Estimating the sample mean and standard deviation from order statistics and sample size in meta-analysis

Abstract

In recent years, a growing number of researchers have attempted to overcome the constraints of size and scope in different medical studies to find out the overall treatment effects. As a widespread technique to combine results of multiple studies, commonly used meta-analytic approaches for continuous outcomes demand sample means and standard deviations of primary studies, which are absent sometimes, especially when the outcome is skewed. Instead, the median, the extrema, and/or the quartiles are reported. One feasible solution is to convert the preceding order statistics to demanded statistics to keep effect measures consistent. In this article, we propose new methods based on maximum likelihood estimation for known distributions with unknown parameters. For unknown underlying distributions, the Box–Cox transformation is applied to the reported order statistics so that the techniques for normal distribution can be utilized. Two approaches for estimating the power parameter in Box–Cox transformation are provided. Both simulation studies and real data analysis indicate that in most cases, the proposed methods outperform the existing methods in estimation accuracy.

Standard error estimation in meta-analysis of studies reporting medians

“…Like the BC approach, the Method for Unknown Non-Normal Distributions (MLN) [11] assumes that the underlying distribution is normal after applying a suitable Box-Cox transformation (i.e., g λ (X i )…”

“…The methodological contributions of this paper are as follows: 1) Focusing on the recently proposed transformation-based approaches of McGrath et al [7] and Cai et al [11], we illustrate that the standard application of transformation-based approaches severely underestimates the within-study SEs and consequently overestimates between-study heterogeneity in random effects meta-analyses. 2) We describe a parametric bootstrap approach to estimate within-study SEs in this context and systematically evaluate its performance at both the study-and meta-analytic level.…”

“…In the following subsections, we describe the transformation-based approaches proposed by McGrath et al [7] and Cai et al [11]. We focus on these approaches because they were developed for situations when the data were suspected to be non-normal -which is often the case when primary studies report the median of the outcome -and these approaches yielded excellent performance for estimating the mean and standard deviation in simulation studies under non-normal distributions.…”

“…Then, one applies standard meta-analytic approaches with inverse-variance weighting based on the (imputed) study-specific sample means and standard deviations. Such approaches, which we refer to as transformation-based 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]. Reflecting their widespread application, Google Scholar lists over 10,000 articles citing these transformation-based approaches (i.e., [2,3,4,5,6,7,8,9,10,11,12]) as of May 1 2022.…”

“…Even more explicitly, Hozo et al [2], Luo et al [6], McGrath et al [7], and Shi et al [9] used the the naïve SE estimator in data applications illustrating how to apply their proposed transformation-based approaches, for instance. While some have raised concerns over potential issues of the naïve SE estimator (e.g., see the open peer review report of Hozo et al [2]), the downstream consequences of using naïve SE estimators has been largely unexplored in the literature because most simulation studies evaluating the transformation-based approaches (e.g., see [2,3,4,5,6,7,8,9,10,11]) have almost entirely focused on the performance of these methods at the study-level (i.e., for estimating the mean and standard deviation of the outcome distribution for a given study) rather than at the meta-analytic level.…”