Calculation of sample size guaranteeing the required width of the empirical confidence interval with predefined probability

TL;DR

This paper discusses a method for calculating the sample size needed to ensure empirical confidence intervals have a specified width with a certain probability, highlighting its implementation and differences from traditional expected interval methods.

Contribution

It provides a concise description and framework for implementing a probability-guaranteed sample size calculation for empirical confidence intervals across various distributions.

Findings

Sample size for empirical intervals can be over 20% larger than for expected intervals.

The approach is applicable to Normal, Poisson, and Binomial distributions.

Implementation details are provided for practical use in statistical software.

Abstract

The goal of any estimation study is an interval estimation of a the parameter(s) of interest. These estimations are mostly expressed using empirical confidence intervals that are based on sample point estimates of the corresponding parameter(s). In contrast, calculations of the necessary sample size usually use expected confidence intervals that are based on the expected value of the parameter(s). The approach that guarantees the required probability of the required width of empirical confidence interval is known at least since 1989. However, till now, this approach is not implemented for most software and is not even described in many modern papers and textbooks. Here we present the concise description of the approach to sample size calculation for obtaining empirical confidence interval of the required width with the predefined probability and give a framework of its general implementation. We illustrate the approach in Normal, Poisson, and Binomial distributions. The numeric results showed that the sample size necessary to obtain the required width of empirical confidence interval with the standard probability of $0.8$ or $0.9$ may be more than 20\% larger than the sample size calculated for the expected values of the parameters.