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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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MethodYassin, A., Nawaiseh, M. B., Shaban, A., Alsherbini, K. A., El‐Salem, K., Soudah, O., & Abu-Rub, M. (2021) · BMC Neurology · doi:10.1186/s12883-021-02161-4 · Cited by 109
Appel, M., Forsthuber, M., Ramos, R., Widhalm, R., Granitzer, S., Uhl, M., Hengstschläger, M., Stamm, T., & Gundacker, C. (2022) · Journal of Toxicology and Environmental Health, Part B · doi:10.1080/10937404.2021.2009946 · Cited by 104
Kwon, S., Park, J. K., & Koh, Y. H. (2023) · Journal of NeuroEngineering and Rehabilitation · doi:10.1186/s12984-023-01219-3 · Cited by 105
Falz, R., Bischoff, C., Thieme, R., Lässing, J., Mehdorn, M., Stelzner, S., Busse, M., & Gockel, I. (2022) · Journal of Cancer Research and Clinical Oncology · doi:10.1007/s00432-022-04088-w · Cited by 104
“For continuous variables, data for meta-analysis were obtained directly from the study results or on request from articles’ authors or calculated from the mean, variance 95% confidence intervals or median and Interquartile range (Higgins et al. 2021; Luo et al. 2018; Shi et al. 2020; Wan et al. 2014).”
MethodShao, M., Li, X.-M., Liu, F., Tian, T., Luo, J., & Yang, Y. (2020) · Pharmacological Research · doi:10.1016/j.phrs.2020.105107 · Cited by 105
Cheng, I., Sasegbon, A., & Hamdy, S. (2020) · Neuromodulation Technology at the Neural Interface · doi:10.1111/ner.13327 · Cited by 104
“For studies that reported median and interquartile range, the mean and standard deviations were estimated using methods previously described (32,33).”
MethodAntonelli, M., Donelli, D., & Fioravanti, A. (2018) · Rheumatology International · doi:10.1007/s00296-018-4081-6 · Cited by 103
Li, G., Feng, H., Mao, X.-L., Deng, Y., Wang, X.-B., Zhang, Q., Guo, Y., & Xiao, S. (2023) · Journal of Translational Medicine · doi:10.1186/s12967-023-04306-0 · Cited by 115
“If the study presented data in medians and quartiles, the mean and SD values were estimated [20, 21].”
MethodLex, J. R., Edwards, T. C., Packer, T., Jones, G. G., & Ravi, B. (2020) · The Journal of Arthroplasty · doi:10.1016/j.arth.2020.10.010 · Cited by 98
Woldendorp, K., Indja, B., Bannon, P. G., Fanning, J. P., Plunkett, B., & Grieve, S. M. (2021) · European Heart Journal · doi:10.1093/eurheartj/ehab002 · Cited by 102
Omega-3 world map: 2024 update
Schuchardt, J. P., Beinhorn, P., Hu, X. F., Chan, H. M., Roke, K., Bernasconi, A. A., Hahn, A., Sala‐Vila, A., Stark, K. D., & Harris, W. S. (2024) · Progress in Lipid Research · doi:10.1016/j.plipres.2024.101286 · Cited by 107
Nielsen, S., Picco, L., Murnion, B. P., Winters, B. L., Matheson, J., Graham, M., Campbell, G., Parvaresh, L., Khor, K. E., Betz‐Stablein, B., et al. (2022) · Neuropsychopharmacology · doi:10.1038/s41386-022-01322-4 · Cited by 103
Silent Brain Infarcts Following Cardiac Procedures: A Systematic Review and Meta‐Analysis
Indja, B., Woldendorp, K., Vallely, M. P., & Grieve, S. M. (2019) · Journal of the American Heart Association · doi:10.1161/jaha.118.010920 · Cited by 97
Flükiger, J., Hollinger, A., Speich, B., Meier, V., Tontsch, J., Zehnder, T., & Siegemund, M. (2018) · Annals of Intensive Care · doi:10.1186/s13613-018-0437-z · Cited by 92
Bakaloudi, D. R., Chrysoula, L., Kotzakioulafi, E., Theodoridis, X., & Chourdakis, M. (2021) · Nutrients · doi:10.3390/nu13051514 · Cited by 92
Maski, K. P., Trotti, L. M., Kotagal, S., Auger, R. R., Swick, T. J., Rowley, J. A., Hashmi, S. D., & Watson, N. F. (2021) · Journal of Clinical Sleep Medicine · doi:10.5664/jcsm.9326 · Cited by 92
Wang, J., Hu, K., Wang, Y., Wu, Y., Bao, E., Wang, J., Tan, C., & Tang, T. (2023) · Journal of Robotic Surgery · doi:10.1007/s11701-023-01714-8 · Cited by 102
“When continuous variables were reported as median and quartiles, we calculated the mean ± standard deviation using appropriate mathematical methods [31, 32].”
MethodBenefits of physical activity in children and adolescents with type 1 diabetes: A systematic review
Absil, H., Baudet, L., Robert, A. R., & Lysy, P. A. (2019) · Diabetes Research and Clinical Practice · doi:10.1016/j.diabres.2019.107810 · Cited by 99
Su, M., Jia, Y., Li, Y., Zhou, D., & Jia, J. (2020) · Respiratory Care · doi:10.4187/respcare.07097 · Cited by 91
Effects of ocean acidification on Antarctic marine organisms: A meta‐analysis
Hancock, A. M., King, C. K., Stark, J. S., McMinn, A., & Davidson, A. T. (2020) · Ecology and Evolution · doi:10.1002/ece3.6205 · Cited by 96
Qin, R., He, L., Yang, Z., Jia, N., Chen, R., Xie, J., Fu, W., Chen, H., Lin, X., Huang, R., et al. (2022) · Clinical Reviews in Allergy & Immunology · doi:10.1007/s12016-021-08908-8 · Cited by 97
“For the meta-analysis, we transformed the format of laboratory variables presented as “median [interquartile range (IQR)]” into that of “mean [standard deviation (SD)]” [18, 19].”
MethodLowe, E. B., Groves, R. L., & Gratton, C. (2021) · Agriculture Ecosystems & Environment · doi:10.1016/j.agee.2020.107290 · Cited by 100
“If a study reported the median and interquartile range (IQR), we used an online calculator based on methods developed by Luo et al. (2018) and Shi et al. (2020) which are accurate for skewed data, to backcalculate an estimate of the mean and standard deviation.”
MethodHuang, Y., Yin, X.-S., & Li, Z. (2022) · Frontiers in Immunology · doi:10.3389/fimmu.2022.1090305 · Cited by 93
Factors related to the resilience and mental health of adult cancer patients: a systematic review
Tamura, S., Suzuki, K., Ito, Y., & Fukawa, A. (2021) · Supportive Care in Cancer · doi:10.1007/s00520-020-05943-7 · Cited by 89
“Further, when the median and quartile were described, these values were converted into the mean and standard deviation by the conversion formula of previous studies [15, 16] and such articles were included in the meta-analysis.”
MethodProtein intake and physical function in older adults: A systematic review and meta-analysis
Coelho-Júnior, H. J., Calvani, R., Tosato, M., Landi, F., Picca, A., & Marzetti, E. (2022) · Ageing Research Reviews · doi:10.1016/j.arr.2022.101731 · Cited by 86
Brain Selenium in Alzheimer’s Disease (BRAIN SEAD Study): a Systematic Review and Meta-Analysis
Varikasuvu, S. R., Prasad, S., Kothapalli, J., & Munikumar, M. (2018) · Biological Trace Element Research · doi:10.1007/s12011-018-1492-x · Cited by 86
Louwen, C., Reidlinger, D. P., & Milne, N. (2023) · BMC Medical Education · doi:10.1186/s12909-023-04003-y · Cited by 117
Triglyceride-glucose index and obstructive sleep apnea: a systematic review and meta-analysis
Behnoush, A. H., Khalaji, A., Ghondaghsaz, E., Masrour, M., Varniab, Z. S., Khalaji, S., & Cannavò, A. (2024) · Lipids in Health and Disease · doi:10.1186/s12944-024-02005-3 · Cited by 86
“In the case of median and interquartile range reporting, they were converted to the mean and standard deviation using methods developed by Luo et al. [27] and Wan et al. [28].”
MethodMalignant transformation of oral leukoplakia: Systematic review and comprehensive meta‐analysis
de Barros, L. A. P., Ramos‐García, P., González‐Moles, M. Á., Urízar, J. M. A., & Warnakulasuriya, S. (2024) · Oral Diseases · doi:10.1111/odi.15140 · Cited by 100
Carbonara, U., Srinath, M., Crocerossa, F., Ferro, M., Cantiello, F., Lucarelli, G. T., Porpiglia, F., Battaglia, M., Ditonno, P., & Riccardo Autorino, B. (2021) · World Journal of Urology · doi:10.1007/s00345-021-03687-5 · Cited by 85
“[11] was used, while, for time-based data or other data suspected of being skewed, the Box–Cox and quantile estimation methods described by McGarth et al.”
MethodRawat, D., Roy, A., Maitra, S., Gulati, A., Khanna, P., & Baidya, D. K. (2021) · Diabetes & Metabolic Syndrome Clinical Research & Reviews · doi:10.1016/j.dsx.2021.102324 · Cited by 84
Distribution of delirium motor subtypes in the intensive care unit: a systematic scoping review
la Cour, K. N., Andersen‐Ranberg, N. C., Weihe, S., Poulsen, L. M., Mortensen, C. B., Kjer, C. K. W., Collet, M. O., Estrup, S., & Mathiesen, O. (2022) · Critical Care · doi:10.1186/s13054-022-03931-3 · Cited by 98
Jayedi, A., Zeraattalab-Motlagh, S., Ganjeh, B. J., Hosseini, Y., Jibril, A. T., Shahinfar, H., Mirrafiei, A., Hosseini, F. Z., & Shab‐Bidar, S. (2022) · American Journal of Clinical Nutrition · doi:10.1093/ajcn/nqac066 · Cited by 89
van Reijen, N. S., Ponchant, K., Ubbink, D. T., & Koelemay, M. J. W. (2019) · European Journal of Vascular and Endovascular Surgery · doi:10.1016/j.ejvs.2019.03.040 · Cited by 86
Qu, Y., Ma, Y., Huang, Y., Ou, Y., Shen, X., Chen, S.-D., Dong, Q., Tan, L., & Yu, J. (2021) · Neuroscience & Biobehavioral Reviews · doi:10.1016/j.neubiorev.2021.07.007 · Cited by 85
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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