On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach
Abstract
I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small to intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, active galactic nucleus fraction, supermassive black hole fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widelyused techniques for estimating binomial confidence intervals  the `normal approximation' and the Clopper & Pearson approach  are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (i) to reveal the ease with which (Bayesian) binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g. r, matlab, mathematica, idl, python); and (ii) to demonstrate convincingly the major flaws of both the `normal approximation' and the Clopper & Pearson approach for error estimation.
 Publication:

Publications of the Astronomical Society of Australia
 Pub Date:
 June 2011
 DOI:
 10.1071/AS10046
 arXiv:
 arXiv:1012.0566
 Bibcode:
 2011PASA...28..128C
 Keywords:

 methods: data analysis;
 methods: statistical;
 Astrophysics  Instrumentation and Methods for Astrophysics;
 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 13 pages, 6 figures, accepted for publication in PASA