Testing normality using the summary statistics with application to meta-analysis

01/29/2018
by   Dehui Luo, et al.
Hong Kong Baptist University
0

As the most important tool to provide high-level evidence-based medicine, researchers can statistically summarize and combine data from multiple studies by conducting meta-analysis. In meta-analysis, mean differences are frequently used effect size measurements to deal with continuous data, such as the Cohen's d statistic and Hedges' g statistic values. To calculate the mean difference based effect sizes, the sample mean and standard deviation are two essential summary measures. However, many of the clinical reports tend not to directly record the sample mean and standard deviation. Instead, the sample size, median, minimum and maximum values and/or the first and third quartiles are reported. As a result, researchers have to transform the reported information to the sample mean and standard deviation for further compute the effect size. Since most of the popular transformation methods were developed upon the normality assumption of the underlying data, it is necessary to perform a pre-test before transforming the summary statistics. In this article, we had introduced test statistics for three popular scenarios in meta-analysis. We suggests medical researchers to perform a normality test of the selected studies before using them to conduct further analysis. Moreover, we applied three different case studies to demonstrate the usage of the newly proposed test statistics. The real data case studies indicate that the new test statistics are easy to apply in practice and by following the recommended path to conduct the meta-analysis, researchers can obtain more reliable conclusions.

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1 Introduction

Meta-analysis is the most important tool to provide high-level evidence in evidence-based medicine. By conducting meta-analysis, researchers can statistically summarize and combine data from multiple studies with some pre-determined 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 meta-analysis 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 meta-analysis about the association between B-type natriuretic peptides (BNP) and chronic obstructive pulmonary disease (COPD). Totally there were 51 studies included in the meta-analysis 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 meta-analysis.

Index
Study
Sample size
BNP Levels

1
Anderson et al. (2013) Case 93 296
Control 93 26(20-32)
2 Gemici et al. (2008) Case 17 2116
Control 17 1311
3 Boschetto et al. (2013) Case 23 121(59-227)
Control 23 50(43-51)
4 Wang et al. (2013) Case 80 245(196-336)
Control 80 101(56-150)
5 Beghé et al. (2013) Case 70 115(50-364)
Control 70 50(43-51)
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).
Table 1: Summary of included studies

Selected studies

Pre-test if data is symmetric ?

Estimate the sample mean using methods of Luo et al. (2017)

Estimate the sample standard deviation using methods of Wan et al. (2014)

Exclude this data

Compute the effect size

Further analysis and make decision

Yes

No

Figure 1: Recommended procedure of computing effect sizes for meta-analysis

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 follow-up analysis. In view of the above situation, we propose some new test statistics to pre-test 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 meta-analysis. In this case, we suggest when conducting meta-analysis, 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 meta-analysis. 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 log-normal distribution with location parameter

and scale parameter for the disease cases,

for the controls; the chi-square 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.

Data
Distribution
Size
(Cases)
Size (Controls)
Summary statistics
(Cases)
Summary statistics
(Controls)
Log-normal 350 350
Chi-square 200 200
Exponential 150 150
Beta 300 300
Weibull 400 400
Table 2: Example of 5 skewed distributed data
Data
Distribution
(Cases) True
mean (SD)
(Controls) True
mean (SD)
(Cases)
Estimated
Mean (SD)
(Controls)
Estimated
Mean (SD)
Effect Size
(use estimated
mean)
Effect Size
(use true
mean)
Log-normal 1.72 (2.70) 4.05 (4.64) 1.61 (4.82) 3.20 (5.72) 0.30 0.61
Chi-square 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
Table 3: Effect sizes comparison of 5 skewed distributed data

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 log-normal distribution and chi-square 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 5-number 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 mid-range, as the interquartile range, and as the mid-quartile range. Note that according to Wan et al. (2014) and Luo et al. (2017), the mid-range and the mid-quartile 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 5-number summary statistics. Otherwise, if the null hypothesis is rejected, we suggest researchers not to include the tested data when proceeding meta-analysis.

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 5-number 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 meta-analysis.

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 log-normal distribution with ; the standard exponential distribution with ; the beta distribution with shape parameters ; the Chi-square 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.

Figure 2: Type I error for scenario .
Figure 3: The statistical power for scenario .

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 real-life application. In the next section, we will conduct real data analysis to evaluate the performance of .

Figure 4: Type I error for scenario .
Figure 5: The statistical power for scenario .

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.

Figure 6: Type I error for scenario .
Figure 7: The statistical power for scenario .

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 HIV-infected 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 meta-analysis 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
Type of asthma
Sample size
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[0.4-27.4] NS
Healthy 51 8.8[0.3-31.3] 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
Non-obese 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(30-60) 4.5(3.5-8.5)
Healthy 50 39(25-50) 4(3-7.8)
Giouleka et al. (2011) Asthma 100 9.6(7.6-16.25) 6.2(5.4-7.3)
Healthy 60 7.2(4.6-10.3) 8.2(5.8-13.5)
Leivo-Korpela et al. (2011) Asthma 35 0.5(0.5-1.1) 1659.5
Healthy 32 0.6(0.4-0.8) 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(0.65-5.03) 7.603.84
Non-atopic asthma 37 2.22(0.96-3.29) 8.104.73
Healthy 54 2.10(0.71-4.49) 7.324.19
Guler et al. (2004) Asthma 102 3.53(2.06-7.24) NS
Control 33 2.26(1.26-4.71) NS
Observations are expressed as mean SD, median (interquartile range) or median [minimum-maximum].
indicates the information is not specified in the original study.
Table 4: Summary of included studies in Zhang et al. (2017)

5.1.2 Results of symmetry test and meta-analysis

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), Leivo-Korpela et al. (2011), Kim et al. (2008) and Guler et al. (2004). To further proceed meta-analysis, 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 p-value 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, non-obese asthma and healthy individuals. In this case, only when the p-values 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 p-value Test statistic p-value
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
Leivo-Korpela 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
Non-atopic 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
p-value indicates the underlying data is not symmetric.
NS indicates the reported data do not need to conduct the symmetry test.
Table 5: Results of normality test on reported data from Zhang et al. (2017)

It is obvious that for leptin serum level, all the studies, except for study da Silva et al. (2012), have p-values less than 0.05 on either asthma samples or healthy samples. That is, these 5 studies (Cobanoglu et al. (2013),Giouleka et al. (2011), Leivo-Korpela 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 meta-analysis of investigating the association between leptin serum level and asthma.

(a) Leptin level
(b) Adiponectin level
Figure 8: Forest plots for association between asthma versus leptin (8(a)) and adiponectin levels (8(b)), respectively.

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 meta-analysis. 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 meta-analysis, 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 HIV-infected patients

The second data is a meta-analysis to investigate the effects of statin therapy on plasma lipid concentrations in HIV-infected patients (Banach et al., 2016). In this article, meta-analysis 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 (LDL-C), total cholesterol, high density lipoprotein cholesterol (HDL-C), non-high density lipoprotein cholesterol (non-HDL-C) and triglycerides. Note that in Banach et al. (2016), only two studies reports data of non-HDL-C and both these two studies provided the sample mean and standard deviation. Hence in the following discussion, we will remove variable non-HDL-C 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 p-value 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 meta-analysis in order to obtain more precise conclusion.

Study
Sample
size
Total
Cholesterol
(mmol/L)
LDL-C
(mmol/L)
HDL-C
(mmol/L)
Triglycerides
(mmol/L)

Bonnet et al. (2007)
Case 12 6.1(5.8-6.3) 4.1(3.7-4.6) 0.9(0.8-1.1) 2.0(1.1-3.3)
Control 9 6.4(6.1-7.7) 3.9(3.7-4.8) 1.0(0.8-1.1) 3.2(2.1-4.4)
Calmy et al. (2010) Case 10 5.6(4.5-6.4) 2.8(2.4-3.3) 1.0(0.9-1.3) 3.9(2.0-6.2)
Control 12 5.6(4.6-6.2) 3.5(2.5-4.1) 1.1(0.81-1.2) 2.3(1.5-3.5)
Eckard et al. (2014) Case 67 NS 2.48(1.96-2.77) NS NS
Control 69 NS 2.50(1.99-3.13) NS NS
Funderburg et al. (2015) Case 72 NS NS 1.21(0.98-1.49) NS
Control 75 NS NS 1.19(0.96-1.47) NS
Ganesan et al. (2011) Case 22 4.34(3.72-4.45) 2.50(2.25-2.82) NS NS
Control 22 4.34(3.72-4.45) 2.50(2.25-2.82) NS NS
Hürlimann et al. (2006) Case 29 6.4(6.0-7.4) 3.7(2.8-4.2) 1.2(1.1-1.6) 3.0(2.1-4.0)
Control 29 6.4(6.0-7.4) 3.7(2.8-4.2) 1.2(1.1-1.6) 3.0(2.1-4.0)
Lo et al. (2015) Case 17 5.140.98 3.200.95 1.340.50 1.36(1.10-2.31)
Control 20 4.970.70 3.230.83 1.310.39 1.28(1.04-1.53)
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(2.2-4.9) 1.7(1.6-1.8) 1.6(1.1-2.4)
Control 15 NS 4.9(2.4-6.7) 1.7(1.5-2.0) 2.0(1.4-3.2)
Montoya et al. (2012) Case 51 NS 2.63(2.20-3.28) NS NS
Control 53 NS 2.53(2.30-3.10) NS NS
Moyle et al. (2001) Case 14 7.5(6.7-8.3) 4.65(4.1-5.2) 0.94(0.79-1.08) 3.96(2.84-6.52)
Control 13 7.4(6.8-7.9) 4.68(3.89-5.47) 0.87(0.72-1.02) 4.06(2.20-5.97)
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.
Table 6: Summary of included studies in Banach et al. (2016)

5.2.2 Results of symmetry test and meta-analysis

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 p-value less than 0.05, i.e. there is significant evidence to reject the symmetry null hypothesis for this study. For LDL-C, the control samples in study Montoya et al. (2012) has p-value less than 0.05. For HDL-C, both case samples and control samples in study Hürlimann et al. (2006) have small p-values () and for triglycerides, all the studies have non-significant test results.

Study
Sample
size
Total
Cholesterol
LDL-C
HDL-C
Triglycerides

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 Test-statistic (p-value).
NS indicates the original data do not need to conduct the symmetry test.
p-value indicates the underlying data is not symmetric.
Table 7: Results of normality test on reported data from Banach et al. (2016)
(a) Total cholesterol
(b) LDL-C
Figure 9: Forest plots of effects in plasma concentrations of total cholesterol (9(a)), LDL-C (9(b)) in HIV-infected patients.
(a) HDL-C
(b) Triglycerides
Figure 10: Forest plots of effects in plasma concentrations of HDL-C (10(a)) and triglycerides (10(b)) in HIV-infected patients.

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, LDL-C, HDL-C and triglycerides levels in HIV-infected 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, HIV-infected patients may have moderate growth in total cholesterol level (SMD with CI from -0.32 to 1.17), slight growth in HDL-C and triglycerides levels (SMD with CI from -0.17 to 0.25 for HDL-C; SMD with CI from -0.23 to 0.26 for triglycerides) and, slight reduction in LDL-C 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 LDL-C levels. In contrast, we found that total cholesterol level has a moderate growth trend and LDL-C level only has a slight reduction. As for HDL-C and triglycerides levels, nearly zero pooled effect sizes declared that these two plasma concentration levels do not have significant change in HIV-infected 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 MMP-3, MMP-9 and TIMP-I concentrations

In this section, we will discuss the third real case as introduced in Ferretti et al. (2017). This data is a meta-analysis about whether statin therapy has influence on plasma MMP-3, MMP-9 and TIMP-I 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 follow-up 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 MMP-9, MMP-3 and TIMP-I 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 p-value greater than 0.05, we regard it as appropriate to further conduct the meta-analysis. If a study has p-value smaller than 0.05, it is recommended to be excluded from further analysis.

Study
Sample
size
MMP-9
(ng/mL)
MMP-3
(ng/mL)
TIMP-I
(ng/mL)

Andrade et al. (2013)
Case 25 11369 NS 281231
Control 8 14789 NS 354287
Koh et al. (2002) Case 32 28(19-34) 1614 7423
Control 31 26(17-41) 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(106-367) NS NS
Control 35 354(162-467) 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(169-310) 21(16-28) 155(143-174)
Control 39 184(141-256) 20(14-24) 149(135-166)
Observations are expressed as mean SD or median (interquartile range).
NS indicates the information is not specified in the original study.
Table 8: Summary of included studies in Ferretti et al. (2017)

5.3.2 Results of symmetry test and meta-analysis

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 (p-value 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 MMP-3, MMP-9 and TIMP-I concentrations (Figures 11(a), 11(b) and 11(c)).

Study
Sample
size
MMP-9
(ng/mL)
MMP-3
(ng/mL)
TIMP-I
(ng/mL)

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 Test-statistic (p-value).
NS indicates the original data do not need to conduct the symmetry test.
p-value indicates the underlying data is not symmetric.
Table 9: Results of symmetry test on reported data from Ferretti et al. (2017)

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 MMP-9 and MMP-3 levels at the end of follow-up period may have a slightly increase (SMD with CI from -0.25 to 0.56 for MMP-9 and SMD with CI from -0.31 to 0.50 for MMP-3) than healthy individuals. On the other hand, patients’ TIMP-I levels at the end of follow-up 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 MMP-9 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 MMP-3 and TIMP-I levels, respectively ( for MMP-3 and for TIMP-I). However, the values of statistics for these two plasma levels are very small and p-values are greater than 0.05 ( with for MMP-3; with for TIMP-I), which imply that both MMP-3 and TIMP-I 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 MMP-3 and TIMP-I levels, more studies and information are needed.

(a) MMP-9
(b) MMP-3
(c) TIMP-I
Figure 11: Forest plots of effects in follow-up plasma MMP-9 (11(a)), MMP-3 (11(b)) and TIMP-I (11(c)) concentration levels in patients that use statin therapy.

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

Meta-analysis is a useful tool in evidenced-based 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
Table 10: Summary table of the test statistics under three scenarios

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 meta-analysis 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 meta-analysis in Section 4. By conducting the symmetry test, the p-values show that some of the selected studies of these three cases should be excluded before carry on to meta-analysis. As a result, the results of meta-analysis 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 HIV-infected patients (Banach et al., 2016), compare to the original article, we have opposite conclusion about the pooled effect sizes of total cholesterol level and LDL-C 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 evidence-based medicine.


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