Unified approach to optimal estimation of mean and standard deviation from sample summaries

Abstract

Recently, various methods have been developed to estimate the sample mean and standard deviation when only the sample size, and other selected sample summaries are reported. In this paper, we provide a unified approach to optimal estimation that can be easily adopted when only some summary statistics are reported. We show that the proposed estimators have the lowest variance among linear unbiased estimators. We also show that in the most commonly reported cases, that is, when only a three-number or five-number summary is reported, the newly proposed estimators match the previously developed estimators. Finally, we demonstrate the performance of the estimators numerically.

Standard error estimation in meta-analysis of studies reporting medians

“…For simpler transformation-based approaches whose mean and standard deviation estimators are linear combinations of sample quantiles, the SE can be analytically derived under parametric assumptions (e.g. see Yang et al 13 and Balakrishnan et al 14 ). Deriving the SE is more challenging for the methods of McGrath et al 7 and Cai et al 12 because their estimators do not have simple analytic expressions, as they involve model selection and are based on solutions to ad-hoc optimization problems.…”

“…Such approaches, which we refer to as transformationbased approaches, were first proposed and systematically evaluated by Hozo et al 2 and have been further developed by a number of authors in recent years. [3][4][5][6][7][8][9][10][11][12][13][14] Reflecting their widespread application, Google Scholar lists over 10,000 articles citing these transformation-based approaches [2][3][4][5][6][7][8][9][10][11][12][13][14] as of 1 May 2022.…”

metamedian: An R package for meta-analyzing studies reporting medians

“…The most commonly applied approach to meta-analyze studies reporting sample medians involves imputing sample means and standard deviations of the outcome from studies reporting medians [2,3,4,5,6,7,8,9,10,11,12,13,14]. Then, data analysts may apply standard meta-analytic methods based on the (imputed) study-specific sample means and standard deviations.…”

“…Motivated by the observation that studies often report sample medians instead of sample means because the distribution of the outcome is skewed, Kwon and Reis [5], McGrath et al [7], Shi et al [9], and Cai et al [12] developed estimators for skewed data. Other estimators have been developed which require the data analysts to specify the assumed parametric distribution of the outcome, such as the normal or log-normal distribution [13,14].…”

“…Mean-based methods impute the sample means and standard deviations of the outcome from primary studies that report sample medians in order to apply standard meta-analytic methods based on the (imputed) study-specific sample means and standard deviations. Many authors have proposed sample mean and standard deviation estimators for this context [2,3,4,5,6,7,8,9,10,11,12,13,14]. Median-based methods directly meta-analyze the study-specific sample medians in order to estimate a pooled median of the outcome or the difference of medians across groups [15,16,17].…”