1 Introduction
Metaanalysis is the most important tool to provide highlevel evidence in evidencebased medicine. By conducting metaanalysis, researchers can statistically summarize and combine data from multiple studies with some predetermined summary measure. In most of the studies, mean difference is one of the frequently used effect size measurements for continuous data. For example, Cohen’s statistic (Cohen, 2013) and Hedges’ statistic (Hedges, 1981) are two of the most famous mean difference measurements. In order to calculate the mean difference based effect sizes, the sample mean and standard deviation are two essential summary measures. In practical research, most of the medical studies provide the sample mean and standard deviation directly but, some of the clinical studies tend to report the summary statistics such as the sample median, quartiles and extremas. As a result, researchers had developed a few methods to transform the reported information to the sample mean and standard deviation for further analysis. In particular, Hozo et al. (2005)
was the first to establish estimators for the sample mean and standard deviation.
Wan et al. (2014) further improved Hozo et al.’s estimators of the sample standard deviation and Luo et al. (2017) developed the optimal estimators of the sample mean.These estimation methods, especially the methods proposed by Wan et al. were widely adopted in medical research area and had been frequently cited after published. In particular, Wan et al.’s methods have already gained 274 citations in Google Scholar. However, among the current metaanalysis studies, researchers usually apply the estimation methods to all kind of reported data directly, without considering the symmetry of original data. For instance, we consider an example with the data obtained from Hawkins et al. (2017), a metaanalysis about the association between Btype natriuretic peptides (BNP) and chronic obstructive pulmonary disease (COPD). Totally there were 51 studies included in the metaanalysis and the authors discussed situations about stable disease cases, exacerbation cases and comparison between these two phases of patients. For illustration purpose, we only reported 7 studies that recorded information of both patients and healthy individuals within stable disease phase in Table 1. In Hawkins et al. (2017), the authors used the methods in Wan et al. (2014)
to estimate the sample mean and standard deviation. The authors claimed that they found most of the recorded data were possibly not symmetric and they realized all the existing transformation methods were developed based on the symmetric assumption. For example for Study 5, the median BNP level for the control group is 50 but the third quartile is just 51, which indicates that the underlying data are more likely to be right skewed. In spite of this, the authors still conducted the transformation and they believed the major error of the statistical results were generated by the normality assumption of transformation methods. Although Luo et al.’s methods of estimating the sample mean and Wan et al.’s standard deviation estimators were proved to have very good performance for both symmetric and skewed data, the authors in
Hawkins et al. (2017) still had concern about the reliability of the transformation methods. Under such circumstances, we believe it is essential to seek for a better solution to help people further filter the studies before performing the metaanalysis.





1 
Anderson et al. (2013)  Case  93  296  
Control  93  26(2032)  
2  Gemici et al. (2008)  Case  17  2116  
Control  17  1311  
3  Boschetto et al. (2013)  Case  23  121(59227)  
Control  23  50(4351)  
4  Wang et al. (2013)  Case  80  245(196336)  
Control  80  101(56150)  
5  Beghé et al. (2013)  Case  70  115(50364)  
Control  70  50(4351)  
6  Bando et al. (1999)  Case  14  133  
Control  14  71  
7  Bozkanat et al. (2005)  Case  38  2110  
Control  38  93  
Observations are expressed as mean SD or median (interquartile range). 
As a matter of fact, in both Wan et al. (2014) and Luo et al. (2017)
, the estimators of the sample mean and standard deviation are proposed based on the assumption that the underlying data is normally distributed. Therefore, if the original data is skewed or very skewed, it might be inappropriate to treat this kind of data as normally distributed. Furthermore, it is natural to consider that in clinical trial studies, when the underlying data is not symmetric, reporting the sample median rather than the sample mean is reasonable. In this case, when the original data is skewed, transforming the reported information (such as the sample median, minimum and maximum values) to the sample mean and standard deviation may lead to lack of accuracy in the followup analysis. In view of the above situation, we propose some new test statistics to pretest whether the underlying data is normally distributed or not. If there is no significant evidence to prove that the selected study is skewed, researchers may consider to transform the reported information to the sample mean and standard deviation. Otherwise, we suggest that researchers may consider to not to include the tested study when conducting metaanalysis. In this case, we suggest when conducting metaanalysis, researchers can follow the procedures as shown in the below flowchart.
Based on the above motivation, in this paper, we propose new test statistics for three most frequently used scenarios in clinical trial reports as mentioned in both Wan et al. (2014) and Luo et al. (2017), which may help researchers better choose the included studies for conducting metaanalysis. For the proposed test statistics for each scenario, we conduct simulation studies to check whether the new test methods perform well in practice. We also apply a few real data examples to illustrate the usefulness of the new test statistics. Eventually, we summarize our new test statistics and discuss some future directions.
2 Motivation
To better illustrate the issue we mentioned in previous section, we choose five popular skewed distributions as examples to show how skewed data may affect the medical decision. Suppose we are computing the effect size for some paired experiments about a certain disease. For the sake of consistency, we follow the notations as used in Luo et al. (2017): using letters , , to denote the sample minimum value, median and maximum value for a study with size , respectively.
In Table 2
, let the lognormal distribution with location parameter
and scale parameter for the disease cases,for the controls; the chisquare distribution with degrees of freedom
for the disease cases,for controls; the exponential distribution with rate parameter
for the disease cases,for controls; the beta distribution with shape parameters
and for the disease cases, for controls; and the Weibull distribution with shape parameter and scale parameter for the disease cases, for controls. These distributions are used to generate the true sample mean, minimum (), maximum () and median (). We then use the method in Luo et al. (2017) to estimate the sample mean and use the method in Wan et al. (2014) to estimate the standard deviation for further analysis. Eventually, we compute the effect size (Cohen’s value) using the estimated sample mean as well as the actual sample mean and make comparison. Note that for ease of computation, we assume the sample sizes for both the disease cases and controls are the same and the sample sizes for different distributions are arbitrarily chosen.







Lognormal  350  350  
Chisquare  200  200  
Exponential  150  150  
Beta  300  300  
Weibull  400  400 









Lognormal  1.72 (2.70)  4.05 (4.64)  1.61 (4.82)  3.20 (5.72)  0.30  0.61  
Chisquare  3.08 (2.17)  4.71 (3.82)  2.81 (1.74)  4.48 (5.27)  0.42  0.53  
Exponential  1.07 (1.13)  0.70 (0.73)  0.90 (1.29)  0.59 (0.12)  0.34  0.39  
Beta  0.28 (0.16)  0.23 (0.12)  0.27 (0.13)  0.22 (0.11)  0.42  0.35  
Weibull  0.93 (0.58)  0.89 (0.33)  0.88 (0.50)  0.87 (0.30)  0.02  0.08 
From Tables 2 and 3, we clearly find that if the underlying distribution is very skewed, estimating the sample mean and standard deviation using the summary statistics may lead to an incorrect conclusion. In particular, for the lognormal distribution and chisquare distribution in Table 3, the effect size computed by the actual sample mean is within the median effect level but the effect size computed by the estimated sample mean is in the low effect level. As a result, testing the normality of the underlying data before estimating the sample mean and standard deviation is of crucial importance. The normality test may help researchers to filter out the skewed data and further reduce the estimation error when computing the effect sizes. Motivated by this circumstance, we will propose some new test statistics for three frequently used scenarios in Section 3. The simulation results and the real data case studies in the later sections indicate the good performance of the new test methods.
3 Test methods
For the sake of consistency, we follow the same notations as those in Luo et al. (2017). Let be the sample size and denote the 5number summary for the data as
In this work, we consider the three most frequently occurred scenarios in clinic trial reports:
According to (Triola, 2009), we refer to as the range, as the midrange, as the interquartile range, and as the midquartile range. Note that according to Wan et al. (2014) and Luo et al. (2017), the midrange and the midquartile range are used to estimate the sample mean, while the range and the interquartile range are used to estimate the standard deviation.
For the following sections, our general assumption is letting be a random sample of size from the normal distribution , and be the ordered statistics of the sample. According to Chen (2004), we use to denote the sample
th quantile, where
and represents the integer part of . With the above notations, we have , , , , and .For convenience, let also , or equivalently, for . Then follows the standard normal distribution , and are the ordered statistics of the sample .
3.1 Hypothesis test for
Note that the proposed estimators of the sample mean in Luo et al. (2017) and the estimators of the standard deviation in Wan et al. (2014) are developed based on the normality assumption. To ensure the accuracy of the estimation results, it is natural to test whether the population data are normally distributed or not. That is, if the data does not pass the test, we may conclude the underlying data is not normally distributed and hence it is not appropriate to be used in the sample mean and standard deviation estimation. Therefore, the hypothesis we proposed for this scenario is:
As the most important property of normal distribution is the symmetry property, we will use it to identify the normality of the underlying distribution. Under the situation of scenario , only the sample median and extremes are reported. Therefore, it is reasonable to consider comparing the distances between the sample median to the minimum as well as it to the maximum, i.e. computing the value of . If the difference is very close to zero, we may conclude that the original data is normally distributed. Based on the above hypothesis, we introduce the following test statistic, :
(1) 
By Theorem 1
in Appendix B, the simplified test statistic under the null hypothesis
is(2) 
Note that is unknown and need to be estimated. Based on Wan et al. (2014), the estimation of the sample standard deviation for scenario is
(3) 
Plugging (3) into (2), we can easily get the modified test statistic for this scenario:
(4) 
where is the coefficient function related to the sample size .
Based on the above test statistic, if the observed is within the interval , we accept the null hypothesis and conclude that the underlying data is normally distributed. Thus, we can continue to conduct the data transformation, which is to estimate the sample mean and standard deviation from the 5number summary statistics. Otherwise, if the null hypothesis is rejected, we suggest researchers not to include the tested data when proceeding metaanalysis.
3.2 Hypothesis test for
Similar to the previous section, the hypothesis is defined as:
Under the case of scenario , only the sample quartiles are provided. Based on the symmetry property of the normal distribution, in population aspect, the distance between the first quartile and the median is equal to that from the median to the third quartile. As a result, we introduce a test statistic which compares these two distances, i.e. computing the value of . By the above setting, the test statistic for this scenario is:
(5) 
By Theorem 2 in Appendix B, the simplified test statistic under the null hypothesis is
(6) 
Similar to Section 2.1, is unknown and by Wan et al. (2014), the estimation of the sample standard deviation for scenario is
(7) 
Plugging (7) into (6), we can easily obtain the updated test statistic for this scenario:
(8) 
where is the coefficient function related to the sample size .
Similarly, if the observed is within the interval , we accept the null hypothesis and regard the underlying data as applicable to perform the estimation of the sample mean and standard deviation. Otherwise, we suggest that researchers should remove the tested data from further analysis procedure.
3.3 Test statistic for
In this section, we discuss the case of all 5number summary statistics are provided. Similar to the previous two subsections, the hypothesis is defined as:
Recall that in the previous subsections, the symmetry property of normal distribution is used to test the normality of underlying data. In Section 2.1, the test statistic is built by comparing the difference in distances between the sample extremes to the median, i.e. computing and , respectively. In Section 2.2, the value of is computed to measure the difference in distances between the sample quartiles to the median. Hence, taking both previous ideas into account, we introduce the following test statistic:
(9) 
By Theorem 3 in Appendix B, the simplified test statistic under the null hypothesis is
(10) 
where is unknown. Therefore, the next task is to determine an estimator of the sample standard deviation .
Following the similar idea as in Wan et al. (2014), by Lemma 1, we have , which leads to the estimator . According to Blom’s method of approximating (Blom, 1958), the estimator of the sample standard deviation is defined as
(11) 
Consequently, plugging (11) into (10), the updated test statistic has the following expression:
(12) 
where is the coefficient function related to the sample size .
Based on the above definition, if the observed is within the interval , we accept the null hypothesis. Otherwise, the null hypothesis is rejected and we suggest researchers not to include the tested data into metaanalysis.
4 Simulation studies
In this section, we conduct some simulation studies to evaluate the performance of the test statistics in Section 2. The type I error and the statistical power will be computed. To compute the statistical power of test statistics (
4) and (8), we choose the following skewed distributions as the alternative distribution: the standard lognormal distribution with ; the standard exponential distribution with ; the beta distribution with shape parameters ; the Chisquare distribution with degree of freedom and the Weibull distribution with scale parameter , shape parameter .The simulation results indicate that all three test statistics can provide an acceptable statistical power, as well as the type I error under control.
4.1 Simulation study for
Figure 2 shows the type I error of test statistic in scenario (Eq. (4)). Since in practical research, the sample size rarely drop below 30, we tend to focus on the situation when is large. Hence, although the type I error of is not very close to 0.05 for small sample sizes, we consider it acceptable because the type I error is able to maintain a value around 0.05 as increases to 200 or larger. Figure 3 reports the statistical power of test statistic . The simulation studies were conducted by assuming the alternative distribution is not normal distribution. It is obvious that for all the five skewed distributions, the statistical power increase to 1 rapidly (mostly increase to 1 before reaches 100). Consequently, with the type I error very close to 0.05 and a statistical power close to 1, we consider that test statistic may have impressive performance in practical application and we will conduct a real data analysis to evaluate its performance in the next section.
4.2 Simulation study for
Similar to section 3.1, Figure 4 shows the type I error of test statistic in scenario (Eq. (8)). It is evident that the type I error is able to maintain a value around 0.05 especially when the sample size increases to more than 200. Figure 5 reports the statistical power of test statistic when the alternative distributions are skewed distributions. Compare to section 3.1, the statistical power of test statistic increases to 1 in a slower motion, it reaches 1 only when the sample size is larger than 400. However, since in practice, most of the studies have large sample sizes, it is acceptable that test statistic may provide a statistical power that reaches 1 in a slightly slower speed. Also, as all the value of type I error is around 0.05, we consider test statistic may have a very good performance in reallife application. In the next section, we will conduct real data analysis to evaluate the performance of .
4.3 Simulation study for
In this section, Figure 6 shows the type I error of test statistic in scenario (Eq. (12)). In Figure 6, it is obvious that the type I error of drop within the range of [0.045,0.055] and most of the points are very close to 0.05. Figure 7 reports the statistical power of test statistic with skewed alternative distributions. Similar to scenario , the statistical power of reaches toward 1 rapidly. Therefore, with a type I error around 0.05 and statistical power close to 1, we expect that would have very good performance in practice.
5 Real data analysis
In this section, we tend to apply 2 real data analysis as examples to demonstrate the usage of our proposed test statistics. The first case is about investigating the association between asthma and leptin and adiponectin serum levels, respectively (Zhang et al., 2017). The second case is to identify the effects of statin therapy on four indicators of plasma lipid concentrations in HIVinfected patients (Banach et al., 2016). In both cases, some data are recorded as the sample median and interquartile ranges, or median with the sample extremum. In this section, test statistics (4) and (8) are used to conduct the symmetry test of underlying data.
5.1 Case study of investigating the association of asthma diagnosis with leptin and adiponectin
The first data obtained from a metaanalysis about the effects of leptin and adiponectin serum levels on the diagnosis of asthma (Zhang et al., 2017). The article is published on the Journal of Investigative Medicine (JIM), one of the BMJ journals. It includes 13 studies and all the analysis is divided into 2 parts, one focus on leptin serum level and the other focus on adiponectin serum level. For leptin serum level, 5 of the studies report the sample median and interquartile range (satisfies scenario ), 1 reports median, minimum and maximum values (satisfies scenario ). For adiponectin serum level, two of the studies report the sample median and interquartile range. Hence, in this section, test statistics and will be applied to evaluate whether the underlying data is symmetry and can be further used to estimate the sample mean and standard deviation.
5.1.1 Data description
The reported information of the 13 studies in Zhang et al. (2017) are displayed in the following Table 4. There are 12 studies that record the values of leptin serum level and 11 studies that record the values of adiponectin serum level. Since for both leptin and adiponectin serum levels, some of the selected studies only provide summary measurements (the sample median, interquartile range or extremum), it is essential to estimate the sample mean and standard deviation (SD) from recorded information. However, all the existing sample mean and SD estimation methods are developed based on normality assumption. As a result, before conducting the transformation, we will apply the symmetry test as proposed in Section 2 to check whether the selected studies are reasonable to be transformed.
Study 


Leptin  Adiponection  

Hayashikawa et al. (2015) 
Asthma  23  NS  13.479.08  
Healthy  68  NS  14.048.82  
Haidari et al. (2014)  Asthma  47  1.410.50  6.682.07  
Healthy  47  0.590.19  7.552.10  
Sood et al. (2014)  Asthma  44  34032.827597.8  4180.62671.1  
Healthy  44  33263.427874.9  3987.43106.0  
Cobanoglu et al. (2013)  Asthma  23  5.3[0.427.4]  NS  
Healthy  51  8.8[0.331.3]  NS  
Tsaroucha et al. (2013)  Asthma  32  24.814.8  13.59.2  
Healthy  22  13.710.0  10.16.4  
Yuksel et al. (2012)  Obese asthma  40  11.87.9  12586.23724.1  
Nonobese asthma  51  5.36.8  18089.36452.3  
Healthy  20  2.12.4  20297.53680.7  
Sideleva et al. (2012)  Asthma  11  0.30510.047  0.34710.037  
Healthy  15  0.12560.016  0.86660.134  
da Silva et al. (2012)  Asthma  26  38(3060)  4.5(3.58.5)  
Healthy  50  39(2550)  4(37.8)  
Giouleka et al. (2011)  Asthma  100  9.6(7.616.25)  6.2(5.47.3)  
Healthy  60  7.2(4.610.3)  8.2(5.813.5)  
LeivoKorpela et al. (2011)  Asthma  35  0.5(0.51.1)  1659.5  
Healthy  32  0.6(0.40.8)  17613  
Jang et al. (2009)  Asthma  60  2.310.04  1.900.17  
Healthy  30  2.220.06  1.950.04  
Kim et al. (2008)  Atopic asthma  149  2.27(0.655.03)  7.603.84  
Nonatopic asthma  37  2.22(0.963.29)  8.104.73  
Healthy  54  2.10(0.714.49)  7.324.19  
Guler et al. (2004)  Asthma  102  3.53(2.067.24)  NS  
Control  33  2.26(1.264.71)  NS  
Observations are expressed as mean SD, median (interquartile range) or median [minimummaximum].  
indicates the information is not specified in the original study. 
5.1.2 Results of symmetry test and metaanalysis
The results of the symmetry tests are recorded in Table 5. Note that test statistic is applied on study Cobanoglu et al. (2013) as it reports the sample median and extremum, test statistic is applied to studies da Silva et al. (2012), Giouleka et al. (2011), LeivoKorpela et al. (2011), Kim et al. (2008) and Guler et al. (2004). To further proceed metaanalysis, it is more reliable to only transform the studies with symmetric data, to the sample mean and standard deviation via methods proposed by Luo et al. (2017) and Wan et al. (2014). Thus, if the studies in Table 5 have pvalue greater than 0.05 on both asthma samples and healthy samples, these studies can be used to estimate the sample mean and standard deviation. Note also that in Kim et al. (2008), patients were divided into three groups: obese asthma, nonobese asthma and healthy individuals. In this case, only when the pvalues of all these three groups are greater than 0.05, we would compute the sample mean and standard deviation for Kim et al. (2008) and further combine the information of the first two groups.
Study  Type of asthma  Sample size  Leptin  Adiponection  

Test statistic  pvalue  Test statistic  pvalue  
Cobanoglu et al. (2013)  Asthma  23  3.022  0.003  NS  NS 
Healthy  51  2.935  0.003  NS  NS  
da Silva et al. (2012)  Asthma  26  1.653  0.098  2.126  0.034 
Healthy  50  0.606  0.545  2.945  0.003  
Giouleka et al. (2011)  Asthma  100  3.895  1.144  0.253  
Healthy  60  0.488  0.626  2.093  0.036  
LeivoKorpela et al. (2011)  Asthma  35  4.171  NS  NS  
Healthy  32  2.205  1  NS  NS  
Kim et al. (2008)  Atopic asthma  149  2.313  0.021  NS  NS 
Nonatopic asthma  37  0.351  0.726  NS  NS  
Healthy  54  1.391  0.164  NS  NS  
Guler et al. (2004)  Asthma  102  3.189  0.001  NS  NS 
Control  33  1.697  0.090  NS  NS  
pvalue indicates the underlying data is not symmetric.  
NS indicates the reported data do not need to conduct the symmetry test. 
It is obvious that for leptin serum level, all the studies, except for study da Silva et al. (2012), have pvalues less than 0.05 on either asthma samples or healthy samples. That is, these 5 studies (Cobanoglu et al. (2013),Giouleka et al. (2011), LeivoKorpela et al. (2011), Kim et al. (2008) and Guler et al. (2004)) is not suitable to further conduct the sample mean and standard deviation estimation. We suggest to exclude these five studies from the metaanalysis of investigating the association between leptin serum level and asthma.
Note that in studies Yuksel et al. (2012) and Kim et al. (2008), asthma patients were separated into two different levels. Therefore, before computing the standardized mean difference (SMD, i.e. the Cohen’s value (Cohen, 2013)) as the effect size for these two studies, we need to combine the information of those two levels of asthma as one. Note also that since there are 5 studies did not pass the symmetry test, we excluded those studies in the metaanalysis. As a result, we have the following forest plots for leptin and adiponectin levels. During computation, we used Luo et al. (2017)’s methods to estimate the sample mean value and Wan et al. (2014)’s methods to estimate the standard deviations for studies that only provided median and interquartile range or extremum. Based on Figure 8, there is extreme heterogeneity among the studies for both leptin and adiponectin levels ( for leptin; for adiponectin). The overall results indicate that patients with asthma have significant higher levels of leptin than healthy individuals (pooled standardized mean difference (SMD), with CI from 0.63 to 2.23), and lower levels of adiponectin than healthy individuals (pooled SMD, with CI from 0.95 to 0.06). Since this section is aimed to provide a demonstration of how to use the proposed test statistics before conducting metaanalysis, we would not further proceed the sensitivity analysis and the publication bias analysis.
The pooled conclusion is similar to Zhang et al. (2017). However, in the original article, the authors did not conduct the symmetry test and they used Hozo et al.’s methods(Hozo et al., 2005) to estimate the sample means and standard deviations. As we had mention in introduction section, blindly estimate the sample mean and standard deviation before testing the symmetry of underlying data may yield an inaccurate conclusion. Therefore, we suggest researchers to follow the procedure in Figure 1: conduct symmetry test for selected studies, exclude studies that may have skewed underlying data and then compute the individual and pooled effect sizes.
5.2 Case study of investigating the effects of statin therapy on plasma lipid concentrations in HIVinfected patients
The second data is a metaanalysis to investigate the effects of statin therapy on plasma lipid concentrations in HIVinfected patients (Banach et al., 2016). In this article, metaanalysis contains 12 randomized control trials (RCT) with 697 participants. The reported data included 5 types of cholesterol level that may influence plasma concentration: low density lipoprotein cholesterol (LDLC), total cholesterol, high density lipoprotein cholesterol (HDLC), nonhigh density lipoprotein cholesterol (nonHDLC) and triglycerides. Note that in Banach et al. (2016), only two studies reports data of nonHDLC and both these two studies provided the sample mean and standard deviation. Hence in the following discussion, we will remove variable nonHDLC and focus on the other four types of cholesterol level.
5.2.1 Data description
The data of included studies are displayed in the following Table 6. For each type of cholesterol level, data are reported either as the sample mean with standard deviation, or as median with interquartile range. In this case, the test statistic will be used to conduct the symmetry test of the underlying data. Similar to the previous case study, selected studies that have pvalue greater than 0.05 are recommended to further proceed the sample mean and standard deviation estimation. Otherwise, we regard that there is evidence to conclude the underlying data of selected studies are not symmetric and hence, we suggest researchers to remove those studies from the metaanalysis in order to obtain more precise conclusion.








Bonnet et al. (2007) 
Case  12  6.1(5.86.3)  4.1(3.74.6)  0.9(0.81.1)  2.0(1.13.3)  
Control  9  6.4(6.17.7)  3.9(3.74.8)  1.0(0.81.1)  3.2(2.14.4)  
Calmy et al. (2010)  Case  10  5.6(4.56.4)  2.8(2.43.3)  1.0(0.91.3)  3.9(2.06.2)  
Control  12  5.6(4.66.2)  3.5(2.54.1)  1.1(0.811.2)  2.3(1.53.5)  
Eckard et al. (2014)  Case  67  NS  2.48(1.962.77)  NS  NS  
Control  69  NS  2.50(1.993.13)  NS  NS  
Funderburg et al. (2015)  Case  72  NS  NS  1.21(0.981.49)  NS  
Control  75  NS  NS  1.19(0.961.47)  NS  
Ganesan et al. (2011)  Case  22  4.34(3.724.45)  2.50(2.252.82)  NS  NS  
Control  22  4.34(3.724.45)  2.50(2.252.82)  NS  NS  
Hürlimann et al. (2006)  Case  29  6.4(6.07.4)  3.7(2.84.2)  1.2(1.11.6)  3.0(2.14.0)  
Control  29  6.4(6.07.4)  3.7(2.84.2)  1.2(1.11.6)  3.0(2.14.0)  
Lo et al. (2015)  Case  17  5.140.98  3.200.95  1.340.50  1.36(1.102.31)  
Control  20  4.970.70  3.230.83  1.310.39  1.28(1.041.53)  
Stein et al. (2004)  Case  20  5.580.40  3.470.32  0.940.07  3.780.67  
Control  20  5.580.40  3.470.32  0.940.07  3.780.67  
Nakanjako et al. (2015)  Case  15  NS  3.1(2.24.9)  1.7(1.61.8)  1.6(1.12.4)  
Control  15  NS  4.9(2.46.7)  1.7(1.52.0)  2.0(1.43.2)  
Montoya et al. (2012)  Case  51  NS  2.63(2.203.28)  NS  NS  
Control  53  NS  2.53(2.303.10)  NS  NS  
Moyle et al. (2001)  Case  14  7.5(6.78.3)  4.65(4.15.2)  0.94(0.791.08)  3.96(2.846.52)  
Control  13  7.4(6.87.9)  4.68(3.895.47)  0.87(0.721.02)  4.06(2.205.97)  
Mallon et al. (2006)  Case  14  7.61.7  NS  1.10.4  3.84.1  
Control  17  7.61.4  NS  1.10.4  4.97.8  
Observations are expressed as mean SD or median (interquartile range).  
NS indicates the information is not specified in the original study. 
5.2.2 Results of symmetry test and metaanalysis
The test results are reported in Table 7. For total cholesterol level, both case samples and control samples in study Ganesan et al. (2011) have pvalue less than 0.05, i.e. there is significant evidence to reject the symmetry null hypothesis for this study. For LDLC, the control samples in study Montoya et al. (2012) has pvalue less than 0.05. For HDLC, both case samples and control samples in study Hürlimann et al. (2006) have small pvalues () and for triglycerides, all the studies have nonsignificant test results.








Bonnet et al. (2007) 
Case  12  0.450(0.653)  0.250(0.802)  0.750(0.453)  0.409(0.682)  
Control  9  1.168(0.243)  1.190(0.234)  0.623(0.533)  0.081(0.935)  
Calmy et al. (2010)  Case  10  0.316(0.752)  0.223(0.824)  1.002(0.316)  0.191(0.849)  
Control  12  0.563(0.574)  0.563(0.574)  1.097(0.273)  0.450(0.653)  
Eckard et al. (2014)  Case  67  NS  1.672(0.095)  NS  NS  
Control  69  NS  0.629(0.529)  NS  NS  
Funderburg et al. (2015)  Case  72  NS  NS  0.599(0.549)  NS  
Control  75  NS  NS  0.612(0.540)  NS  
Ganesan et al. (2011)  Case  22  2.254(0.024)  0.396(0.692)  NS  NS  
Control  22  2.254(0.024)  0.396(0.692)  NS  NS  
Hürlimann et al. (2006)  Case  29  1.613(0.107)  1.075(0.282)  2.258(0.024)  0.198(0.843)  
Control  29  1.613(0.107)  1.075(0.282)  2.258(0.024)  0.198(0.843)  
Lo et al. (2015)  Case  17  NS  NS  NS  1.585(0.113)  
Control  20  NS  NS  NS  0.062(0.950)  
Stein et al. (2004)  Case  20  NS  NS  NS  NS  
Control  20  NS  NS  NS  NS  
Nakanjako et al. (2015)  Case  15  NS  0.860(0.390)  5.73(1.000)  0.596(0.551)  
Control  15  NS  0.420(0.674)  0.516(0.606)  0.860(0.390)  
Montoya et al. (2012)  Case  51  NS  1.039(0.299)  NS  NS  
Control  53  NS  2.213(0.027)  NS  NS  
Moyle et al. (2001)  Case  14  0.000(1.000)  0.000(1.000)  0.085(0.932)  0.969(0.333)  
Control  13  0.215(0.830)  2.366(0.018)  0.000(1.000)  0.031(0.975)  
Mallon et al. (2006)  Case  14  NS  NS  NS  NS  
Control  17  NS  NS  NS  NS  
Results are expressed as Teststatistic (pvalue).  
NS indicates the original data do not need to conduct the symmetry test.  
pvalue indicates the underlying data is not symmetric. 
Similar to Section 4.1.2, after removing the studies with significant test results (i.e. underlying data is more likely to be skewed), we obtain the following four forest plots of effects of statin therapy on total cholesterol, LDLC, HDLC and triglycerides levels in HIVinfected patients, respectively. Likewise, we only provide overall results in this section due to demonstration purpose and will not further proceed sensitivity analysis and publication bias analysis. The pooled results showed that there was very high heterogeneity among studies for total cholesterol level ( and ) and nearly no heterogeneity among studies for the other three plasma concentration levels ( and ). The overall effect sizes imply that by using statin therapy, HIVinfected patients may have moderate growth in total cholesterol level (SMD with CI from 0.32 to 1.17), slight growth in HDLC and triglycerides levels (SMD with CI from 0.17 to 0.25 for HDLC; SMD with CI from 0.23 to 0.26 for triglycerides) and, slight reduction in LDLC level (SMD with CI from 0.33 to 0.07). Note that the pooled conclusions we obtained are different from Banach et al. (2016). In the original article, the authors concluded that there are significant reductions in both total cholesterol and LDLC levels. In contrast, we found that total cholesterol level has a moderate growth trend and LDLC level only has a slight reduction. As for HDLC and triglycerides levels, nearly zero pooled effect sizes declared that these two plasma concentration levels do not have significant change in HIVinfected patients. The opposite results we obtained indeed imply that it is essential to conduct a symmetry test on the selected studies before estimating the sample mean and standard deviation. If the included studies are blindly used to estimate the sample mean and standard deviation without considering the symmetry of original data, it may increase computational error and eventually lead to an inaccurate conclusion.
5.3 Case study of exploring the impact of statin therapy on plasma MMP3, MMP9 and TIMPI concentrations
In this section, we will discuss the third real case as introduced in Ferretti et al. (2017). This data is a metaanalysis about whether statin therapy has influence on plasma MMP3, MMP9 and TIMPI concentration levels. Total 10 studies were selected to conduct the systematic review. Note that in Ferretti et al. (2017), the authors compared the net changes (mean difference) in measurements between pretreatment and posttreatment. Since we tend to provide a demonstration for the usage of our proposed test statistics, we will not compute the net changes as in Ferretti et al. (2017). Instead, the followup measurements, i.e. posttreatment measurements are used to conduct the test and further analysis. Note also that in the original data set, study Hanefeld et al. (2007) does not record any information of the control set and hence, we will remove it from the following analysis.
5.3.1 Data description
The basic information of the selected data is reported in Table 8. Similar to the previous two cases, for plasma MMP9, MMP3 and TIMPI levels, data were recorded as the sample mean with standard deviation, or the sample median with interquartile range. As a result, test statistic (Eq.(8)) will be used to test the symmetry of underlying data. Likewise, if a study has pvalue greater than 0.05, we regard it as appropriate to further conduct the metaanalysis. If a study has pvalue smaller than 0.05, it is recommended to be excluded from further analysis.







Andrade et al. (2013) 
Case  25  11369  NS  281231  
Control  8  14789  NS  354287  
Koh et al. (2002)  Case  32  28(1934)  1614  7423  
Control  31  26(1741)  1817  8626  
Mohebbi et al. (2014)  Case  21  164.95126.68  NS  NS  
Control  21  180.81115.93  NS  NS  
Singh et al. (2008)  Case  23  106  NS  NS  
Control  24  26  NS  NS  
Singh et al. (2008)  Case  22  96  NS  NS  
Control  24  26  NS  NS  
Broch et al. (2014)  Case  36  243(106367)  NS  NS  
Control  35  354(162467)  NS  NS  
Kalela et al. (2001)  Case  24  35.18.20  NS  NS  
Control  26  40.425.30  NS  NS  
Leu et al. (2005)  Case  32  0.390.22  NS  NS  
Control  19  0.420.22  NS  NS  
Nilsson et al. (2011)  Case  37  212(169310)  21(1628)  155(143174)  
Control  39  184(141256)  20(1424)  149(135166)  
Observations are expressed as mean SD or median (interquartile range).  
NS indicates the information is not specified in the original study. 
5.3.2 Results of symmetry test and metaanalysis
The test results are provided in Table 9. Since in the original data set, only 3 studies reported the sample median and interquartile range, Table 9 only report the test results of these three studies. Fortunately, the reported data in all three studies had passed the symmetry test (pvalue greater than 0.05). That is, we can apply the methods provided by Luo et al. (2017) and Wan et al. (2014) to estimate the sample mean and standard deviation for these studies, respectively. Similar to the previous two cases, we generated forest plots of overall effects from the statin therapy on patients’ plasma MMP3, MMP9 and TIMPI concentrations (Figures 11(a), 11(b) and 11(c)).







Koh et al. (2002) 
Case  32  0.795(0.427)  NS  NS  
Control  31  0.976(0.329)  NS  NS  
Broch et al. (2014)  Case  36  0.211(0.833)  NS  NS  
Control  35  1.080(0.280)  NS  NS  
Nilsson et al. (2011)  Case  37  1.677(0.094)  0.716(0.474)  0.971(0.332)  
Control  39  1.115(0.265)  0.884(0.376)  0.428(0.669)  
Results are expressed as Teststatistic (pvalue).  
NS indicates the original data do not need to conduct the symmetry test.  
pvalue indicates the underlying data is not symmetric. 
Similar to the previous two case studies, we only compute overall effect sizes of the included data for demonstration purpose and will not conduct further analysis such as sensitivity analysis. The effect sizes of each study and the overall results are shown in the three graphs of Figure 11. Based on these three forest plots, the pooled results imply that by using the statin therapy, patients’ plasma MMP9 and MMP3 levels at the end of followup period may have a slightly increase (SMD with CI from 0.25 to 0.56 for MMP9 and SMD with CI from 0.31 to 0.50 for MMP3) than healthy individuals. On the other hand, patients’ TIMPI levels at the end of followup period may decrease very slightly (SMD with CI from 0.68 to 0.42). Figure 11(a) also indicates that there might be high heterogeneity among studies for MMP9 levels (, and ). Note that in Figures 11(b) and 11(c), the values of imply that there are low or moderate heterogeneity among studies for MMP3 and TIMPI levels, respectively ( for MMP3 and for TIMPI). However, the values of statistics for these two plasma levels are very small and pvalues are greater than 0.05 ( with for MMP3; with for TIMPI), which imply that both MMP3 and TIMPI levels hardly have heterogeneity among selected studies. Such conflict conclusion about heterogeneity between studies may be caused by the small number of studies included. As a result, if researchers aim to obtain an accurate conclusion about the effects of statin therapy on patients’ plasma MMP3 and TIMPI levels, more studies and information are needed.
Based on the above three real data analysis, it is obvious that our proposed test statistics are necessary for medical researchers, especially when we need to transform the intermediate summary statistics (median, minimum, maximum or interquartile range) to the sample mean and standard deviation. We believe by conducting the proposed symmetry tests, researchers can efficiently find out and exclude the potential skewed data to reduce the errors in computing the effect sizes.
6 Conclusion
Metaanalysis is a useful tool in evidencedbased medicine to statistically combine and analyze clinical results from two or more independent trials. Researchers use different effect size measurements to statistically compare the effectiveness of some particular medicine or therapy. Mean difference based effect sizes are used to analyze continuous data in clinical research. To compute the mean difference measurements, the sample mean and standard deviation are two indispensable information. However, most of the medical studies do not record these two statistics directly. Instead, the summary measurements such as the sample median, minimum and maximum or interquartile range are more likely to be reported. In this case, researchers have to transform the reported data into the sample mean and standard deviation. Our major concern is that even for the optimal estimation methods of these two statistics (introduced by Luo et al. (2017) and Wan et al. (2014)), the estimators are developed based on the symmetry assumption. Thus we recommend that researchers should conduct a symmetry test for the underlying data before transform them into the sample mean and standard deviation, in order to improve the accuracy of the estimation. As a result, we introduce test statistics based on the summary measurements for three most frequently appeared scenarios.
Test statistic  Coefficient  

Scenario  
Scenario  
Scenario 
For the three most popular scenarios, the corresponding test statistics and their coefficient functions are summarized in Table 10. All these three test statistics are theoretically proved and have simple formulation which are easy for researchers to adopt in practice. The simulation studies in Section 3 show that our proposed test statistics have statistical power close to one and for both scenarios, which indicates that the test statistics should have very good performance in detecting potential skewed data. Moreover, to help researchers to make more convincing conclusions in medical research, we suggest a proper path to conduct the metaanalysis in Figure 1 and the first step is to conduct the symmetry test via our test statistics first. To further illustrate the usage of the newly proposed test statistics, we applied them on three different real metaanalysis in Section 4. By conducting the symmetry test, the pvalues show that some of the selected studies of these three cases should be excluded before carry on to metaanalysis. As a result, the results of metaanalysis with skewed data removed may be different from the original conclusion. In particularly, for the case study about the effects of statin therapy on plasma lipid concentrations in HIVinfected patients (Banach et al., 2016), compare to the original article, we have opposite conclusion about the pooled effect sizes of total cholesterol level and LDLC level. Since the simulation studies indicate the reliability of the proposed test statistics, we expect that our test statistics and the recommended procedure in Figure 1 can help researchers to obtain faithful conclusion in evidencebased medicine.
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