Estimating the sample mean and standard deviation from commonly reported quantiles in meta-analysis (1903.10498v1)

Abstract: Researchers increasingly use meta-analysis to synthesize the results of several studies in order to estimate a common effect. When the outcome variable is continuous, standard meta-analytic approaches assume that the primary studies report the sample mean and standard deviation of the outcome. However, when the outcome is skewed, authors sometimes summarize the data by reporting the sample median and one or both of (i) the minimum and maximum values and (ii) the first and third quartiles, but do not report the mean or standard deviation. To include these studies in meta-analysis, several methods have been developed to estimate the sample mean and standard deviation from the reported summary data. A major limitation of these widely used methods is that they assume that the outcome distribution is normal, which is unlikely to be tenable for studies reporting medians. We propose two novel approaches to estimate the sample mean and standard deviation when data are suspected to be non-normal. Our simulation results and empirical assessments show that the proposed methods often perform better than the existing methods when applied to non-normal data.

• The paper introduces two novel methods, QE and BC, that accurately estimate sample means and SDs from quantile data in skewed distributions.
• Simulation studies demonstrate the BC method's minimal bias and consistent performance across various skewness levels.
• These methods offer practical tools for meta-analysts to robustly synthesize non-normal data in medical research and other fields.

Estimating the Sample Mean and Standard Deviation from Commonly Reported Quantiles in Meta-Analysis: A Summary

This paper addresses the critical challenge in meta-analysis—integrating studies that report medians instead of means for skewed data distributions. The authors propose two novel methods (QE and BC) to estimate the sample mean and standard deviation (SD) from reported quantiles, particularly in the context of skewed data distributions, overcoming the limitations of commonly assumed normality in existing estimation methods.

Methods Overview

The authors review widely-used methods, such as those by Luo et al. (2017) and Wan et al. (2014), which provide formula-based estimators under the assumption of normality. Kwon and Reis's simulation-based method, known as ABC, offers an approach under various parametric assumptions, presenting computational challenges and sensitivity to outliers.

The proposed Quantile Estimation (QE) Method employs parametric distribution fitting to observed quantiles, optimizing parameter estimation to minimize distances between observed and model-predicted quantiles. Additionally, the Box-Cox (BC) method uses transformations to align skewed data closer to normality, applying the Luo and Wan methodologies post-transformation to then derive inverse-transformed estimates. The BC method showcases robustness across various skewness levels without prior distributional assumptions.

Simulation Study and Empirical Findings

Simulation studies, designed to benchmark existing versus proposed methods, incorporate various skewness degrees across scenarios $S_1, S_2, S_3$ with non-normal and normal distributions. The proposed methods generally outperform existing ones in scenarios with increased skewness. Specifically:

• The BC method stood out, displaying minimal bias in estimating sample means and SDs irrespective of skewness or sample size, maintaining ARE values comfortably within practical thresholds.
• Unlike the ABC and formula-based methods which exhibited rising bias at larger $n$ or extreme skewness conditions, the QE and BC methodologies retained performance consistency, even adapting to diverse distributional contexts during empirical evaluations with real-world data, such as PHQ-9 from the DEPRESsion Screening Data Collaboration.

Theoretical and Practical Implications

The contributions of this research are multi-faceted. Theoretically, the approaches enhance the estimation domain, enabling accurate mean and SD retrieval without stringent normality confines. Practically, they equip meta-analysts with advanced, computationally feasible tools for synthesizing diverse datasets, crucial when handling skewness-intensive data in medical meta-analyses, such as treatment durations, physiological measurements, or diagnostic indices.

Future Directions

Future research is encouraged to extend the BC and QE methods' applicability to diverse real-world datasets and meta-analytic contexts. Additionally, exploring their integration within broader Bayesian frameworks could provide robust, scalable solutions for combined effect estimation and population heterogeneity assessment within meta-analyses.

Through robust empirical validation and methodological innovation, this paper's proposed paradigms mark a significant evolution in acceptable practices for mean and standard deviation approximation in scenarios where typical assumptions of normality are inadequate. The accompanying R package 'estmeansd' and online app offer convenient access to these advanced methods, fostering widespread adoption and potential synergistic development across meta-analytic domains.