Biases in galaxy cluster velocity dispersion and mass estimates in the small Ngal regime

Aims. We present a study of the statistical properties of three velocity dispersion and mass estimators: biweight, gapper, and standard deviation for a small number of galaxies (Ngal ≤ 75). Methods. Using a set of 73 numerically simulated galaxy clusters, we first characterised the statistical bias and the variance for each one of the three estimators (biweight, gapper, and standard deviation) in the determination of the velocity dispersion and the dynamical mass of the clusters through the σ–M relation. These results were used to define a new set of unbiased estimators that are able to correct for these statistical biases with a minimum increase in associated variance. We also used the same set of numerical simulations to characterise two other physical biases that affect the estimates: the effect of velocity segregation on the selection of cluster members, and the effect of using cluster members within different physical radii from the cluster centre. Results. The standard deviation (and its unbiased counterpart) is the estimator with the lowest variance estimator after the biweight and gapper. The effect of velocity segregation in the selection of galaxies within the sub-sample of the most massive galaxies in the cluster introduces a bias of 2% in the velocity dispersion estimate when it is calculated using a quarter of the most massive cluster members. We also find a dependence of the velocity dispersion estimate on the aperture radius as a fraction of R200. This is consistent with previous results in the literature. Conclusions. The proposed set of unbiased estimators effectively provides a correction of the velocity dispersion and mass estimates from the statistical and physical effects discussed above for small numbers of cluster members. When these new estimators are applied to a subset of simulated observations, they can retrieve bias-corrected values for the mean velocity dispersion and the mean mass; the standard deviation has the lowest variance. Although for a single galaxy cluster the statistical and physical effects discussed here are comparable to or slightly smaller than the bias introduced by interlopers, they are relevant when ensemble properties and scaling relations for large number of clusters are studied.

Variability in the Reporting of Serum Urate and Flares in Gout Clinical Trials: Need for Minimum Reporting Requirements

Objective.To describe the ways in which serum urate (SU) and gout flares are reported in clinical trials, and to propose minimum reporting requirements.Methods.This analysis was done as part of a systematic review aiming to validate SU as a biomarker for gout. The ways in which SU and flares were reported were extracted from each study by 2 reviewers.Results.A total of 22 studies (10 randomized controlled trials, 3 open-label extension studies, and 9 observational studies) were identified. There were 3 broad categories of SU reporting: percentage at target SU, mean SU, and change in SU. A median of 2 (range 1–3) categories were reported across all studies. The most common method of reporting SU was percentage at target in 17/22 (77.3%) studies, with all studies reporting a target of SU < 6 mg/dl. There were 12/22 (54.5%) studies reporting mean SU at some time after study entry, with 7 (58.3%) of these reporting at more than just the final study visit. Two ways of reporting gout flares were identified: mean flare rate and percentage of participants with flares. There was variability in time periods over which flares rates were reported.Conclusion.There is inconsistent reporting of SU and flares in gout studies. Reporting the percentage of participants who achieve a target SU reflects international treatment guidelines. SU should also be reported as a continuous variable with a relevant central and dispersion estimate. Gout flares should be reported as both percentage of participants and mean flare rates at each timepoint.