechanics, Yarkovsky effect, invariance