# An investigation into the statistical precision attainable with a distribution-free method of constructing age-dependent reference centiles

## TL;DR

This paper evaluates how accurately age-dependent reference centiles can be constructed using a distribution-free method, focusing on sample size requirements.

## Contribution

The study fills a gap by analyzing the statistical precision of a distribution-free method for age-dependent reference centiles using Monte Carlo simulations.

## Key findings

- Sample sizes required for the distribution-free method are generally within a reasonable range for practical studies.
- The distribution-free method is more efficient than quantile regression in terms of sample size requirements.
- Results cover various conditional distributions and relationships between standard deviation and age.

## Abstract

The distribution-free approach to the construction of age-dependent reference centiles which has been originally published by this author in 1995 and applied since then in a multitude of large-scale studies has never been investigated from a sample-size planning perspective. In the present paper, this gap is filled using the precision criterion introduced by Jennen-Steinmetz and Wellek (2005) for the estimation of reference centiles for quantitative diagnostic markers being independent of other variables, and extended by Jennen-Steinmetz (2014) to the study of age-dependent markers. In the age-dependent case, that criterion does not admit an exact representation as a function of the sample size, even when interest is in estimating a one-sided reference limit. Hence, all sample-size results presented here are based on Monte Carlo simulation. The computations cover a broad range of conditional distributions of the marker at given age including both symmetric and positively skewed distributions. For the relationship between the conditional standard deviation and age, a linear function of different slopes was assumed. Except for the most extreme settings investigated, the sample sizes shown in the tables summarizing our numerical results do not exceed the order of magnitude which has been available for a recent, potentially very influential reference-value study of basic parameters making-up the normal fetal growth profile. Furthermore, our results suggest that in terms of sample-size requirements, the distribution-free approach of Wellek & Merz (1995) to the construction of age-dependent reference ranges is typically a good bit more efficient than reference-range determination by means of quantile regression.