0e9d988710b1a8b117dd5e3c2782c949 mathematics_14_02052.pdf e3ab2bf0ad61539f84e49f16dd540acefb496201 mathematics_14_02052.pdf d011f836444918c364b4a1a9e022d1a84afa11d1ac2c255f3dd06d6538ac5dd2 mathematics_14_02052.pdf Title: The Same Problem by Different Names: Unifying Regression Dilution and Regression to the Mean Subject: Regression to the Mean (RTM) and Regression Dilution are traditionally treated as unrelated issues in the clinical and ecological studies. In this work, we demonstrate that within a linear errors-in-variables framework where baseline variables are subject to transient temporal or measurement noise, these two phenomena share an identical underlying mathematical signature. We unify these disparate traditions by comparing specialized clinical tools, such as the Berry shrinkage correction, with standard sign-agnostic structural estimators like Major Axis (MA) and Reduced Major Axis (RMA) regression. Using an analytical framework, we evaluate the closed-form population limits and finite-sample performance of these methods across various noise-to-signal ratios and sample sizes. Our results show that the Berry method is a specialized tool designed for clinical scenarios where a 1:1 relationship is expected. However, applying it to ecological trade-offs with negative slopes can lead to severe errors. We provide maps of optimality to identify which estimator most accurately recovers the true biological signal under different conditions. By reconciling these disparate methods, we offer a principled guide for researchers to choose the correct tool based on their data's noise profile rather than their disciplinary tradition. Keywords: structural regression; regression to the mean; regression dilution; estimator optimality Author: José F. Fontanari and Mauro Santos Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25; modified using OpenPDF 1.4.2 CreationDate: Tue Jun 9 10:51:05 2026 CEST ModDate: Tue Jun 9 10:54:43 2026 CEST Custom Metadata: yes Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 22 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 381337 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- XZGLVJ+URWPalladioL-Bold Type 1 Custom yes yes yes 126 0 DKPPLL+URWPalladioL-Roma Type 1 Custom yes yes yes 127 0 EKHUAH+URWPalladioL-Ital Type 1 Custom yes yes yes 128 0 NVVLXS+PazoMath-Italic Type 1 Builtin yes yes yes 169 0 PQOUBI+CMSY10 Type 1 Builtin yes yes yes 207 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 208 0 SYFPBV+CMMI10 Type 1 Builtin yes yes yes 248 0 DSUBXR+CMEX10 Type 1 Builtin yes yes yes 274 0 IKXQUG+PazoMath Type 1 Builtin yes yes yes 299 0 Symbol Type 1 Symbol no no yes 366 0 HMMSBE+Serif-Oblique Type 1C WinAnsi yes yes yes 367 0 ZCFGQQ+Serif Type 1C WinAnsi yes yes yes 368 0 Symbol Type 1 Symbol no no yes 439 0 HMMSBE+Serif-Oblique Type 1C WinAnsi yes yes yes 440 0 AVACLZ+Serif Type 1C WinAnsi yes yes yes 441 0 HDXTYU+Serif-Oblique Type 1C WinAnsi yes yes yes 469 0 Symbol Type 1 Symbol no no yes 470 0 QYCKKN+Serif Type 1C WinAnsi yes yes yes 471 0 HDXTYU+Serif-Oblique Type 1C WinAnsi yes yes yes 476 0 Symbol Type 1 Symbol no no yes 477 0 QYCKKN+Serif Type 1C WinAnsi yes yes yes 478 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2026-07-17 04:44:08 CEST RepresentationInformation: mathematics_14_02052.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2026-07-16 11:24:48 CEST Size: 381337 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 615 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: ViewerPreferences: HideToolbar: false HideMenubar: false HideWindowUI: false FitWindow: true CenterWindow: false DisplayDocTitle: false NonFullScreenPageMode: UseNone Direction: L2R ViewArea: CropBox ViewClip: CropBox PrintArea: CropBox PageClip: CropBox PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Item: Title: Statement of the Problem Item: Title: Classic Structural Estimators Children: Item: Title: The Major Axis (MA) Estimator Item: Title: The Reduced Major Axis (RMA) Estimator Item: Title: The Exact Solution: Recovering the True Structural Slope Item: Title: Comparative Analysis of the Estimators Children: Item: Title: Pathological Behavior Under the Null Hypothesis Item: Title: Domains of Optimality: Phase Diagram Analysis Children: Item: Title: The OLS-RMA Frontier Item: Title: The OLS-MA Frontier Item: Title: The Triple Point Item: Title: The RMA-MA Frontier Item: Title: Reconciling Clinical RTM Corrections with Structural Estimators Item: Title: Sample Size Effects on the Domains of Optimality Item: Title: Conclusions Item: Title: Appendix A Item: Title: References Info: Title: The Same Problem by Different Names: Unifying Regression Dilution and Regression to the Mean Author: José F. Fontanari and Mauro Santos Subject: Regression to the Mean (RTM) and Regression Dilution are traditionally treated as unrelated issues in the clinical and ecological studies. In this work, we demonstrate that within a linear errors-in-variables framework where baseline variables are subject to transient temporal or measurement noise, these two phenomena share an identical underlying mathematical signature. We unify these disparate traditions by comparing specialized clinical tools, such as the Berry shrinkage correction, with standard sign-agnostic structural estimators like Major Axis (MA) and Reduced Major Axis (RMA) regression. Using an analytical framework, we evaluate the closed-form population limits and finite-sample performance of these methods across various noise-to-signal ratios and sample sizes. Our results show that the Berry method is a specialized tool designed for clinical scenarios where a 1:1 relationship is expected. However, applying it to ecological trade-offs with negative slopes can lead to severe errors. We provide maps of optimality to identify which estimator most accurately recovers the true biological signal under different conditions. By reconciling these disparate methods, we offer a principled guide for researchers to choose the correct tool based on their data's noise profile rather than their disciplinary tradition. 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