The DOI matches its Crossref record; 21 of the 21 references it cites were found in a registry or an index.
Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range
Open accessAbstract
The era of big data is coming, and evidence-based medicine is attracting increasing attention to improve decision making in medical practice via integrating evidence from well designed and conducted clinical research. Meta-analysis is a statistical technique widely used in evidence-based medicine for analytically combining the findings from independent clinical trials to provide an overall estimation of a treatment effectiveness. The sample mean and standard deviation are two commonly used statistics in meta-analysis but some trials use the median, the minimum and maximum values, or sometimes the first and third quartiles to report the results. Thus, to pool results in a consistent format, researchers need to transform those information back to the sample mean and standard deviation. In this article, we investigate the optimal estimation of the sample mean for meta-analysis from both theoretical and empirical perspectives. A major drawback in the literature is that the sample size, needless to say its importance, is either ignored or used in a stepwise but somewhat arbitrary manner, e.g. the famous method proposed by Hozo et al. We solve this issue by incorporating the sample size in a smoothly changing weight in the estimators to reach the optimal estimation. Our proposed estimators not only improve the existing ones significantly but also share the same virtue of the simplicity. The real data application indicates that our proposed estimators are capable to serve as "rules of thumb" and will be widely applied in evidence-based medicine.
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“When studies reported a median and interquartile range, we estimated the mean and standard deviation (SD) using conventional formulae (Luo et al., 2018;Wan et al., 2014).When both of these were absent, we estimated the standard deviation using prognostic imputation (Ma et al., 2008).”
From Decision making capacity for treatment in psychiatric inpatients: a systematic review and meta-analysis (2024) · cites this paper as Luo et al., 2018
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Citations
Cites
Statistical power analysis for the behavioral sciences
Cohen, J. (1990) · Computers Environment and Urban Systems · doi:10.1016/0198-9715(90)90050-4 · Cited by 65,417
Measuring inconsistency in meta-analyses
Higgins, J. P. T., Thompson, S. G., Deeks, J. J., & Altman, D. G. (2003) · BMJ · doi:10.1136/bmj.327.7414.557 · Cited by 63,139 · cited 1×
“ods seem to be close to each other, we found that the I2 indices for heterogeneity between studies are quite different (i.e., 48.54% from Hozo et al.’s method and 34.85% from our method). According to Higgins et al. (2003), the value 48.54% of I2 is very close to moderate heterogeneity level while the value 34.85% is c…”
MethodInfluentialStatistical Power Analysis for the Behavioral Sciences
Cohen, J. (2013) · doi:10.4324/9780203771587 · Cited by 25,373
Wan, X., Wang, W., Liu, J., & Tong, T. (2014) · BMC Medical Research Methodology · doi:10.1186/1471-2288-14-135 · Cited by 11,735 · cited 3×
“Finally, by plugging the approximation formula (6) into the estimator (4), we propose the estimator for Scenario S1 as XðwÞ 4 4þ n0:75 aþ b 2 þ n 0:75”
BackgroundMethodSelf-citationEstimating the mean and variance from the median, range, and the size of a sample
Hozo, S. P., Djulbegovic, B., & Hozo, I. (2005) · BMC Medical Research Methodology · doi:10.1186/1471-2288-5-13 · Cited by 9,062
Guyatt, G. H. (1992) · JAMA · doi:10.1001/jama.1992.03490170092032 · Cited by 4,577 · cited 4×
“’s method has a change point near n1⁄4 25 (as suggested in (1)), especially for log-normal distribution and exponential distribution.”
BackgroundMethodInfluentialWilkinson, R. J., Llewelyn, M., Toossi, Z., Patel, P., Pasvol, G., Lalvani, A., Wright, D. J., Latif, M., & Davidson, R. N. (2000) · The Lancet · doi:10.1016/s0140-6736(99)02301-6 · Cited by 768
Low serum vitamin D levels and tuberculosis: a systematic review and meta-analysis
Nnoaham, K. E., & Clarke, A. (2008) · International Journal of Epidemiology · doi:10.1093/ije/dym247 · Cited by 682
Relations, Bounds and Approximations for Order Statistics
Arnold, B. C., & Balakrishnan, N. (1989) · Lecture notes in statistics · doi:10.1007/978-1-4612-3644-3 · Cited by 243
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Cited by
Sorted by citations
Estimating the sample mean and standard deviation from commonly reported quantiles in meta-analysis
McGrath, S., Zhao, X., Steele, R. J., Thombs, B. D., Benedetti, A., Levis, B., Riehm, K. E., Saadat, N., Levis, A. W., Azar, M., et al. (2020) · Statistical Methods in Medical Research · doi:10.1177/0962280219889080 · Cited by 869
Association of inflammatory markers with the severity of COVID-19: A meta-analysis
Zeng, F., Huang, Y., Guo, Y., Yin, M., Chen, X., Liang, X., & Deng, G. (2020) · International Journal of Infectious Diseases · doi:10.1016/j.ijid.2020.05.055 · Cited by 676
“Stratified data or interquartile range (IQR) were converted to mean ( SD) using mathematical formulas for meta-analysis (Luo et al., 2018; Wan et al., 2014).”
MethodOptimally estimating the sample standard deviation from the five‐number summary
Shi, J., Luo, D., Weng, H., Zeng, X., Lin, L., Chu, H., & Tong, T. (2020) · Research Synthesis Methods · doi:10.1002/jrsm.1429 · Cited by 622
“To have an approximation formula for the optimal weight, we numerically compute the true values of J(n) and wopt(n) for different values of n using formulas (7) and (8).”
BackgroundMethodSelf-citationFernández‐de‐las‐Peñas, C., Palacios‐Ceña, D., Gómez‐Mayordomo, V., Florêncio, L. L., Cuadrado, M. L., Plaza‐Manzano, G., & Navarro‐Santana, M. J. (2021) · European Journal of Internal Medicine · doi:10.1016/j.ejim.2021.06.009 · Cited by 607
Hassan, C., Spadaccini, M., Iannone, A., Maselli, R., Jovani, M., Chandrasekar, V. T., Antonelli, G., Yu, H., Areia, M., Dinis-Ribeiro, M., et al. (2021) · Gastrointestinal Endoscopy · doi:10.1016/j.gie.2020.06.059 · Cited by 529
“…mean and standard deviation were not reported for continuous outcomes, we calculated these statistics from median and interquartile range, according to the methods described by Luo et al 16 and Wan et al. 17 We calculated pooled estimates using the DerSimonian and Laird random-effects model 18 .”
Method
Metadata
- Authors
- Luo, D., Wan, X., Liu, J., & Tong, T.
- Affiliations
- Hong Kong Baptist University (HK)
- Corresponding author
- Wan, Xiang
- Venue
- Statistical Methods in Medical Research
- Publisher
- SAGE Publications
- Published
- 2016-09-27
- Volume
- 27
- Issue
- 6
- Pages
- 1785-1805
- ISSN
- 0962-2802 (ISSN-L)
- 1477-0334
- OpenAlex
- W1534181333
- Cited by
- 4,066
- References
- 21
- Citations by year
- 2026: 789 · 2025: 833 · 2024: 700 · 2023: 601 · 2022: 462 · 2021: 379 · 2020: 188 · 2019: 91 · 2018: 20 · 2017: 1
- FWCI
- 34.66
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Statistical Methods in Medical Research (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Hong Kong Baptist University (FRG2/14-15/077, FRG1/14-15/084, FRG2/15-16/019)
- Topics
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Clinical Trials
- Keywords
- sample mean estimation, meta-analysis, median, sample size, standard deviation estimation, optimal estimators, evidence-based medicine
- MeSH terms
- Algorithms, Sampling Studies, Meta-Analysis as Topic, Sample Size, Evidence-Based Medicine
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