On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach

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 widely-used 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 estimatio

The star cluster mass--galactocentric radius relation: Implications for cluster formation

Whether or not the initial star cluster mass function is established through a universal, galactocentric-distance-independent stochastic process, on the scales of individual galaxies, remains an unsolved problem. This debate has recently gained new impetus through the publication of a study that concluded that the maximum cluster mass in a given population is not solely determined by size-of-sample effects. Here, we revisit the evidence in favor and against stochastic cluster formation by examining the young ($\lesssim$ a few $\times 10^8$ yr-old) star cluster mass--galactocentric radius relation in M33, M51, M83, and the Large Magellanic Cloud. To eliminate size-of-sample effects, we first adopt radial bin sizes containing constant numbers of clusters, which we use to quantify the radial distribution of the first- to fifth-ranked most massive clusters using ordinary least-squares fitting. We supplement this analysis with an application of quantile regression, a binless approach to rank-based regression taking an absolute-value-distance penalty. Both methods yield, within the $1\sigma$ to $3\sigma$ uncertainties, near-zero slopes in the diagnostic plane, largely irrespective of the maximum age or minimum mass imposed on our sample selection, or of the radial bin size adopted. We conclude that, at least in our four well-studied sample galaxies, star cluster formation does not necessarily require an environment-dependent cluster formation scenario, which thus supports the notion of stochastic star cluster formation as the dominant star cluster-formation process within a given galaxy.Comment: ApJ, in press, 39 pages in AAS preprint format, 10 multi-panel figures (some reduced in size to match arXiv compilation routines