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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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“[43], and the standard deviation was estimated as described by Wan et al.”
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Chang, T., Chou, C., & Chang, L. (2020) · Obesity Reviews · doi:10.1111/obr.13089 · Cited by 63
“If studies reported only median values with interquartile ranges alone, we derived means and standard deviations using statistical methods described by Chen et al.(14) and Gao et al.”
MethodSanfilippo, F., Huang, S., Messina, A., Franchi, F., Oliveri, F., Vieillard‐Baron, A., Cecconi, M., & Astuto, M. (2020) · Journal of Critical Care · doi:10.1016/j.jcrc.2020.12.026 · Cited by 63
Diagnostic Value of sST2 in Cardiovascular Diseases: A Systematic Review and Meta-Analysis
Zhang, T., Xu, C., Zhao, R., & Cao, Z. (2021) · Frontiers in Cardiovascular Medicine · doi:10.3389/fcvm.2021.697837 · Cited by 63
Haussen, D. C., Al‐Bayati, A. R., Mohammaden, M. H., Sheth, S. A., Salazar‐Marioni, S. A., Linfante, I., Dabus, G. C., Starosciak, A. K., Hassan, A. E., Tekle, W. G., et al. (2022) · Stroke Vascular and Interventional Neurology · doi:10.1161/svin.121.000234 · Cited by 63
Scott, H. A., Ng, S. H., McLoughlin, R. F., Valkenborghs, S. R., Nair, P. K., Brown, A. C., Carroll, O. R., Horvat, J. C., & Wood, L. G. (2023) · Thorax · doi:10.1136/thorax-2022-219268 · Cited by 76
Cai, S., Xu, S., Zhao, Y., & Zhang, L. (2025) · Allergy · doi:10.1111/all.16499 · Cited by 63
“If only the whole or part of the five-number summary (minimum, quartiles, and maximum) were reported, the estimated mean and SD were treated as true sample values, regardless of skewness [8, 9].”
Liu, L., Wang, T., & Lei, B. (2021) · European Radiology · doi:10.1007/s00330-021-08156-6 · Cited by 63
Guan, Z., Chen, C., Li, Y., Yan, D., Zhang, X., Jiang, D., Yang, S., & Li, L. (2021) · Frontiers in Public Health · doi:10.3389/fpubh.2021.773130 · Cited by 64
Lee, D., Chiavaroli, L., Ayoub‐Charette, S., Khan, T. A., Zurbau, A., Au‐Yeung, F., Cheung, A., Liu, Q., Qi, X., Ahmed, A., et al. (2022) · Nutrients · doi:10.3390/nu14142846 · Cited by 72
“When median data were provided, they were converted to mean data with corresponding variances using methods developed by Luo et al. (2018) [16] and Wan et al. (2014) [17].”
MethodResection Techniques During Robotic Partial Nephrectomy: A Systematic Review
Bertolo, R. G., Pecoraro, A., Carbonara, U., Amparore, D., Diana, P., Muselaers, S., Marchioni, M., Mir, M. C., Antonelli, A., Badani, K. K., et al. (2023) · European Urology Open Science · doi:10.1016/j.euros.2023.03.008 · Cited by 62
“tor was used [9], while the Box-Cox and quantile estimation methods were used for time-based data or data suspected of being skewed [10].”
Methodmetamedian : An R package for meta‐analyzing studies reporting medians
McGrath, S., Zhao, X., Öztürk, Ö., Katzenschlager, S., Steele, R., & Benedetti, A. (2023) · Research Synthesis Methods · doi:10.1002/jrsm.1686 · Cited by 61
“Since then, Luo et al. [6] and Shi et al. [8] developed new estimators to achieve certain optimality properties under the assumption that the outcome is normally distributed.”
BackgroundMethodInfluentialXu, M., Tang, Y., Wang, J., & Yang, J. Y. (2020) · International Journal of Obstetric Anesthesia · doi:10.1016/j.ijoa.2020.02.005 · Cited by 62
Neto, M. G., Lopes, I. A. D. S., Araujo, A. C. C. L. M., Oliveira, L. S., & Saquetto, M. B. (2020) · European Journal of Pediatrics · doi:10.1007/s00431-020-03640-5 · Cited by 62
An evolution perspective on the urban land carrying capacity in the urbanization era of China
Luo, W., Ren, Y., Shen, L., Zhu, M., Jiang, Y., Meng, C., & Zhang, P. (2020) · The Science of The Total Environment · doi:10.1016/j.scitotenv.2020.140827 · Cited by 62
Effect of preparative rehabilitation on recovery after cardiac surgery: A systematic review
Yau, D. K. W., Underwood, M. J., Joynt, G. M., & Lee, A. S. Y. (2020) · Annals of Physical and Rehabilitation Medicine · doi:10.1016/j.rehab.2020.03.014 · Cited by 70
Lipid profile as an indicator of COVID-19 severity: A systematic review and meta-analysis
Mahat, R. K., Rathore, V., Singh, N., Singh, N., Singh, S. K., Shah, R. K., & Garg, C. (2021) · Clinical Nutrition ESPEN · doi:10.1016/j.clnesp.2021.07.023 · Cited by 63
Beran, A., Matar, R., Jaruvongvanich, V., Rapaka, B., Alalwan, A. I., Portela, R. C., Ghanem, O. M., & Dayyeh, B. K. A. (2022) · Obesity Surgery · doi:10.1007/s11695-022-06254-y · Cited by 61
“The median and interquartile ranges were converted to mean and SD where applicable [24].”
MethodHübel, C., Yılmaz, Z., Schaumberg, K., Breithaupt, L. E., Hunjan, A. K., Horne, E., García‐González, J., O’Reilly, P. F., Bulik, C. M., & Breen, G. D. (2019) · International Journal of Eating Disorders · doi:10.1002/eat.23158 · Cited by 61
“We used additional formulas to calculate missing values (Hozo, Djulbegovic, & Hozo, 2005; Luo, Wan, Liu, & Tong, 2018; Wan, Wang, Liu, & Tong, 2014).”
Pregernig, A., Müller, M. M., Held, U., & Beck‐Schimmer, B. (2019) · Annals of Intensive Care · doi:10.1186/s13613-019-0600-1 · Cited by 61
“[23], and conducted a sensitivity analysis to compare the results of the two methods.”
BackgroundResultLuo, J., Wang, X., Li, Y., & Guo, L. (2021) · Frontiers in Endocrinology · doi:10.3389/fendo.2021.629831 · Cited by 69
“Some studies provided data as median and interquartile range (or maximum and minimum), and we calculated their mean and standard deviation according to formulas (18, 19).”
MethodMeng, X., Chen, K., Yang, C., Li, H., & Wang, X. (2021) · Frontiers in Medicine · doi:10.3389/fmed.2021.694385 · Cited by 60
Xiong, C., Han, C., Zhao, D., Peng, W., Xu, D., & Lan, Z. (2021) · PLoS ONE · doi:10.1371/journal.pone.0256611 · Cited by 60
Lorentzen, T., Heidemann, L. N., Möller, S., & Bille, C. (2021) · European Journal of Surgical Oncology · doi:10.1016/j.ejso.2021.09.007 · Cited by 60
Inflammation in Cerebral Venous Thrombosis
Ding, J., Song, B., Xie, X., Li, X., Chen, Z., Wang, Z., Pan, L., Lan, D., & Meng, R. (2022) · Frontiers in Immunology · doi:10.3389/fimmu.2022.833490 · Cited by 60
“(4) Proinflammatory cytokines such as IL-1, IL-6, IL-8, IL-10, and TNF-a modulate the inflammatory reactions to affect the evolution or resolution of thrombosis.”
BackgroundLi, H., Xu, X., Wang, D., Lin, Y., Lin, N., & Lu, H. (2019) · Clinical Microbiology and Infection · doi:10.1016/j.cmi.2019.06.005 · Cited by 61
“For continuous outcomes (inverse variance method), we extracted the actual number of patients who completed the follow-up when considering the difficulty to calculate MDs and standard deviation (SDs) in an ITT model.”
MethodLi, M.-L., Zhang, F., Luo, H.-Y., Quan, Z.-W., Wang, Y.-F., Huang, L.-T., & Wang, J.-H. (2024) · The Journal of nutrition, health and aging · doi:10.1016/j.jnha.2024.100184 · Cited by 62
“When the mean and SD data were missing from the original study, they were obtained using formula calculations or data conversion tools [36,41,42].”
Marques, A. L. S., Andres, M. P., Kho, R. M., & Abrão, M. S. (2019) · Journal of Minimally Invasive Gynecology · doi:10.1016/j.jmig.2019.07.029 · Cited by 60
Are the steroids a blanket solution for COVID‐19? A systematic review and meta‐analysis
Sarkar, S., Khanna, P., & Soni, K. D. (2020) · Journal of Medical Virology · doi:10.1002/jmv.26483 · Cited by 59
Li, B. V., & Jiang, B. (2021) · Global Change Biology · doi:10.1111/gcb.15781 · Cited by 60
“For studies with only quartiles and median values of the response variables, we used the method provided by Luo et al. (2018) and Wan et al. (2014). b. Livestock type and density: We classified livestock types into Cattle, Horses, Sheep, Goats, Reindeer, and Mixed grazing groups.”
MethodDiaconu, A., Holte, M. B., Berg‐Beckhoff, G., & Pinholt, E. M. (2023) · Journal of Personalized Medicine · doi:10.3390/jpm13010125 · Cited by 59
“After checking for normal distribution, the standard deviation was calculated using the interquartile range [28].”
MethodHill, A., Heyland, D. K., Ortiz‐Reyes, L., Laaf, E., Wendt, S., Elke, G., & Stoppe, C. (2021) · Journal of Parenteral and Enteral Nutrition · doi:10.1002/jpen.2125 · Cited by 66
Association of thyroid disorders with gestational diabetes mellitus: a meta-analysis
Luo, J., Wang, X., Yuan, L., & Guo, L. (2021) · Endocrine · doi:10.1007/s12020-021-02712-2 · Cited by 59
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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