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.

When reporting the results of clinical studies, some researchers may choose the five‐number summary (including the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation (SD), particularly for skewed data. For these studies, when included in a meta‐analysis, it is often desired to convert the five‐number summary back to the sample mean and SD. For this purpose, several methods have been proposed in the recent literature and they are increasingly used nowadays. In this article, we propose to further advance the literature by developing a smoothly weighted estimator for the sample SD that fully utilizes the sample size information. For ease of implementation, we also derive an approximation formula for the optimal weight, as well as a shortcut formula for the sample SD. Numerical results show that our new estimator provides a more accurate estimate for normal data and also performs favorably for non‐normal data. Together with the optimal sample mean estimator in Luo et al., our new methods have dramatically improved the existing methods for data transformation, and they are capable to serve as “rules of thumb” in meta‐analysis for studies reported with the five‐number summary. Finally for practical use, an Excel spreadsheet and an online calculator are also provided for implementing our optimal estimators.

For clinical studies with continuous outcomes, when the data are potentially skewed, researchers may choose to report the whole or part of the five-number summary (the sample median, the first and third quartiles, and the minimum and maximum values), rather than the sample mean and standard deviation. For the studies with skewed data, if we include them in the classical meta-analysis for normal data, it may yield misleading or even wrong conclusions. In this paper, we develop a flow chart and three new tests for detecting the skewness of data from the sample size and the five-number summary. Simulation studies demonstrate that our new tests are able to control the type I error rates, and meanwhile provide good statistical power. A real data example is also analyzed to demonstrate the usefulness of the skewness tests in meta-analysis and evidence-based practice.

Hand-foot-mouth disease (HFMD) is a common infectious disease caused by a group of enteroviruses, with enterovirus 71 (EV-71) and Coxsackie virus A16 (CA-V16) being the most prevalent in China 1, 2 . Over the last decade, outbreaks of HFMD that were associated with EV71 have been reported in countries in the Western Pacific Region, including Japan, Malaysia, Singapore, and China 3, 4 . The cumulative total of the reported cases in China has reached approximately 1.7 million, 1.9 million, and 2.7 million in 2010, 2013, and 2014, respectively 5, 6 . The clinical manifestations of most HFMD cases were mild and limited to fever, rash, or herpes on hand, foot, and mouth 7 . In general, mild infections are self-limited and not life-threatening, while severe HFMD are often associated with neurological and systemic complications, such as aseptic meningitis, brainstem encephalitis, acute flaccid paralysis, myocarditis and pulmonary oedema that requires hospitalization, or even causing death 2, 8 . Unfortunately, the incidence of severe HFMD in mainland China is high.Evidences from global reports on HFMD epidemics have substantiated that the incidence of severe HFMD is elevating gradually, along with mortality rate 9 . Thus, identifying potential early indicators for severe HFMD is essential, which enable early medical interventions and alleviating the disease severity, subsequently reducing the mortality rate. Previous studies have found that the following conditions or practices were associated with the increased risk of severe HFMD: a duration of fever ≥3 days, body temperature ≥37.5 °C, fatigue, the use of glucocorticoids, the use of dehydrant drugs, maculopapular rash, hyperglycemia, vomiting, EV71 infection, attending home care, neutrophilia, and young age [10][11][12][13][14][15][16][17][18] . Although specific clinical manifestations have been identified using the Magnetic Resonance Imaging (MRI), the potential indicators of the disease severity were not validated.

For clinical studies with continuous outcomes, when the data are potentially skewed, researchers may choose to report the whole or part of the five-number summary (the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation. In the recent literature, it is often suggested to transform the five-number summary back to the sample mean and standard deviation, which can be subsequently used in a meta-analysis. However, if a study contains skewed data, this transformation and hence the conclusions from the meta-analysis are unreliable. Therefore, we introduce a novel method for detecting the skewness of data using only the five-number summary and the sample size, and meanwhile, propose a new flow chart to handle the skewed studies in a different manner. We further show by simulations that our skewness tests are able to control the type I error rates and provide good statistical power, followed by a simulated meta-analysis and a real data example that illustrate the usefulness of our new method in meta-analysis and evidence-based medicine.

When reporting the results of clinical studies, some researchers may choose the five-number summary (including the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation, particularly for skewed data. For these studies, when included in a meta-analysis, it is often desired to convert the five-number summary back to the sample mean and standard deviation. For this purpose, several methods have been proposed in the recent literature and they are increasingly used nowadays. In this paper, we propose to further advance the literature by developing a smoothly weighted estimator for the sample standard deviation that fully utilizes the sample size information. For ease of implementation, we also derive an approximation formula for the optimal weight, as well as a shortcut formula for the sample standard deviation. Numerical results show that our new estimator provides a more accurate estimate for normal data and also performs favorably for non-normal data. Together with the optimal sample mean estimator in Luo et al., [1] our new methods have dramatically improved the existing methods for data transformation, and they are capable to serve as "rules of thumb" in meta-analysis for studies reported with the five-number summary. Finally for practical use, an Excel spreadsheet and an online calculator are also provided for implementing our optimal estimators.

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