# Standardization sensitivity specification freeze

**Frozen:** 12 August 2026, before implementation or estimation of the alternative standardization specifications described below.  
**Status:** Internal analysis-stage decision record; this is not an external preregistration. The primary analysis had already been estimated. Its estimand and implementation are unchanged, and exact identity checks will guard against drift.

## Primary specification retained unchanged

The primary Grade 4-to-5 estimand remains system-specific recovery from below both 475 benchmarks at baseline (BB) to at/above both at follow-up (AA). Within each of five matched plausible-value (PV) draws, the primary standardization uses:

- all nine participating systems;
- a separate six-parameter quadratic logistic response surface in each system: intercept, mathematics deficit, science deficit, both squared terms, and their interaction;
- the intersection of system-specific weighted 1st–99th percentile intervals in baseline mathematics and science among baseline-BB students;
- an empirical equal-system target within that rectangle, with each system normalized to mass 1/9; and
- stacked jackknife repeated replication (JRR), rebuilding the shared support and target in every replicate.

The existing primary point estimates and standard errors must be reproduced to absolute tolerance `1e-10` after refactoring. The equal-system target is a synthetic “one-system, one-vote” target, not a world-student population.

## Frozen one-factor-at-a-time alternatives

All alternatives retain the 475 BB-to-AA outcome and differ from the primary in only the listed component.

| ID | Component changed | Exact definition | Interpretation |
|---|---|---|---|
| `population_target` | Target weighting | Retain all nine systems, the primary quadratic surface, and the primary rectangle. Give each retained record weight proportional to its original TIMSS student weight across the pooled retained BB population; do not normalize systems to equal mass. | Synthetic pooled population of baseline-BB students represented by the nine participating systems, conditional on common support. It is not a global target. |
| `leave_korea_out` | System set and target | Exclude the Republic of Korea from both support construction and target construction; fit and report the primary quadratic specification for the remaining eight systems, each contributing target mass 1/8. Do not report a Korea estimate under this specification. | Influence analysis for the sparse, weakest-overlap system, not evidence that Korea should be omitted from the primary analysis. |
| `strict_common_cells` | Integration support | Begin with the primary nine-system 1st–99th percentile rectangle. Within each PV draw or JRR replicate, form a 4 × 4 grid using weighted quartiles of mathematics and science under that replicate’s primary equal-system rectangular target. Retain only cells in which **every** system has at least five unweighted BB records and Kish effective sample size at least three. Renormalize the equal-system target over retained cells. | A stricter common-cell estimand that reduces reliance on sparsely represented joint score regions. |
| `additive_linear` | Response surface | Retain all nine systems, the primary rectangle, and equal-system target; fit intercept plus linear mathematics and science deficits, without squares or interaction. | Low-complexity response-surface sensitivity. |
| `linear_interaction` | Response surface | Retain all nine systems, the primary rectangle, and equal-system target; fit intercept, linear mathematics and science deficits, and their product, without squared terms. | Tests whether the primary findings require domain-specific quadratic curvature. |

The strict-cell rule was frozen after examining risk-set size and overlap diagnostics but before calculating strict-cell recovery estimates. In the existing full-sample geometry check, the rule retained 10–11 of 16 cells and 75.8%–85.9% of the primary equal-system target mass across the five PV draws. These figures justify computational feasibility; they are not outcomes from the alternative analysis.

No generalized-overlap target, spline surface, convex-hull intersection estimator, or additional threshold will be introduced as a post hoc alternative. Convex-hull coverage remains a diagnostic. This minimal set directly tests target weighting, influence of the weak-overlap system, joint support, and response-surface form without creating an uncontrolled specification search.

## Estimand components and language

For every system and applicable specification, retain four separately labeled quantities:

1. `raw_full`: observed weighted recovery in the complete baseline-BB risk set;
2. `raw_support`: observed weighted recovery among that system’s records in the specification’s integration support;
3. `model_own_support`: fitted recovery averaged over that same system-specific supported empirical distribution; and
4. `standardized_common`: fitted recovery averaged over the specification’s shared target distribution.

The identity

`standardized_common - raw_full = (raw_support - raw_full) + (model_own_support - raw_support) + (standardized_common - model_own_support)`

will be reported as support-restriction, in-support model-calibration, and modeled target-distribution components. The total must not be called a pure “composition effect” or causal decomposition.

## PV and JRR inference freeze

The official TIMSS 2023 Longitudinal documentation states that five achievement PVs were imputed simultaneously from country-specific four-dimensional models covering mathematics 2023, mathematics 2024, science 2023, and science 2024. The database user guide identifies the terminal digit 1–5 as the PV index. Accordingly, draw `j` is the matched tuple `(ASMMAT1j, ASMMAT2j, ASSSCI1j, ASSSCI2j)` for Grades 4–5 and the analogous `B` variables for Grades 8–9. Of the 25 ordered index pairs, only the five equal-index tuples are valid primary imputations; the 20 unequal-index pairs are excluded from primary inference and remain only in the explicitly labeled same-wave measurement-crossing diagnostic.

Evidence locations:

- [TIMSS 2023 Longitudinal Technical Report](https://doi.org/10.6017/lse.tpisc.timss.de4812), Section 6, printed pp. 32 and 36–37 (PDF pp. 37 and 41–42).
- [TIMSS 2023 Longitudinal User Guide for the International Database](https://timss2023.org/wp-content/uploads/data/T23L_UG-International-Database.pdf), Chapter 2, printed pp. 51–52 (PDF pp. 57–58), Exhibit 2.5.

Within each matched PV draw, one stacked parameter vector will contain the raw quantities and every applicable standardized specification. Each JRR perturbation will rebuild all affected support bounds, cell boundaries and eligibility rules, target weights, fitted model for the perturbed system, and standardized estimates. This yields cross-system and cross-specification covariance. Unperturbed system models remain fixed when only another system’s weights change, while all estimates affected by the shared target are recomputed. The five draw-level covariance matrices will be pooled with the same PV combining rule as the primary analysis.

A replicate must never be silently dropped. Any empty support, invalid target, rank deficiency, nonfinite coefficient or estimate, model nonconvergence, or covariance failure stops canonical artifact replacement.

## Frozen diagnostics

Full-sample and replicate diagnostics will cover:

- risk-set and supported unweighted counts, weighted retention, and Kish effective sample size;
- common bounds, retained cell count, and retained target mass;
- model matrix columns and rank, weighted-design condition number, iterations, convergence, and maximum absolute coefficient;
- weighted observed event rate, weighted mean fitted probability, their difference, and weighted Brier score;
- minimum and maximum linear predictors and predicted probabilities, plus separate pooled-evaluation, own-system, and common-target counts and weighted shares requiring the frozen numerical linear-predictor clamp at ±30; and
- expected, attempted, and completed replicate/support/refit counts, together with retained warning counts, by specification.

Because execution is fail-fast, no successful artifact reports fictional zero-valued failure counters: an incomplete or failed stage prevents canonical replacement. All probability estimates must be finite and in `[0,1]`; covariance matrices must be finite, symmetric, and positive semidefinite to numerical tolerance.

## Novelty boundary

A focused, nonsystematic search of TIMSS/large-scale-assessment and methodological literature identified prior work on transition descriptions, benchmark classification, plausible-value inference, direct standardization, common support, and JRR. Because that search was not a systematic review and cannot establish absence or priority, the manuscript does not claim a “first,” a new estimator, or novelty of any component. It describes the contribution as an integrated descriptive application combining a joint four-state mathematics–science longitudinal transition, BB-to-AA attainment, explicit shared baseline-severity targets, matched four-dimensional longitudinal/cross-domain PVs, and replicate-wise rebuilding of shared support and target with full covariance. Key nearby literature and checks included DOI records `10.1186/s40536-016-0026-2`, `10.1177/016146811711900507`, `10.1016/j.sbspro.2016.02.134`, `10.1111/j.1745-3984.1995.tb00462.x`, `10.2307/2530043`, `10.1093/pan/mpj004`, `10.1093/biomet/asn055`, and `10.1186/s40536-016-0017-3`.

## Journal and packaging constraints retained

Requirements were checked on 12 August 2026 against the official [Large-scale Assessments in Education submission guidance](https://largescaleassessmentsineducation.springeropen.com/submission-guidelines/preparing-your-manuscript): structured abstract of at most 350 words using Background, Methods, Results, and Conclusions; 3–10 keywords; double-anonymous review; APA references; disclosure of generative-AI use in Methods or another suitable section; additional files no larger than 20 MB each; and figure files no larger than 10 MB each.

Deliverables will use the labels “Additional file 1,” “Additional file 2,” and “Additional file 3.” Public guidance does not expressly confirm ZIP acceptance, so archive acceptance must be checked in the live submission portal. Unknown author identities, affiliations, declarations, funding, contributions, acknowledgements, repository identifiers, and reviewer suggestions remain author-supplied and must not be fabricated.
