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 Table of Contents  
ORIGINAL ARTICLE
Year : 2020  |  Volume : 40  |  Issue : 3  |  Page : 119-126

A superior odds ratio compared to the risk ratio when estimating moderator effects in meta-regression analyses of randomized controlled trials


1 Division of Infectious Diseases, Department of Internal Medicine, Taoyuan Armed Forces General Hospital, Taoyuan, Taiwan, ROC, Taiwan
2 Department of Orthopedics, Taichung Armed Forces General Hospital, Taichung, Taiwan, ROC, Taiwan
3 Renal Division, Department of Internal Medicine, Taoyuan Armed Forces General Hospital, Taoyuan, Taiwan, ROC, Taiwan
4 Graduate Institute of Life Sciences, National Defense Medical Center, Taipei, Taiwan, ROC, Taiwan
5 School of Public Health, National Defense Medical Center, Taipei, Taiwan, ROC, Taiwan
6 Department of Family and Community Medicine, Tri-Service General Hospital, National Defense Medical Center, Taipei, Taiwan, ROC, Taiwan
7 Division of Thoracic Medicine, Department of Medicine, Cheng Hsin General Hospital, Taipei, Taiwan, ROC, Taiwan
8 Renal Division, Department of Internal Medicine, Taoyuan Armed Forces General Hospital, Taoyuan, Taiwan, ROC, Taiwan
9 Department of Pediatrics, Taoyuan Armed Forces General Hospital, Taoyuan; Division of Pediatrics, Department of Medicine, Tri-Service General Hospital, National Defense Medical Center, Taipei, Taiwan, ROC, Taiwan
10 Graduate Institute of Life Sciences; School of Pharmacy, National Defense Medical Center, Taipei, Taiwan, ROC, Taiwan
11 Department of Nursing, Tri-Service General Hospital, Taipei, Taiwan, ROC, Taiwan
12 Division of Nephrology, Department of Medicine, Zuoying Branch of Kaohsiung Armed Forces General Hospital, Kaohsiung, Taiwan, ROC, Taiwan
13 Department and Gradute Institute of Pharmacology, National Defense Medical Center, Taipei, Taiwan, ROC, Taiwan

Date of Submission17-Jul-2019
Date of Decision22-Sep-2019
Date of Acceptance25-Oct-2019
Date of Web Publication03-Dec-2019

Correspondence Address:
Dr. Chin Lin
National Defense Medical Center, Graduate Institute of Life Sciences, No. 161, Min-Chun E. Road, Section 6, Neihu, Taipei 114
Taiwan
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Source of Support: None, Conflict of Interest: None


DOI: 10.4103/jmedsci.jmedsci_133_19

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  Abstract 


Background: Moderator effect assessment is important in personalized medicine. We mathematically prove that the average summary value is actually nonlinearly to logRR, and we assess the bias from linear meta-regression on logRR via simulation. Methods: In the meta-analysis of randomized controlled trials, the moderator effect is generally evaluated by the linear meta-regression of the logarithmic risk ratio (RR) versus the average summary value of the entire study population. Conclusions: We recommend using linear meta-regression on logarithmic odds ratio (logOR) since it has been shown that the average summary value is actually linear to logOR.

Keywords: Moderator effect, interaction, personalized medicine, odds ratio, risk ratio, participant-level variable, study-level variable, meta-analysis, meta-regression, randomized controlled trial


How to cite this article:
Chiu CC, Chen CF, Hsiao PJ, Tsai DJ, Chang HL, Fang WH, Chen WT, Chan JS, Liao MT, Ho YJ, Su W, Chen YK, Hu HH, Lai ZZ, Lin C. A superior odds ratio compared to the risk ratio when estimating moderator effects in meta-regression analyses of randomized controlled trials. J Med Sci 2020;40:119-26

How to cite this URL:
Chiu CC, Chen CF, Hsiao PJ, Tsai DJ, Chang HL, Fang WH, Chen WT, Chan JS, Liao MT, Ho YJ, Su W, Chen YK, Hu HH, Lai ZZ, Lin C. A superior odds ratio compared to the risk ratio when estimating moderator effects in meta-regression analyses of randomized controlled trials. J Med Sci [serial online] 2020 [cited 2020 Jul 12];40:119-26. Available from: http://www.jmedscindmc.com/text.asp?2020/40/3/119/272328




  Introduction Top


Detecting a moderator effect (or interaction) is important in personalized medicine. In randomized controlled trials (RCTs), the moderator effect is typically tested in post hoc subgroup analyses.[1] The list of prespecified subgroups could be selected from previous mechanistic studies, epidemiological studies, or RCTs. To explore potential unknown moderate factors was also important; they might help us to generate new hypotheses. When exploring unknown factors in multivariable studies, the significance levels in moderator effect detection must be reduced by adjusting P value. However, the power of a single trial is often insufficient after multicomparison correction.[2],[3] Meta-analysis is a common method for increasing this power, and subgroup analysis may also help detect the moderator effect. The biggest challenge in detecting the moderator effect in a meta-analysis is data collection. Analyses of the subsets of participants within studies are rarely reviewed in the literature because most reports lack sufficient details to extract the data on participant-level variables, such as gender and comorbidity.[4] Thus, most meta-analyses are limited to the moderator effects of the study-level variables, such as geographical location. However, most of potential moderator effects were hidden in participant-level variables, such as diabetes in folic acid treatment effect.[5] Researches needed to explore these potential issues for reducing heterogeneity.

Because participant-level variables are generally reported as their average summary values, there are two common methods for investigating their moderator effects. The first method selects a cutoff point that dichotomizes the included studies into “higher group ” and “lower group. ” However, this method increases the likelihood of Type 1 errors[6],[7] and yields imprecise risk estimates. The second method investigates the relation between the average summary value and the logarithmic risk ratio (RR) by meta-regression.[8],[9],[10] As shown in [Figure 1] and [Figure S1], this relation is nonlinear; thus, this method may bias the estimates.

In a previous study, we proved a linear relationship between the average summary value of a case group and the logarithmic odds ratio (OR).[11] Thus, we inferred that the proposed method would improve moderator effect detection in the meta-analysis of RCTs. However, acknowledging the difficulty of accessing the average summary value of a case group, we also considered replacing this variable with the average summary value of the entire population. Therefore, this study compares the Type 1 error, power and 95% confidence interval coverage rate (CICR) of the following methods: (1) meta-regression of the logarithmic RR versus the average summary value of the entire population (the most common method); (2) meta-regression of the logarithmic OR versus the average summary value of the entire population; and (3) meta-regression of the logarithmic OR versus the average summary value of a case group.


  Methods Top


Derivations

The moderator effect of participant-level variables in the meta-analyses of RCTs is usually detected by using the average summary value, in which the association between the aggregated summary values of the factor and RR was based on multiple factors. For better understanding of this principle, we hereby describe an example.

When the independent variable (x) is the intervention encoded with value 0 for control group and 1 for treatment group, and the moderator (m) is diabetes status encoded with values 0 and 1 for without and with diabetes, respectively. The dependent variable is a binary outcome event (y) (with 0 and 1 signifying nonoccurrence and occurrence, respectively). Probabilities p1, p2, p3, and p4 are then defined as follows:



which represents the outcome incidence of control group without diabetes, treatment group without diabetes, control group with diabetes, and treatment group with diabetes. According to above setting, the RRs of interest for patient without diabetes (RR0[m = 0]) and patient with diabetes (RR1[m = 1]) are calculated as follows, respectively:



In an RCT, the pooled or combined RR (RRcombine) is a function of the diabetes prevalence in the entire study population,and can be expressed as:



Let the π denote the RR of diabetes in the untreated individuals (π = p3/p1). We can express p1, p2, p3, and p4 in terms of p1 and π alone including RRs as follows:



This simplifies the calculation of RRcombine. Let the ME denote the moderator effect (ME = RR1/RR0), RRcombine can be calculated as follows [for details, Text S1]:



and the logarithm of RRcombine can be expressed as follows [for details, Text S1]:

(2.1-1)

Let log (RRi) denotes the observed effect size of i th individual RCT, qtotal, i the proportion of the population with moderator status, ηi the random effects, and εi the residuals. In general, awareness studies, β0 is the expected logarithmic RR of individuals without diabetes (log[RR0]), and β1 is the expected logarithmic moderator effect (log[ME]). Thus, the traditional meta-regression (here called Method 1) can be written as.

(2.1-2)

However, following Equation 2.1-1, the relation between log (RRi) and qtotal, i is actually nonlinear:



and π is the RR of diabetes in the untreated individuals.

Note that π is an unknown population parameter not provided by most papers. That is why the traditional studies often used qtotal, i to replace κi. Obviously, this may cause bias since κi= qtotal,i only when π = 1 and ME = 1 [Figure 1] and [Figure S1]. Plot the true relation between the average summary value of the entire population (prevalence of diabetes in example) and the logarithmic RR. We observe that the bias is larger when the directions of π and ME are the same, and their effects leave l (null effect). The nonlinear relation will cause biased estimates of β0 and β1, and the bias extent in different situations is shown in simulation part.

The average summary value of a case group is approximately linearly related to the logarithmic OR under two assumptions: (1) rare disease and (2) independence between the independent variable and moderator, as in an earlier study.[11],[12] The independence assumption holds in the meta-analysis of RCTs, and the proposed method remains robust when the rare disease assumption is violated.[11] Thus, we expect that the meta-regression of the logarithmic OR versus the average summary value of the case group in study i (qcase, i, i.e., Method 3) will best detect the moderator effect in the meta-analysis of RCTs.

As the average summary value of a case group may be difficult to access, we investigate the viability of replacing the value with the average summary value of the entire population. Let RR0 denote the RR of the treatment in individuals without the moderator; then, the variables qtotal, i and qcase, i are related through Equation 2.1-4 [for this derivation, Text S2]:



The difference between qtotal, i and qcase, i is affected by RR0,ME, and π. The Pearson correlation between qtotal and qcase exceeds 0.95 when RR0,ME, and π are between 0.5 and 2.0 [Table S1]; thus, we considered that qtotal can effectively represent qcase. Thus, the meta-regression of the logarithmic OR versus the average summary value of the entire population (i.e., Method 2) is a useful alternative when the average summary value of a case group is unavailable.

Simulations

In this subsection, we simulate a meta-analysis of RCTs. The simulation code is written in the R programming language; this code is provided in [Appendix S1]. As the meta-analysis data are summarized from individual data, we generated individual simulation studies. We generated 20 studies with each simulation using randomly generated sample sizes from a uniform distribution of (200, 1000). The probability of receiving treatment was set equal to 0.5 (the simple randomized design). The proportions of individuals with moderators were randomly generated from a uniform distribution (0, 1), and the treatment (t) was assumed to be independent of the moderator (m). The above steps were used to generate the information of the treatment and moderator.

The second step was to generate the disease information. The disease incidence of each individual was based on the relative risk model with four parameters: (1) the disease incidence of the entire population (p), which is not the incidence in the patients without treatment or the moderator; (2) the RR of treatment in individuals without the moderator (RR0); (3) the RR of the moderator in individuals without treatment (π); and (4) the moderator effect (ME). Using these parameters, we calculated the disease incidence of untreated individuals without the moderator (Incidence [t = 0 ∩ m = 0]) as follows (the detailed derivation was shown in the previous study):[11]



We also calculated the disease incidences of individuals with other conditions as follows:



The disease statuses of the individuals were randomly generated based on the above incidences. The values of p, RR0, π, and ME used in the simulations are listed in [Table 1].
Table 1: Simulation conditions of this study

Click here to view


The third step was to summarize the individual data into meta-analysis data. Each study provided four pieces of information for the following meta-regression analysis: (1) the logarithmic RR of treatment without stratification and its variance (i.e., log[RRi] and var [log[RRi], respectively); (2) the logarithmic OR of treatment without stratification and its variance (i.e., log [ORi] and var [log [ORi], respectively); (3) the proportion of individuals with the moderator throughout the study (qtotal); and (4) the proportion of individuals with the moderator in the case group (qcase). The meta-regression analyses proceeded as follows, and we used the random effect model for the simulation. We used the “rma ” command in the “metafor ” package[13] in R to calculate the following meta-regression.

1. Meta-regression of the logarithmic RR versus the average summary value of the entire population

Regression formula:



2. Meta-regression of the logarithmic OR versus the average summary value of the entire population

Regression formula:



3. Meta-regression of the logarithmic OR versus the average summary value of the case group

Regression formula:



where log (RRi) is the observed logarithmic RR in study i; qtotal, i is the proportion of the population with moderator status in study i; qcase, i is the proportion of the population with moderator status of case group in study i; ηi is the random effects; εi is the residuals; i is an identifier of an individual RCT ranging from 1 to 20; β0 is the expected logarithmic RR of individuals without moderator status; β1 is the expected logarithmic moderator effect; and τ[2] is the variance of random effects, which is estimated by the restricted maximum likelihood method.

The primary outcomes were the 95% CICRs of the moderator effect (β1) and the intercept (β0). The CICR defines the proportion of the 95% CIs that include the real parameter. The appropriate CI coverage was 95%. In addition, Type 1 errors were assessed in the null moderator effect model (ME = 1). As nonsignificant results are often ignored, we also computed the power of the moderator-effect assessment as the secondary outcome. Data under each condition were acquired from 10,000 simulations.


  Results Top


Figure 2 shows selected simulations relating to Equation 2.1-3 and [Figure 1]. Conditions A, B, and C differ only by the value of π. Method 3 is shown to produce the most robust results, and the 95% CICRs of the slope and intercept approximate 0.95 under all conditions. Methods 1 and 2 introduce varying degrees of bias, which is higher in the intercept than in the slope. However, Method 2 is more robust than Method 1. As expected, the conditions with larger bias (π/ME = 2.0/0.5) provide the same results as the earlier derivation [Figures 1] and [Figure S1]. In all methods, the 95% CICRs approximate 0.95 when ME = 1.0, which indicates that the false-positive rates of all methods are acceptable. However, the methods differ in their statistical power. Overall, Methods 2 and 3 exhibit higher statistical powers than Method 1. As described above, the parameters π and ME affect the bias in Method 1 because they damage the linear relation between the logarithmic RR and the average summary value throughout the study.
Figure 1: Relation between the average summary value and the logarithmic RR. Based on Equation 2.1-3, the true and linear relations differ only with respect to the RR of the moderator (π) and the ME. This figure plots the true and linear relations in several selected scenarios (π= 2.0 and ME = 2.0,π = 1.0 and ME = 2.0, π = 1.0 and ME = 1.0, and π = 0.5 and ME = 2.0). The X axis represents the average summary value throughout the study (q). Black and blue lines plot the expected RR and the linear relation, respectively. The bias is defined as the triangular area in the window straddling the two lines. The larger bias appears in that π/ME deviate null effect and in the same direction (π = 2.0 and ME = 2.0). The only scenarios without bias are that π/ME are null effects. There are bias of varying sizes when π or ME is not equal to 1. The details scenarios are shown in Figure S1. The intercept RR0is set equal to 1, and the Y axis is transformed into the logarithmic scale. ME: Moderator effect, RR: Risk ratio

Click here to view
Figure 2: Results from selected simulations. Based on Equation 2.1-3, Figures 1 and S1, the largest bias occurs when π/ME = 2.0/0.5. Thus, under conditions A and C, π is set equal to 2.0 and 0.5, respectively. We also select a condition that produces the smallest expected bias (condition B). The parameters P and RR0are set equal to 0.2 and 1.0, respectively. Condition A: P = 0.2; RR0= 1.0; π =2.0; ME = 2.0-0.5; Condition B:P = 0.2; RR0= 1.0; π = 1.0; ME = 2.0-0.5; Condition C:P = 0.2; RR0= 1.0; π = 0.5; ME = 2.0-0.5. Red line: Method 1; Green line: Method 2; Blue line: Method 3. ME: Moderator effect

Click here to view


[Figure S2] and [Figure S3] in the supplementary material present the detailed 95% CICRs of the slope and intercept, respectively, in the three meta-regression methods. [Figure S4] shows the false-positive rates and the powers of the moderator effect at the 0.05 significance level in the meta-regression methods. The 95% CICR of the slope is marginally reduced for P = 0.2 and π/ME = 2.0/0.5 in Method 3; however, the bias nearly disappears when P = 0.1. This may correspond to the rare disease assumption of Method 3. Method 2 shows higher bias than Method 3 under all conditions but remains more robust than Method 1. The powers of Methods 2 and 3 are similar under all conditions and frequently exceed that of Method 1. Method 1 is insensitive to the values of RR0 and p, which is also consistent with the earlier derivation. The marginal difference introduced by these parameters might be due to the weights in each study. The effects of the weights are described in the following section.


  Discussion Top


Meta-analysis is a powerful tool to determine the consistency of evidence. However, the pooled results often have high heterogeneity because the treatment effect is different in different patients.[14] The mission of personalized medicine is to advocate for the practice of personalized health care; therefore, investigating the source of heterogeneity is important. Meta-regression is a common method to explain heterogeneity, but the previous study suggested that meta-regression was only suitable on study-level variables.[4] However, personalized medicine is focused on effects at the individual patient level rather than the study level. Thus, investigating the source of heterogeneity of participant-level variables is important. Although certain studies investigated the relation between the average summary participant-level value and the logarithmic RR by meta-regression,[8],[9],[10] this relation is nonlinear, as shown in [Figure 1] and [Figure S1]. Thus, recent research has criticized this method.[14]

As an earlier study proved a linear relationship between the average summary value of a case group and the logarithmic OR,[11] we expected that the performance of the meta-regression using OR would exceed that when using RR. Consistent with the expected results, Method 1 exhibited higher bias than Methods 2 and 3 in all conditions investigated in this study. Thus, although certain conditions improved the power of Method 1 above those of Methods 2 and 3, we inferred that Method 1 is not a robust method. Meta-analysis attempts to determine the P value and to estimate the accuracy of a treatment effect.[4] Thus, a low-bias method is required and is achieved by Methods 2 and 3.

The larger bias of Method 2 compared to Method 3 might arise from the attenuation bias because we replace the average summary value of the case group with the average summary value of the entire population. 15,16 However, the Pearson correlations between qtotal and qcase exceeded 0.95 for appropriate values of RR0,ME and π (i.e., between 0.5 and 2.0) [Table S1]. The Pearson correlations between qtotal and qcase decrease as ME increases, but a large ME is easily detected in a single study.[17] The meta-analysis should focus on a smaller ME, which is difficult to detect in a single study. Thus, we considered that the average summary value of the entire population is an acceptable substitute for the average summary value of the case group. The lower bias in Method 2 compared to Method 1 indicates the superiority of OR over RR in the meta-regression.

The results with different simulation conditions were not entirely consistent with the expected bias. In [Figure 1] and [Figure S1], the biases are equal when π/ME = 2.0 and π/ME = 0.5; however, differences emerge in [Figure 2]. These differences are introduced by the weights in each study, which depend on several factors: the sample size (N), the random effect variance (τ[2]), the disease incidence of untreated individuals without the moderator (p1), the RR of treatment in individuals without the moderator (RR0), the RR of the moderator in untreated individuals (π), the moderator effect (ME), and the proportions of individuals with the moderator throughout the study population in study i (qtotal, i). The dependences of the study weights (Weighti) in study i on these parameters are given by Equations 4-1.1 and 4-1.2 (for detailed derivations), [Text S1 and S3]. Thus, the Weighti changes when π/ME decreases from 2.0 to 0.5 [Figure S5]. This phenomenon also explains the marginal dependence of the simulation results on P and RR0. Specifically, we have:{Figure 2}



The bias introduced by the weights is not easily simplified because it involves the distribution of qtotal among the included studies. However, if the average summary value is a linear function of OR/RR, the studies' weights will not bias the estimates. Thus, the linear relation is important and again highlights the superior mathematical behavior of OR, whose linear relation with the average summary value has been proven.[11]

The selection of summary statistics for the meta-analysis of RCTs has been addressed in previous studies.[15],[16] The authors concluded that OR and RR were both acceptable selections due to their heterogeneity under different risks.[18],[19]RR is typically preferred because it is simpler to interpret than OR.[20],[21],[22] This preference might underlie the common use of RR to detect the moderator effect in meta-regression. Although RR might be bias-free when the candidate moderator is a study-level variable, it may not be suitable for participant-level variables. We found that the mathematical properties of OR are superior to those of RR; RR might also introduce serious bias. Although OR is more difficult to interpret,[20],[21],[22] it can be converted into RR via Equation 4-2,[4] where ACR is the assumed control risk. Thus, we consider OR as the better selection in the meta-regression analysis of RCTs:



Although OR reduces bias in the meta-regression analysis, it retains the nature of the meta-regression. The limitations of meta-regression analysis[14],[16] are important to understand. First, the results of RCT meta-regression must be considered to be epidemiological data, which cannot be randomized to underpin causality.[16] Thus, the associations found in a meta-regression should be considered to be hypothesis-generating rather than proof of causality.[14] Second, individual patient data provide better results than summary data;[11],[23] thus, the average data must be considered to be a suboptimal choice. Third, meta-regression must frequently include more than ten studies, and the moderator of interest should be preproposed and backed by an adequate theoretical basis.[4],[14] Multiple comparisons are also problematic in meta-regression analysis.[24] Fourth, the average summary values in each included study are calculated from a small sample size and thus may include significant random errors. The resulting attenuation bias[15],[16] will steer the result toward a null association. In this study, we confirmed only that OR exhibits stronger mathematical properties than RR in meta-regression analyses; the inherent limitations of meta-regression remain.


  Conclusions Top


This study shows the superior linear properties of OR-based meta-regression compared to RR-based meta-regression. If most of the included studies report the average summary values of case groups, the accuracy and power of the OR-based meta-regression increase. However, we suggested researchers can present OR- and RR-based results simultaneously because the explanation of RR is better than OR. The OR and RR-based results were similar in most of the situation; the authors will not do different conclusions based on these two methods. Moreover, this mathematical improvement does not reduce the inherent limitations of meta-regression. Researchers need to understand that meta-regression results are useful for hypothesis-generating but not for causality inference.

Financial support and sponsorship

Nil.

Conflicts of interest

There are no conflicts of interest.


  Supporting Information Top


Text S1: Derivation of the bias in the linear relation between the logarithmic risk ratio and the average summary value.

When the independent variable (x) is the intervention encoded with value 0 for control group and 1 for treatment group, and the moderator (m) is diabetes status encoded with values 0 and 1 for without and with diabetes. The dependent variable is a binary outcome event (y) (with 0 and 1 signifying non-occurrence and occurrence, respectively). Probabilities p1, p2, p3 and p4 are then defined as follows:



which represent the outcome incidence of control group without diabetes, treatment group without diabetes, control group with diabetes and treatment group with diabetes. According to above setting, the risk ratios (RR) of interest for patient without diabetes [RR0 (m = 0)] and patient with diabetes [RR1 (m = 1)] are calculated as follows:



In an RCT, the pooled or combined RR (RRcombine) is a function of the diabetes prevalence in the entire studied population,and can be expressed as



Let the p denote the RR of diabetes in the untreated individuals (p = p3/p1). We can express p1, p2, p3 and p4 in terms of p1 and p alone including RRs as follows:



This simplifies the calculation of RRcombine. Let the ME denote the moderator effect (ME = RR1/RR0), RRcombine can be calculated as follows:



Let log (RRi) denotes the observed effect size of ith individual RCT, qtotal, i the proportion of the population with moderator status, ηi the random effects, and ei the residuals. In general awareness studies, β0 is the expected logarithmic RR of individuals without diabetes [log (RR0)], and β1 is the expected logarithmic moderator effect [log (ME)]. Thus, the traditional meta-regression (here called Method 1) can be written as.



However, following Equation 2.1-1, the relation between log (RRi) and qtotal, i is actually nonlinear:





and π is the RR of diabetes in the untreated individuals. Note that p is an unknown population parameter not provided by most papers. That is why the traditional studies often used qtotal, i to replace κi. Obviously, this may cause bias since only when p = 1 and ME = 1.

Text S2: Relation between the average summary values of the case group (qcase) and the entire population (qtotal).

The independent variable (x) and the moderator (m) are assumed to be binary variables, where 0 and 1 signify nonexposure and exposure, respectively. The dependent variable is a binary outcome event (y), where 0 and 1 signify nonoccurrence and occurrence, respectively. p1, p2, p3, and p4 are then defined as follows:



Defining RR0 as the risk ratio of the independent variable when m = 0; p as the risk ratio of the moderator when x = 0; and ME as the moderator effect, the parameters p1, p2, p3, and p4 can be rewritten in terms of p1 and p alone:



Setting the expected proportion of x = 0 in an RCT equal to 0.5 and assuming that the independent variable does not depend on the moderator because the process is randomized, we can relate the average summary value of the case group (qcase) to the average summary value of the entire population (qtotal) as follows:



Text S3: Weights of the included RCTs.

The independent variable (x) is assumed to be a binary variable, where 0 and 1 signify nonexposure and exposure, respectively. The dependent variable is a binary outcome event (y), where 0 and 1 signify non-occurrence and occurrence, respectively. The parameters p1 and p2 are then defined as follows:



Because the probability of the treatment group is often set equal to 0.5 in randomized controlled trails, the expected sample size in each group is N/2, where N is the total sample size. Thus, the variance of a specific risk ratio (RR) can be expressed as follows:



Finally, the weights of each study are computed by the inverse variance method (i.e. the most common weighting method):

















 
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