

ORIGINAL ARTICLE 

Year : 2020  Volume
: 40
 Issue : 3  Page : 119126 

A superior odds ratio compared to the risk ratio when estimating moderator effects in metaregression analyses of randomized controlled trials
ChihChien Chiu^{1}, ChienFu Chen^{2}, PoJen Hsiao^{3}, DungJang Tsai^{4}, HsuehLu Chang^{5}, WenHui Fang^{6}, WeiTeing Chen^{7}, JenqShyong Chan^{8}, MinTser Liao^{9}, YiJung Ho^{10}, Wen Su^{11}, YingKai Chen^{12}, HuiHan Hu^{5}, ZhengZong Lai^{13}, Chin Lin^{4}
^{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, TriService 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, TriService 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, TriService 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 Submission  17Jul2019 
Date of Decision  22Sep2019 
Date of Acceptance  25Oct2019 
Date of Web Publication  03Dec2019 
Correspondence Address: Dr. Chin Lin National Defense Medical Center, Graduate Institute of Life Sciences, No. 161, MinChun E. Road, Section 6, Neihu, Taipei 114 Taiwan
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/jmedsci.jmedsci_133_19
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 metaregression on logRR via simulation. Methods: In the metaanalysis of randomized controlled trials, the moderator effect is generally evaluated by the linear metaregression of the logarithmic risk ratio (RR) versus the average summary value of the entire study population. Conclusions: We recommend using linear metaregression 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, participantlevel variable, studylevel variable, metaanalysis, metaregression, 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 metaregression analyses of randomized controlled trials. J Med Sci 2020;40:11926 
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 metaregression analyses of randomized controlled trials. J Med Sci [serial online] 2020 [cited 2020 Jul 12];40:11926. Available from: http://www.jmedscindmc.com/text.asp?2020/40/3/119/272328 
Introduction   
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]} Metaanalysis 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 metaanalysis 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 participantlevel variables, such as gender and comorbidity.^{[4]} Thus, most metaanalyses are limited to the moderator effects of the studylevel variables, such as geographical location. However, most of potential moderator effects were hidden in participantlevel variables, such as diabetes in folic acid treatment effect.^{[5]} Researches needed to explore these potential issues for reducing heterogeneity.
Because participantlevel 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 metaregression.^{[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 metaanalysis 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) metaregression of the logarithmic RR versus the average summary value of the entire population (the most common method); (2) metaregression of the logarithmic OR versus the average summary value of the entire population; and (3) metaregression of the logarithmic OR versus the average summary value of a case group.
Methods   
Derivations
The moderator effect of participantlevel variables in the metaanalyses 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 p_{1}, p_{2}, p_{3}, and p_{4} 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 (RR_{0}[m = 0]) and patient with diabetes (RR_{1}[m = 1]) are calculated as follows, respectively:
In an RCT, the pooled or combined RR (RR_{combine}) 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 = p_{3}/p_{1}). We can express p_{1}, p_{2}, p_{3}, and p_{4} in terms of p_{1} and π alone including RRs as follows:
This simplifies the calculation of RR_{combine}. Let the ME denote the moderator effect (ME = RR_{1}/RR_{0}), RR_{combine} can be calculated as follows [for details, Text S1]:
and the logarithm of RR_{combine} can be expressed as follows [for details, Text S1]:
(2.11)
Let log (RR_{i}) denotes the observed effect size of i th individual RCT, q_{total, i} the proportion of the population with moderator status, η_{i} the random effects, and ε_{i} the residuals. In general, awareness studies, β_{0} is the expected logarithmic RR of individuals without diabetes (log[RR_{0}]), and β_{1} is the expected logarithmic moderator effect (log[ME]). Thus, the traditional metaregression (here called Method 1) can be written as.
(2.12)
However, following Equation 2.11, the relation between log (RR_{i}) and q_{total, 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 q_{total, i} to replace κ_{i}. Obviously, this may cause bias since κ_{i}= q_{total,}_{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 β_{0} and β_{1}, 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 metaanalysis of RCTs, and the proposed method remains robust when the rare disease assumption is violated.^{[11]} Thus, we expect that the metaregression of the logarithmic OR versus the average summary value of the case group in study i (q_{case, i}, i.e., Method 3) will best detect the moderator effect in the metaanalysis 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 RR_{0} denote the RR of the treatment in individuals without the moderator; then, the variables q_{total, i} and q_{case, i} are related through Equation 2.14 [for this derivation, Text S2]:
The difference between q_{total, i} and q_{case, i} is affected by RR_{0},ME, and π. The Pearson correlation between q_{total} and q_{case} exceeds 0.95 when RR_{0},ME, and π are between 0.5 and 2.0 [Table S1]; thus, we considered that q_{total} can effectively represent q_{case}. Thus, the metaregression 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 metaanalysis of RCTs. The simulation code is written in the R programming language; this code is provided in [Appendix S1]. As the metaanalysis 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 (RR_{0}); (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, RR_{0}, π, and ME used in the simulations are listed in [Table 1].
The third step was to summarize the individual data into metaanalysis data. Each study provided four pieces of information for the following metaregression analysis: (1) the logarithmic RR of treatment without stratification and its variance (i.e., log[RR_{i}] and var [log[RR_{i}], respectively); (2) the logarithmic OR of treatment without stratification and its variance (i.e., log [OR_{i}] and var [log [OR_{i}], respectively); (3) the proportion of individuals with the moderator throughout the study (q_{total}); and (4) the proportion of individuals with the moderator in the case group (q_{case}). The metaregression 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 metaregression.
1. Metaregression of the logarithmic RR versus the average summary value of the entire population
Regression formula:
2. Metaregression of the logarithmic OR versus the average summary value of the entire population
Regression formula:
3. Metaregression of the logarithmic OR versus the average summary value of the case group
Regression formula:
where log (RR_{i}) is the observed logarithmic RR in study i; q_{total, i} is the proportion of the population with moderator status in study i; q_{case, i} is the proportion of the population with moderator status of case group in study i; η_{i} is the random effects; ε_{i} is the residuals; i is an identifier of an individual RCT ranging from 1 to 20; β_{0} is the expected logarithmic RR of individuals without moderator status; β_{1} 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 (β_{1}) and the intercept (β_{0}). 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 moderatoreffect assessment as the secondary outcome. Data under each condition were acquired from 10,000 simulations.
Results   
Figure 2 shows selected simulations relating to Equation 2.13 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 falsepositive 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 2.13, 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 RR_{0}is set equal to 1, and the Y axis is transformed into the logarithmic scale. ME: Moderator effect, RR: Risk ratio
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 Figure 2: Results from selected simulations. Based on Equation 2.13, 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 RR_{0}are set equal to 0.2 and 1.0, respectively. Condition A: P = 0.2; RR_{0}= 1.0; π =2.0; ME = 2.00.5; Condition B:P = 0.2; RR_{0}= 1.0; π = 1.0; ME = 2.00.5; Condition C:P = 0.2; RR_{0}= 1.0; π = 0.5; ME = 2.00.5. 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 metaregression methods. [Figure S4] shows the falsepositive rates and the powers of the moderator effect at the 0.05 significance level in the metaregression 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 RR_{0} 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   
Metaanalysis 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. Metaregression is a common method to explain heterogeneity, but the previous study suggested that metaregression was only suitable on studylevel 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 participantlevel variables is important. Although certain studies investigated the relation between the average summary participantlevel value and the logarithmic RR by metaregression,^{[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 metaregression 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. Metaanalysis attempts to determine the P value and to estimate the accuracy of a treatment effect.^{[4]} Thus, a lowbias 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 q_{total} and q_{case} exceeded 0.95 for appropriate values of RR_{0},ME and π (i.e., between 0.5 and 2.0) [Table S1]. The Pearson correlations between q_{total} and q_{case} decrease as ME increases, but a large ME is easily detected in a single study.^{[17]} The metaanalysis 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 metaregression.
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 (p_{1}), the RR of treatment in individuals without the moderator (RR_{0}), 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 (q_{total, i}). The dependences of the study weights (Weight_{i}) in study i on these parameters are given by Equations 41.1 and 41.2 (for detailed derivations), [Text S1 and S3]. Thus, the Weight_{i} 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 RR_{0}. Specifically, we have:{Figure 2}
The bias introduced by the weights is not easily simplified because it involves the distribution of q_{total} 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 metaanalysis 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 metaregression. Although RR might be biasfree when the candidate moderator is a studylevel variable, it may not be suitable for participantlevel 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 42,^{[4]} where ACR is the assumed control risk. Thus, we consider OR as the better selection in the metaregression analysis of RCTs:
Although OR reduces bias in the metaregression analysis, it retains the nature of the metaregression. The limitations of metaregression analysis^{[14],[16]} are important to understand. First, the results of RCT metaregression must be considered to be epidemiological data, which cannot be randomized to underpin causality.^{[16]} Thus, the associations found in a metaregression should be considered to be hypothesisgenerating 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, metaregression 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 metaregression 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 metaregression analyses; the inherent limitations of metaregression remain.
Conclusions   
This study shows the superior linear properties of ORbased metaregression compared to RRbased metaregression. If most of the included studies report the average summary values of case groups, the accuracy and power of the ORbased metaregression increase. However, we suggested researchers can present OR and RRbased results simultaneously because the explanation of RR is better than OR. The OR and RRbased 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 metaregression. Researchers need to understand that metaregression results are useful for hypothesisgenerating but not for causality inference.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
Supporting Information   
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 nonoccurrence and occurrence, respectively). Probabilities p_{1}, p_{2}, p_{3} and p_{4} 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 [RR_{0} (m = 0)] and patient with diabetes [RR_{1} (m = 1)] are calculated as follows:
In an RCT, the pooled or combined RR (RR_{combine}) 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 = p_{3}/p_{1}). We can express p_{1}, p_{2}, p_{3} and p_{4} in terms of p_{1} and p alone including RRs as follows:
This simplifies the calculation of RR_{combine}. Let the ME denote the moderator effect (ME = RR_{1}/RR_{0}), RR_{combine} can be calculated as follows:
Let log (RR_{i}) denotes the observed effect size of ith individual RCT, q_{total, i} the proportion of the population with moderator status, η_{i} the random effects, and e_{i} the residuals. In general awareness studies, β_{0} is the expected logarithmic RR of individuals without diabetes [log (RR_{0})], and β_{1} is the expected logarithmic moderator effect [log (ME)]. Thus, the traditional metaregression (here called Method 1) can be written as.
However, following Equation 2.11, the relation between log (RR_{i}) and q_{total, 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 q_{total, i} to replace κ_{i}. 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 (q_{case}) and the entire population (q_{total}).
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. p_{1}, p_{2}, p_{3,} and p_{4} are then defined as follows:
Defining RR_{0} 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 p_{1}, p_{2}, p_{3,} and p_{4} can be rewritten in terms of p_{1} 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 (q_{case}) to the average summary value of the entire population (q_{total}) 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 nonoccurrence and occurrence, respectively. The parameters p_{1} and p_{2} 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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[Figure 1], [Figure 2]
[Table 1]
