[Paper Review] Domain of Inverse Double Arcsine Transformation

Jong‐Hyeon Jeong|arXiv (Cornell University)|Nov 19, 2018

Molecular spectroscopy and chirality4 citations

TL;DR

This paper analyzes the domain and range of the inverse double arcsine transformation, identifying erratic behavior near 0 and 1 in small samples. It proposes sample-size-dependent approximation methods and introduces a maximum percent error (MPE) metric to quantify accuracy, enabling sample size determination for desired precision in meta-analytic proportion back-transformation.

ABSTRACT

To combine the proportions from different studies for meta-analysis, Freeman and Tukey double arcsine tranformation can be useful for normalization and variance stabilization. The inverse function of the double arcsine transformation has been also derived in the literature to recover the original scale of the proportion after aggregation. In this brief note, we present the domain and range of the inverse double arcsine transformation both analytically and graphically. We notice an erratic behavior in the mathematical formula for the inverse double arcsine tranformation at both limits of its domain, and propose approximation methods for both small and large samples. We also propose a simple accuracy measure, the maximum percent error (MPE), of the large sample approximation, which can be used to determine the sample size that would provide a certain accuracy level, and conversely to determine the accuracy level of the approximation given a sample size.

Motivation & Objective

• To clarify the domain and range of the inverse double arcsine transformation, especially near 0 and 1.
• To address numerical instability in the inverse transformation formula at domain boundaries.
• To propose a flexible, sample-size-dependent approximation method for small and large samples.
• To introduce a maximum percent error (MPE) metric for evaluating large-sample approximation accuracy.
• To provide a practical tool for determining required sample size to achieve a target accuracy level in meta-analysis.

Proposed method

• Analytically derives the domain and range of the inverse double arcsine transformation based on the original double arcsine transformation.
• Identifies erratic behavior in the inverse formula near 0 and 1, especially for small n, and proposes boundary clipping: p(θ) = 0 below lower limit, p(θ) = 1 above upper limit.
• Proposes the maximum percent error (MPE) as a function of sample size n to quantify deviation between the double arcsine and its limiting arcsine form.
• Derives a closed-form equation for required sample size n as a function of desired MPE level ε: n = tan²(π(1/2 − ε)).
• Uses the limiting inverse function sin²(θ) for large samples, valid when n → ∞.
• Validates the MPE measure by computing it for specific n values (e.g., n = 200 → MPE = 2.2%, n = 500 → MPE = 1.4%).

Experimental results

Research questions

• RQ1 What is the correct domain and range of the inverse double arcsine transformation for finite sample sizes?
• RQ2 Why does the inverse transformation formula exhibit erratic behavior near 0 and 1, and how can this be corrected?
• RQ3 How can the accuracy of the large-sample approximation (sin²(θ)) be quantitatively measured across different sample sizes?
• RQ4 What sample size is needed to achieve a pre-specified accuracy level in the inverse transformation approximation?
• RQ5 How can the MPE metric be used to determine the accuracy level of the approximation for a given sample size?

Key findings

• The inverse double arcsine transformation exhibits erratic behavior at both ends of its domain (near 0 and 1), especially for small n, necessitating boundary adjustments.
• For small samples, the inverse function should be capped at 0 below the lower domain limit and at 1 above the upper limit to preserve one-to-one recovery of original proportions.
• The maximum percent error (MPE) between the double arcsine and its limiting arcsine form is highest at p = 1, and is given by δ(1) = 1/2 − (1/π)sin⁻¹(√(n/(n+1))).
• The required sample size to achieve a target MPE level ε is n = tan²(π(1/2 − ε)), with n approaching infinity as ε → 0.
• For n = 200, the MPE is 2.2%; for n = 500, it is 1.4%, demonstrating decreasing error with increasing sample size.
• The MPE metric enables both forward planning (determine n for desired accuracy) and backward evaluation (determine accuracy for a given n), providing a practical tool for meta-analysis.