Field Measurement Protocol ZX — TOST Tool for Testing √2−1

An interactive statistical tool running entirely in the browser to test whether a specific geometric ratio in trees (such as the crown ratio (H−h)/H) is equivalent to the Silver Ratio √2−1 = tan(π/8) ≈ 0.41421356.

Statistical Methodology

• TOST (Two One-Sided Tests): To test equivalence against a practical margin Δ instead of a traditional difference test.
• Pre-registration Lock: Locking the hypothesis at the first tree to prevent p-hacking.
• Bonferroni Correction: Correcting α/6 when no ratio is pre-specified.
• Accurate t-distribution: Via logGamma + betacf to calculate CI and p-value accurately.
• Power Approximation: Estimating the current statistical power and the required sample size.

Calculated Ratios

• h/H — Trunk ÷ Total Height
• r/H — Crown radius ÷ Height
• 2r/H — Full crown diameter ÷ Height
• d/H — Trunk diameter ÷ Height
• (H−h)/H — Crown ÷ Height (Primary target)
• h/(H−h) — Trunk ÷ Crown
• L/H — Shadow ÷ Height (Excluded from testing, as it depends on sun angle)

Reference Mango Sample Values

Exploratory sample n=5 mango trees (Sho'ub, Sana'a, 2026-10-14): Mean (H−h)/H = 0.41297, SD = 0.00845, SE = 0.00378, 90% CI = [0.40491, 0.42102], TOST p = 0.000801 with Δ = ±0.03. Result: Equivalence confirmed within margin, but analysis is exploratory (not pre-registered) and not generalizable as a law.

Tool Features

• Input tree measurements (H, h, r, d, L) with site, species, and time.
• Automatic real-time calculation of ratios.
• Pre-registration lock with the ability to unlock (recording the change in history).
• Filtering by tree species.
• Full statistical analysis with verdict (Equivalent / Different / Inconclusive).
• CSV export with pre-registration header + CSV import.
• All data saved locally (localStorage) — nothing is sent to a server.

By: Eng. Abdulla Mohammed Nasser Al-Jabri — Yemen. ORCID: 0009-0003-3319-3822. License: CC BY 4.0.