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A note on closed-form solutions for estimating sample size when externally validating a binary prediction model based on $C$ -statistic precision

This paper introduces seven novel closed-form solutions for calculating the sample size required to precisely estimate the C-statistic during the external validation of binary prediction models, demonstrating that these mathematically equivalent formulas offer identical accuracy to existing iterative methods while being up to 264,000 times faster in execution.

Original authors: Denis A. Shah, Erick D. De Wolf, Pierce A. Paul, Laurence V. Madden

Published 2026-05-25

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CC BY 4.0

Original authors: Denis A. Shah, Erick D. De Wolf, Pierce A. Paul, Laurence V. Madden

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have built a crystal ball that predicts whether a patient will get sick or stay healthy. Before you let doctors use this crystal ball in real life, you need to test it on a new group of people to see if it actually works. This is called "external validation."

But here's the catch: To get a trustworthy test, you need just the right number of people. Too few, and your test is a fluke; too many, and you've wasted time and money.

For a long time, figuring out that "just right" number was like trying to solve a maze in the dark. The math required to calculate the sample size was so twisted that experts couldn't find a direct path. Instead, they had to use a slow, computerized method called an "iterative approach." Think of this like trying to guess the weight of a watermelon by putting it on a scale, guessing, taking it off, adding a little weight, guessing again, and repeating this process a million times until you get close. It works, but it's slow and tedious.

The Big Discovery
The authors of this paper, a team of statisticians from Kansas and Ohio, decided to see if there was a shortcut. They asked: "Is there a direct formula—a straight line through the maze—that gives us the answer instantly?"

They used two types of modern tools to find this shortcut:

Computer Algebra Systems (CAS): These are like super-calculators that can do complex math symbolically (like a robot mathematician).
Artificial Intelligence (AI): They asked several different AI chatbots to solve the same math puzzle.

The Result: Seven New Keys
Surprisingly, they found seven different ways to write the direct formula. It's as if they found seven different keys that all open the exact same door.

The "Robot" Keys: Two of the formulas came from the super-calculators (Mathematica and Maxima). They worked perfectly but looked like messy, tangled spaghetti equations that were hard for a human to read.
The "AI" Keys: The other five came from AI models. Some of these AIs got confused and gave wrong answers (like a student guessing the wrong answer on a test). But the ones that got it right produced formulas that were much cleaner and easier to understand.

Why Does This Matter?
The paper proves that these new formulas give the exact same answer as the old, slow "guessing" method. If the old method says you need 1,154 people, the new formulas say 1,154 people.

However, the difference is in speed.

The old method is like a snail.
The new formulas are like a rocket.

The authors ran a race to see how fast they were. The new formulas were 148,000 to 264,000 times faster than the old method.

The "Best" Key
Since all seven formulas give the same result, which one should you use? The authors recommend the one generated by an AI tool called MathGPT.

Why? It's the most elegant. It breaks the complex math down into two simple, intermediate steps (like building a house with two sturdy bricks instead of a pile of rubble). This makes it much harder for a human to make a typo when typing it into a computer program.

A Warning About AI
The paper also offers a friendly warning: While AI is powerful, it can be unreliable with math. Some of the AI models they used gave answers that looked right but were actually wrong when checked with real numbers. So, you can't just trust the AI blindly; you still need a human to double-check the work.

In a Nutshell
This paper is a victory lap for efficiency. The authors took a math problem that everyone thought required a slow, roundabout path and showed that there is actually a direct highway. They built seven different maps for this highway, proved they all lead to the same destination, and showed that driving the new highway is nearly instant compared to the old road. This allows researchers to plan their studies much faster and more efficiently.

Technical Summary: Closed-Form Solutions for Estimating Sample Size in External Validation of Binary Prediction Models

Problem Statement
External validation is a critical step in determining whether clinical prediction models for binary outcomes are fit for use. A key requirement for proper validation is a sufficient sample size to precisely estimate model performance metrics, specifically the C-statistic (equivalent to the Area Under the ROC Curve). While Riley et al. previously proposed formulas for determining the minimum sample size required to estimate the C-statistic with a desired precision, their approach relied on an iterative numerical solution. This was necessitated by the assumption that a direct algebraic rearrangement of Newcombe's formula for the standard error of the C-statistic, $SE(C)$, was "not possible." Consequently, existing software implementations (in R and Stata) utilize computationally intensive iterative methods to solve for the sample size ( $N$ ).

Methodology
The authors challenge the assumption that a closed-form solution is unattainable. They systematically attempted to algebraically rearrange Newcombe's formula to solve for $N$ as a function of the C-statistic ( $C$ ), the anticipated outcome event proportion ( $\phi$ ), and the target standard error ($SE(C)$).

The methodology employed two distinct categories of computational tools:

Computer Algebra Systems (CAS): The authors utilized commercial (Mathematica 12.0) and open-source (Maxima 23.05.1) software to perform symbolic manipulation, squaring the equation, simplifying fractions, and solving for roots.
Artificial Intelligence (AI) Models: The authors prompted several Large Language Models (LLMs), including Perplexity's Sonar, Anthropic's Claude Sonnet 4.0, OpenAI's GPT 4.1, xAI's Grok 3 Beta, Google's Gemini 2.5 Pro, and MathGPT. The models were instructed to derive a closed-form solution, identify repeated patterns for substitution, and verify the results numerically.

The authors noted that while AI models successfully generated algebraic structures, they frequently required manual verification and correction, as some models produced algebraically plausible but numerically incorrect results or failed to simplify expressions optimally.

Key Contributions
The primary contribution of this work is the presentation of seven novel closed-form solutions for calculating the required sample size ( $N$ ). These solutions were derived using different computational pathways (CAS and various AI models), resulting in structurally distinct but mathematically equivalent formulas.

Key aspects of the derivation include:

Mathematical Equivalence: Despite differing algebraic forms (some utilizing quadratic formula structures, others involving complex substitutions), all seven solutions yield identical numerical estimates for $N$ .
Verification: The authors validated these solutions against the iterative method used by Riley et al. across five illustrative examples and a grid-based sensitivity analysis involving 450 combinations of parameters ( $C$ from 0.55 to 0.95 and $\phi$ from 0.01 to 0.5). In all cases, the closed-form solutions matched the iterative results (with absolute differences $\le 1$ due to integer rounding conventions).
Implementation: The authors provide R code functions implementing all seven closed-form solutions and the original iterative method.

Results

Accuracy: The closed-form solutions produce sample size estimates identical to the established iterative approach.
Computational Efficiency: A benchmarking analysis revealed a massive performance disparity. The closed-form solutions were, on average, 148,000 to 264,000 times faster in median execution time than the iterative implementation.
Performance Variance: Among the closed-form solutions, those derived via AI (specifically GPT 4.1 and MathGPT) demonstrated slightly faster execution times compared to those derived via traditional CAS (Mathematica and Maxima), though all operated within the same "nanosecond" performance ballpark.
Robustness: The solutions remained stable across a wide parameter space, with no observed numerical instability.

Significance and Claims
The paper asserts that the existence of closed-form solutions transforms the calculation of validation sample sizes from a computationally iterative task into a direct, instantaneous calculation. While the speed difference is negligible for a single calculation, the authors highlight that the efficiency and stability of closed-form solutions are critical for computationally intensive tasks, such as:

Sensitivity analyses involving grid searches.
Extensive simulation studies where the function is called millions of times.

The authors recommend the solution derived by MathGPT for routine use. They argue that this specific formulation is the most "elegant," reducing the final equation to a compact quadratic form via two intermediate variables ( $A$ and $\delta$ ). This modularity minimizes the risk of transcription errors during coding and offers superior algebraic clarity compared to the expanded high-degree polynomials found in the CAS-derived solutions.

Finally, the paper notes that while the inversion of Newcombe's formula is unique in its complexity within the Riley et al. framework, the strategy of leveraging CAS and AI to resolve "intractable" algebraic problems is broadly applicable to other biostatistical challenges where mathematical complexity has historically hindered the development of exact solutions. The authors caution, however, that AI tools require careful human evaluation, as they are prone to generating plausible-sounding but mathematically flawed derivations.

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