Bivariate betabinomial model using Gaussian copula for bivariate metaanalysis of two binary outcomes with low incidence
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Abstract
In metaanalysis of rareevent outcomes, an additional statistical consideration is necessary due to the occurrence of studies with no event. The traditional approaches of adding a correction factor or omitting these studies are known to result in misleading conclusions. Furthermore, studies involved in the metaanalysis often report results for more than one outcome. Bivariate metaanalysis is known as a promising approach for jointly combining two outcomes whilst incorporating correlations between outcomes. However, there has not been sufficient discussion on a bivariate extension in the context of metaanalysis for rareevent outcomes. We consider a joint synthesis of two binary outcomes with low incidence, and propose a novel bivariate metaanalysis method using copula. The method assumes marginal betabinomial distributions for the two outcomes, and links these margins by a bivariate copula which identifies an overall dependence structure between outcomes. A simulation study suggested that the method could provide a robust estimation for the incidence of rare events and have potential benefits of bivariate metaanalysis such as an improvement of precision of pooled estimates. We illustrated the method through an application to a metaanalysis of 48 studies that evaluated a potential risk of rosiglitazone on myocardial infection and cardiovascular death.
Keywords
Bivariate metaanalysis Betabinominal model Gaussian copula Binary outcome Rare event1 Introduction
A strength of metaanalysis is its capability of accumulating evidence from several studies to detect benefits or risks of a therapeutic treatment that are difficult to be assessed in a single study alone. In particular, if interest lies in safety assessments of a treatment regarding serious adverse events with very low incidence (e.g., less than 1%), the metaanalysis enables the estimation and quantification of the risk of such rare adverse events in situations where individual studies do not have large enough sample sizes for reliable evaluations (Loke et al. 2011; Berlin et al. 2013; Shuster and Walker 2016). Bennetts et al. (2017) summarized several publications and guidelines on the safety metaanalytic approaches, and advocated that the metaanalysis of safety data is now becoming a requirement in more drug development programs and beyond.
In the metaanalysis of rareevent outcomes, an additional statistical consideration is necessary due to the occurrence of studies with no event. It is not unusual for practitioners to face a situation that some studies involved in the metaanalysis have zero events in one or more treatment groups. The traditional approaches of adding a correction factor or omitting these studies are known to result in misleading conclusions (Sweeting et al. 2004; Bradburn et al. 2007; Friedrich et al. 2007; Cheng et al. 2015). For the handling of studies with no event, a variety of alternative approaches have been discussed broadly, including arcsine difference (Rucker et al. 2009), Poisson–gamma model (Cai et al. 2010), hypergeometric–normal model (Stijnen et al. 2010), bivariate binomial–normal model (Chu et al. 2012; Stijnen et al. 2010), bivariate betabinomial model (Chu et al. 2012), unweighted and studysize weighted methods (Shuster et al. 2012), betabinomial regression model (Kuss 2015), and Poisson regression model (Bohning et al. 2015; Spittal et al. 2015). Among them, we focus on the betabinomial model, since it has attractive features such as capabilities of including information from studies having no event without any corrections, leading to a closedform expression of likelihood function and providing a simple interpretation of model parameters. An extensive simulation study by Kuss (2015) has also revealed that the betabinomial model could outperform the other approaches in terms of estimation bias and accuracy for the treatment effects (odds ratio, relative risk, and risk difference) in the metaanalysis of rareevent outcomes.
Furthermore, studies involved in the metaanalysis often report results for more than one outcome. For example, Nissen and Wolski (2007) performed a metaanalysis of 48 studies to evaluate a potential risk of rosiglitazone on myocardial infection and cardiovascular death. Hammad et al. (2006) performed a metaanalysis of 24 studies to investigate a relationship between antidepressant drugs and suicidality in pediatric patients, where two different endpoints were specified for measuring suicidal behavior and ideation. Bivariate metaanalysis is known as a promising approach for jointly combining two outcomes whilst incorporating correlations between outcomes, which has potential benefits to improve the precision of pooled estimates and to reduce an impact of outcome reporting bias compared with separate univariate metaanalyses of each outcome (Riley et al. 2007, 2008, 2009; Jackson et al. 2011; Kirkham et al. 2012; Mavridis and Salanti 2013). However, there has not been sufficient discussion on a bivariate extension in the context of metaanalysis for rareevent outcomes. Indeed, most of the methods developed for performing metaanalysis of rareevent outcomes were limited to applications to the univariate outcome.
In this article, we consider a joint synthesis of two binary outcomes with low incidence, and propose a novel bivariate metaanalysis method using copula. The method assumes marginal betabinomial distributions for the two outcomes, and links these margins by a bivariate copula which identifies an overall dependence structure between outcomes. This is fundamentally built on a framework of bivariate copulas for discrete data (Joe 2014). The use of copula allows for a flexible modeling of dependences, and several researchers have discussed the copula modeling in the context of metaanalysis of diagnostic test accuracy studies (Kuss et al. 2014; Hoyer and Kuss 2015; Nikoloulopoulos 2015a, b; Chen et al. 2016). They aimed at modeling the relationship between disease groups rather than the relationship between outcomes. Our contribution is to apply the copula to the joint synthesis of two binary rareevent outcomes.
In Sect. 2, we introduce a motivating example of rosiglitazone data, which will be used for illustrating the method. In Sect. 3, we describe the bivariate betabinomial model using the copula. We begin with brief introductions to the betabinomial distribution and the bivariate copula, and then provide detailed explanations of the proposed method. In Sect. 4, we perform a simulation study to examine a relative performance of the method in comparison to the existing approaches. We show that the method is capable of providing a robust estimation for the incidence of rare events and maintaining potential benefits of bivariate metaanalysis such as an improvement of precision of pooled estimates. In Sect. 5, we illustrate the method through an application to the rosiglitazone data. Finally, in Sect. 6, we conclude this article with some discussion.
2 Motivating example
3 Methods
Consider a joint synthesis of two binary outcomes by performing metaanalysis of N studies with two groups (control and treatment group). Let n_{ij} denote the number of subjects in the ith study (i = 1, …, N) and the group j which takes a value of j = 0 for the control group and j = 1 for the treatment group. Let y_{ijk} denote the number of events for the outcome k (k = 1, 2) in the ith study and group j. Then, y_{ijk} is assumed to follow a binomial distribution with an event probability p_{ijk}, i.e., y_{ijk} ~ Binomial(n_{ij}; p_{ijk}) that assumes different true event probabilities underlying different studies. This assumption reflects a fact that the true event probability might be similar across studies, but is unlikely to be identical because of difference in the characteristics of included subjects and the study design, among other reasons. We are now interested in the situation of rareevent outcomes; that is, the p_{ijk} can take very small values such as less than 0.01.
3.1 Betabinomial distribution
3.2 Bivariate copula
A variety of copula families can be used to identify the dependence between variables (Joe 2014). We here focus on a typical bivariate copula family of Gaussian copula. The Gaussian copula interpolates from the Fréchet lower (perfect negative dependence) to the Fréchet upper (perfect positive dependence) bound, and thus, this allows for both situations where positive and negative dependences between the two outcomes are expected (Nikoloulopoulos 2015a).
3.3 Bivariate betabinomial model using copula and its estimation
For convenience of estimation and interpretation, we consider the parameterisations of μ_{jk} = α_{jk}/(α_{jk} + β_{jk}) and γ_{jk} = 1/(α_{jk} + β_{jk} + 1). The μ_{jk} is mean of the beta randomeffects distribution of Beta(α_{jk}, β_{jk}), and is further parametrized by logit(μ_{0k}) = φ_{k} and logit(μ_{1k}) = φ_{k} + δ_{k}. The δ_{k} is a parameter representing a log odds ratio between groups for the outcome k, which can be interpreted as an overall (summary) treatment effect. The γ_{jk} represents the variability (heterogeneity) across studies in the group j for the outcome k. The correlation between outcomes is identified by the bivariate copula and its parameter θ_{j}, and thus, the model allows the two outcomes to be correlated by groups. Then, all the parameters to be estimated consist of (φ_{1}, δ_{1}, γ_{01}, γ_{11}, φ_{2}, δ_{2}, γ_{02}, γ_{12}, θ_{0}, θ_{1}). Maximization of the log likelihood function (4) is performed by numerical optimisation methods; in particular, we use a quasiNewton method with BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm for obtaining maximumlikelihood estimates \( (\hat{\varphi }_{1} ,\hat{\delta }_{1} ,\hat{\gamma }_{01} ,\hat{\gamma }_{11} ,\hat{\varphi }_{2} ,\hat{\delta }_{2} ,\hat{\gamma }_{02} ,\hat{\gamma }_{12} ,\hat{\theta }_{0} ,\hat{\theta }_{1} ) \) and their variance estimates.
4 Simulation study
We conducted a simulation study to examine a relative performance of the method in comparison to the existing approaches under a variety of metaanalytic situations. Suppose that, in each study involved in a metaanalysis, subjects randomly assigned to a control or a treatment group were assessed for two binary outcomes whose potential incidences were assumed to be very low. Since the method allows for the joint synthesis of the two binary outcomes, we considered situations where some studies reported only one outcome.
4.1 Designs
 Step 1.Choose a scenario of the following parameters:

Number of studies reporting both outcomes

Event probabilities in each group and their odds ratio

Correlation between outcomes.

 Step 2.
Generate a set of metaanalysis data under the scenario given in Step 1.
 Step 3.
Perform metaanalyses using five methods.
 Step 4.
Repeat Step 2–3 5000 times, and summarize results obtained in Step 3.
We below describe each step in detail.
Step 1–2: Generating metaanalysis data sets
Scenario of true parameter values in the simulation study
Parameter  No. of scenarios  Value 

Number of studies reporting both outcomes  S1.1  N = 48, Λ_{12} = 48, Λ_{1} = 0, Λ_{2} = 0 
S1.2  N = 48, Λ_{12} = 32, Λ_{1} = 16, Λ_{2} = 0  
S1.3  N = 48, Λ_{12} = 24, Λ_{1} = 24, Λ_{2} = 0  
S1.4  N = 48, Λ_{12} = 16, Λ_{1} = 32, Λ_{2} = 0  
Number of subjects per group in a single study  –  \( n_{i0} \sim {\text{LN}}({\text{meanlog}} = { \log }(300),{\text{sdlog}} = 0.5) \), n_{i1} = n_{i0} 
Event probabilities in each group and their odds ratio  S2.1  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.003, 0.003, 1.0) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.003, 0.003, 1.0) \) 
S2.2  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.003, 0.003, 1.0) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.003, 0.005, 1.8) \)  
S2.3  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.003, 0.005, 1.8) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.003, 0.003, 1.0) \)  
S2.4  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.003, 0.005, 1.8) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.003, 0.005, 1.8) \)  
S2.5  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.006, 0.006, 1.0) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.006, 0.006, 1.0) \)  
S2.6  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.006, 0.006, 1.0) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.006, 0.010, 1.8) \)  
S2.7  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.006, 0.010, 1.8) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.006, 0.006, 1.0) \)  
S2.8  Outcome 1: \( \left( {\mu_{01} ,\mu_{11} ,OR_{1} } \right) = (0.006, 0.010, 1.8) \) Outcome 2: \( \left( {\mu_{02} ,\mu_{12} ,OR_{2} } \right) = (0.006, 0.010, 1.8) \)  
Correlation between outcomes  S3.1  θ_{1} = θ_{2} = 0.40 
S3.2  θ_{1} = θ_{2} = 0.80  
Variability across studies  –  γ_{01} = γ_{11} = γ_{02} = γ_{12} = 0.001 
We gave 8 scenarios of event probabilities in each group and their odds ratio. The true event probability in the control group, μ_{0k}, was set by 0.003 (from scenario S2.1 to S2.4) or 0.006 (from scenario S2.5 to S2.8), which came from the incidences of the MI and the CVD events in the rosiglitazone data. Specifically, the incidences of the MI and the CVD events in the control group were 0.007 and 0.005 on average over 48 studies, respectively (continuity correction factor of 0.5 was added to studies with zero events in either group). The true odds ratio between groups was set by 1.0 or 1.8, where the former signified a null hypothesis of no treatment effect. The true event probability in the treatment group, μ_{1k}, was calculated using the true event probability in the control group and the true odds ratio between groups. The parameter representing the variability across studies, γ_{jk}, was fixed as γ_{01} = γ_{02} = γ_{11} = γ_{12} = 0.001. The correlation parameters in the Gaussian copula, θ_{0} and θ_{1}, were set by 0.4 (scenario S3.1) or 0.8 (scenario S3.2), reflecting the situations of a moderate or a high correlation between outcomes in each group.

Scenario S1.1: All studies reported both of the outcome 1 and 2; i.e., Λ_{12} = 48, Λ_{1} = 0, and Λ_{2} = 0

Scenario S1.2: 32 studies (2/3 of the total number of studies) reported both of the outcome 1 and 2, and 16 studies (1/3 of the total number of studies) reported only the outcome 1; i.e., Λ_{12} = 32, Λ_{1} = 16, and Λ_{2} = 0.

Scenario S1.3: 24 studies (1/2 of the total number of studies) reported both of the outcome 1 and 2, and 24 studies (1/2 of the total number of studies) reported only the outcome 1; i.e., Λ_{12} = 24, Λ_{1} = 24, and Λ_{2} = 0.

Scenario S1.4: 16 studies (1/3 of the total number of studies) reported both of the outcome 1 and 2, and 32 studies (2/3 of the total number of studies) reported only the outcome 1; i.e., Λ_{12} = 16, Λ_{1} = 32, and Λ_{2} = 0.
These scenarios reflect a common situation that some studies report only one outcome. The number of subjects per group in a single study was drawn from a lognormal distribution with logmean of log(300) and scale of 0.5, and the two groups were assumed to have the same number of subjects; i.e., n_{i0} ~ LN(meanlog = log(300), sdlog = 0.5) with rounding to the nearest integer and n_{i0} = n_{i1}. With this lognormal distribution, Q1/median/Q3 of the number of subjects per group was 214/300/420. This setting also came from the rosiglitazone data, where the average numbers of subjects were 270.0 in the control group and 351.2 in the rosiglitazone group.
Given the number of subjects per group, the event probabilities in each group and their odds ratio, the correlation between outcomes, and the variability across studies, we first generated bivariate binomial data of the two outcomes individually for the 48 studies. And then, we randomly selected studies reporting both outcomes in accordance with the corresponding scenario.
Step 3–4: Performing metaanalyses and summarizing their results
 (i)
Yusuf–Peto method (Yusuf et al. 1985): This is currently known as a standard method for performing metaanalysis of the rareevent outcomes. Indeed, the Cochrane collaboration (Higgins and Green 2011) recommended using the Yusuf–Peto method at event rates below 0.01. They advocated that this was the least biased and the most powerful method, provided that there was no substantial imbalance between treatment and control group sizes within studies and treatment effects were not exceptionally large. This finding has been supported by a simulation study conducted in Sweeting et al. (2004). We applied the Yusef–Peto method to the two outcomes separately.
 (ii)
Two separate univariate metaanalyses using normal randomeffects models: The normal randomeffects model is the most common practice in the metaanalysis. The model assumes an asymptotic normality of treatment effect estimates from each study conditioned on different true treatment effects underlying different studies, where the true treatment effects are also assumed to follow a normal distribution. The log odds ratio is here considered as the treatment effect of interest. We applied the normal randomeffects model to the two outcomes separately.
 (iii)
Bivariate metaanalysis using a bivariate normal randomeffects model: The bivariate normal randomeffects model can be used for jointly combining the two outcomes whilst incorporating withinstudy correlations and a betweenstudy correlation. We here supposed a situation that the withinstudy correlations were unknown, and considered using an alternative model proposed by Riley et al. (2008). This alternative model first assumes marginal normal distributions for the two outcomes separately, and then, a single overall correlation is introduced for linking the margins under bivariate normality assumption. We applied the bivariate normal randomeffects model (alternative model) to the two outcomes.
 (iv)
Two separate univariate metaanalyses using betabinomial models: The betabinomial distribution (1) has a potential benefit of naturally including information from studies having no event without any corrections. We applied the betabinomial model to the two outcomes separately.
 (v)
Bivariate metaanalysis using a bivariate betabinomial model with bivariate Gaussian copula: We applied the bivariate betabinomial model described in Sect. 3 to the two outcomes.
In the method (i), studies having zero events in both groups were removed from the analysis. In the methods (ii) and (iii), a continuity correction factor of 0.5 was added to studies with zero events in either group.
In Step 4, we calculated the following quantities for comparing the methods described above: (a) bias and rootmeansquare error (RMSE) of pooled estimates, and coverage probability of 95% confidence interval of the log odds ratio for the two outcomes, and (b) type I error rate and empirical power on odds ratio for the two outcomes.
4.1.1 Misspecification of copula
To assess robustness of the bivariate betabinomial model with the Gaussian copula against misspecification of the correlation structure, the simulation study was repeated under the situations that the Gaussian copula was not a correct model for the dependence. We considered three typical bivariate Archimedean copulas as true correlation structures between outcomes: Clayton (1978), Cook and Johnson (1981), Oakes (1982), Gumbel (1960), Barnett (1980), and Frank (1979) copula. Their copula parameters were provided, so that Kendall’s tau becomes 0.60. Note that the Gaussian copula parameter is 0.81 for the Kendall’s tau of 0.60 (see also Nelsen (2006) and Joe (2014) for further details of each copula).
4.2 Results
4.2.1 Bias and RMSE of pooled estimates, and coverage probability of the log odds ratio
Figures 2, 3 show the biases and the RMSEs of pooled estimates, and the coverage probabilities of the log odds ratio for the outcomes 1 and 2, respectively. The UN and the BN provided negatively biased estimates in almost all the scenarios for both outcomes. The negative biases from the UN and the BN were substantial, especially in the scenarios that the true event probabilities in the treatment group were higher than those in the control group; i.e., scenarios S2.3, S2.4, S2.7, and S2.8 for the outcome 1, and scenarios S2.2, S2.4, S2.6, and S2.8 for the outcome 2. These biases indicate a wellknown concern associated with the continuity correction. When the true event probabilities in the treatment group were higher than those in the control group, zero events were more likely to occur in the control group. Thus, adding the continuity correction factor of 0.5 to studies with zero events obviously pulled the pooled estimates in the negative direction. Although the RMSEs from the UN and the BN were smaller than those from the other methods, the coverage probabilities from the UN and the BN were seriously below or above nominal level of 0.95 in almost all the scenarios for both outcomes. These findings were not depending on the scenarios of the correlation between outcomes and the number of studies reporting both outcomes.
The pooled estimates from the YP were unbiased for both outcomes, except for the scenario that the true event probabilities in the treatment group were higher than those in the control group; i.e., scenario S2.3, S2.4, S2.7, and S2.8 for the outcome 1, and scenario S2.2, S2.4, S2.6, and S2.8 for the outcome 2, where slight negative biases were found. The coverage probabilities from the YP were slightly below the nominal level of 0.95 for the outcome 2, especially in the scenario that the number of studies reporting the outcome 2 was the smallest (scenario S1.4).
The UBB and the BBB provided quite similar results for the outcome 1. We could find no difference between the UBB and the BBB in Fig. 1. The two methods worked well in all the scenarios for the outcome 1. Indeed, their pooled estimates were unbiased, and their coverage probabilities were well controlled around the nominal level of 0.95. For the outcome 2, the pooled estimates from the UBB and the BBB were again unbiased, while we could find a reduction of RMSE from the BBB. In the scenarios of the high correlation (scenario S3.2), the BBB provided smaller RMSEs for the outcome 2 than the UBB as decreasing the number of studies reporting the outcome 2 (scenario S1.2, S1.3, and S1.4), indicating the potential benefit of incorporating the correlation between outcomes. The gain in the RMSE from the BBB was remarkable in the scenario that the number of studies reporting the outcome 2 was the smallest (scenario S1.4). The same trend of reducing the RMSEs was found in results from the UN and the BN for the outcome 2. The coverage probabilities from the UBB and the BBB for the outcome 2 were close to the nominal level of 0.95 in almost all the scenarios. In the scenario that the number of studies reporting the outcome 2 was the smallest (scenario S1.4), the coverage probabilities from the BBB were slightly below the nominal level of 0.95 (around 1%). This is explained by seeing the length of the 95% confidence interval. Figure 4 shows the medians of the length of the 95% confidence interval for the scenarios that the numbers of studies reporting the outcome 2 were 24 and 16 (scenario S1.3 and S1.4, respectively). The medians of the length of 95% confidence interval from the BBB were shorter than those from the UBB in the scenario that the number of studies reporting the outcome 2 was the smallest and the correlation was high. In such situations, the confidence interval of the BBB might be slightly too short to keep the nominal level.
4.2.2 Type I error rate and empirical power on the odds ratio
Figures 5, 6 show the type I error rates and the empirical powers on the odds ratio for the outcome 1 and 2, respectively. The type I error rates from the UN and the BN were substantially below the nominal level of 0.05 for both outcomes in all the scenarios. This again indicates the negative influence of adding the continuity correction factor to studies with zero events. The empirical powers from the two methods were also much lower than the other methods, especially in the scenario of the lower event probabilities in the control group; i.e., scenario S2.3 and S2.4 for the outcome 1, and scenario S2.2 and S2.4 for the outcome 2.
The type I error rates from the YP were slightly above the nominal level of 0.05 for both outcomes (around 1–2%). This inflation would come from the model assumption of the YP. The YP genuinely assumes a common treatment effect across studies (the socalled fixed effect), which might lead to slight underestimation of variance of the pooled estimates for both outcomes.
The UBB and the BBB provided wellcontrolled type I error rates for both outcomes. For the outcome 2, in the scenario that the number of studies reporting the outcome 2 was the smallest (scenario S1.4), the type I error rate from the BBB was slightly above the nominal level of 0.05 (around 1%). The empirical powers from the UBB and the BBB were quite similar for the outcome 1. In the scenario of the high correlation (scenario S3.2) for the outcome 2, the empirical powers from the BBB were higher than those from the UBB as decreasing the number of studies reporting the outcome 2 (scenario S1.2, S1.3, and S1.4), which were remarkable in the scenario that the number of studies reporting the outcome 2 was the smallest (scenario S1.4). This was suggested that the reduction of RMSE in the BBB could yield the gain in the empirical power. The same trend of increasing the empirical power was also found in results from the UN and the BN for the outcome 2.
4.2.3 Robustness against misspecification of copula
Substantial negative biases from the UN and the BN were again observed in almost all the scenarios for both outcomes, as was the case in the simulation study using the Gaussian copula as the true correlation structure. Results from the YP were also consistent with those in the above simulation study.
The UBB and the BBB provided quite similar results for the outcome 1. We could find no difference between the UBB and the BBB for every scenario of the true correlation structure in Fig. 7. Their pooled estimates were unbiased and their coverage probabilities were well controlled around the nominal level of 0.95, indicating that the BBB would be robust against the misspecification of copula in the estimation of the log odds ratio for the outcome 1. For the outcome 2, the pooled estimates from the UBB and the BBB were again unbiased in almost all the scenarios. In particular, despite the presence of misspecification of copula, the BBB could be expected to provide unbiased estimation of the log odds ratio for the outcome 2. The BBB provided smaller RMSEs for the outcome 2 than the UBB for every scenario of the true correlation structure; especially in the scenarios of the Gumbel copula, gains in the reduction of RMSEs were the largest. The coverage probabilities from the UBB and the BBB for the outcome 2 were close to the nominal level of 0.95 in almost all the scenarios. However, in the scenario that the number of studies reporting the outcome 2 was the smallest (scenario S1.4), the coverage probabilities from the BBB were again slightly below the nominal level of 0.95. This trend was more apparent in the scenarios of the Clayton copula (around 3%) than when the Gaussian copula was correctly specified.
In summary, even when the Gaussian copula was not a correct model for the dependence, the BBB could be robust enough to provide unbiased estimation of the log odds ratio for both outcomes. However, it should be also noted that, in some cases (e.g., when assuming the Clayton copula as the true correlation structure), the BBB would provide the underestimation of standard error of the log odds ratio for the outcome 2. Then, the confidence interval from the BBB could become too short to keep the nominal level.
5 Application to rosiglitazone data
Results of application to the rosiglitazone data
YP  UN  BN  UBB  BBB  

Myocardial infection (MI) event  
Incidence in control group (SE) [95% CI]  NA  NA  NA  0.00380 (0.00088) [0.00207, 0.00553]  0.00384 (0.00096) [0.00195, 0.00573] 
Incidence in rosiglitazone group (SE) [95% CI]  NA  NA  NA  0.00491 (0.00090) [0.00315, 0.00668]  0.00492 (0.00090) [0.00316, 0.00668] 
Odds ratio between groups [95% CI]  1.428 [1.031, 1.979]  1.232 [0.909, 1.670]  1.353 [1.037, 1.764]  1.295 [0.723, 2.318]  1.283 [0.697, 2.361] 
Variability across studies*  NA  0.0077  0.0025  0.0039, 0.0034  0.0052, 0.0035 
Cardiovascular death (CVD) event  
Incidence in control group (SE) [95% CI]  NA  NA  NA  0.00133 (0.00043) [0.00049, 0.00216]  0.00129 (0.00044) [0.00044, 0.00215] 
Incidence in rosiglitazone group (SE) [95% CI]  NA  NA  NA  0.00228 (0.00048) [0.00134, 0.00322]  0.00235 (0.00049) [0.00139, 0.00331] 
Odds ratio between groups [95% CI]  1.640 [0.980, 2.744]  1.097 [0.725, 1.657]  1.225 [0.853, 1.758]  1.724 [0.809, 3.676]  1.816 [0.834, 3.956] 
Variability across studies*  NA  0.0095  0.0105  0.0009, 0.0010  0.0018, 0.0010 
Correlation between MI and CVD event†  NA  NA  0.806  NA  0.774, 0.547 
The pooled estimates of odds ratio for the MI events were similar among the five methods. The UBB and the BBB provided wider range of 95% confidence intervals. There was no considerable heterogeneity across studies for the MI event, according to the results from the four methods except for the YP. The incidences of the MI events from the UBB and the BBB were similar for both groups. The pooled estimates from the UN and the BN were much smaller than those from the other methods. This would be because adding the continuity correction factor of 0.5 to studies with zero events pulled the pooled estimates in the negative direction. Indeed, more studies were subjected to zero CVD events in comparison to the MI events. Considering the negative biases from the UN and the BN observed in the simulation study, the YP, the UBB, and the BBB were expected to provide more robust assessments of the risk of rosiglitazone on the CVD events. There were no considerable heterogeneity across studies for the CVD events. The incidences of the CVD events from the UBB and the BBB were similar for both groups. The correlation between the MI and the CVD events was high enough to be considered. In the BBB, the correlation in the control group was higher than that in the rosiglitazone group.
5.1 Illustration with studies reporting MI and/or CVD
To imitate situations that some studies reported only one outcome (either MI or CVD events), we generated 6 scenarios in terms of the number of studies reporting the MI and/or the CVD events. In particular, we assumed the scenarios of: (number of studies reporting MI events, number of studies reporting CVD events) = {(48, 36), (48, 24), (48, 16), (36, 48), (24, 48), (16, 48)}. We first randomly selected studies reporting the MI and/or the CVD events in accordance with the corresponding scenario, and then performed metaanalyses using the five methods (i)–(v) described above. The implementation steps were repeated 1000 times. In each scenario, we took medians of 1000 pooled estimates of the log odds ratio and their 1000 standard errors obtained from each method. The incidences of the MI and the CVD events estimated from the UBB and the BBB were also summarized in the same way.
6 Discussion
We considered the joint synthesis of two binary outcomes with low incidence, and proposed a novel bivariate metaanalysis method using copula. The metaanalysis of rareevent outcomes requires an additional statistical consideration due to the occurrence of studies with no event. While a variety of approaches have been developed for dealing with this matter, our proposed method would be another option for performing metaanalysis of two rareevent outcomes efficiently. The simulation study suggested that the method could be highly robust against the situations including studies with zero events. This advantage came from the use of betabinomial margins, which have been already recognized as a promising alternative for the rareevent outcome. Another advantage of the method indicated from the simulation study was the gain in precision of the pooled estimates. In the situations that some studies reported only one outcome, the method gave smaller RMSEs and higher empirical powers in comparison to the existing univariate approaches such as the Yusuf–Peto method and the betabinomial model. These gains were remarkable in the situation with the higher correlation, and thus, the method could be expected to provide efficient estimation of treatment effects for the two outcomes whilst incorporating the correlation between outcomes. These findings were illustrated through an application to the rosiglitazone data. In addition, even though any realistic situations were not necessarily covered, the simulation study suggested that the method could be robust enough to provide unbiased estimation of the treatment effects in the presence of misspecification of copula. The robustness will need to be enhanced by further examinations.
Our simulation study was designed mainly for examining the benefits of using the bivariate concept in comparison to the univariate betabinomial model which was known as one of the best choices in the metaanalysis of rareevent outcomes. On the one hand, some other forms of bivariate betabinomial models (Kuss et al. 2014; Nikoloulopoulos 2015a, b; Chen et al. 2016) would also be available. A comparison between various forms of bivariate betabinomial models needs to be further addressed in a future research. Note that the existing bivariate approaches were developed for applications to the metaanalysis of diagnostic test accuracy studies; hence, the correlation should be treated and interpreted in different manners for the metaanalysis of rareevent outcomes.
The interpretation of the correlation parameter θ_{j} is not straightforward. Note that, as is the case in bivariate metaanalyses of continuous outcomes, two different sources of correlations, i.e., withinstudy correlations and a betweenstudy correlation, could contribute to the dependence between the two binary outcomes. The withinstudy correlation, measuring the dependence between y_{ij1} and y_{ij2}, occurs, because the two outcomes are assessed using the same set of patients in each study, while the betweenstudy correlation allows the studies’ true underlying effects to be correlated (i.e., the dependence between p_{ij1} and p_{ij2}) due to the characteristics of the two events (Riley et al. 2007; Riley 2009; Jackson et al. 2011). The proposed method first assumes marginal betabinomial distributions for the two outcomes separately, and then introduces a single overall correlation parameter θ_{j} for linking the two margins. This procedure leads to a kind of amalgam of the withinstudy correlations and the betweenstudy correlation, which is the same concept as the alternative model proposed by Riley et al. (2008). Therefore, the estimated Gaussian copula parameter value should be interpreted with caution, and some possible situations such as discrepant withinstudy correlations across studies (i.e., the case that there exists the betweenstudy heterogeneity of correlation) cannot be directly handled by the proposed method. Further investigations will be necessary for confirming the robustness of the proposed method against these situations.
There are some limitations of the proposed method and further research is required. We, in this article, used the Gaussian copula for identifying the correlation structure between the two outcomes; however, a variety of bivariate copula families are now available, and thus, the choice of bivariate copulas and its optimisation might lead to the better option depending on the situation faced by users. Indeed, our simulation study indicated that, in some cases with the presence of misspecification of copula, the confidence interval of the treatment effect from the bivariate betabinomial model with the Gaussian copula could be too short to keep the nominal level. Furthermore, the more investigations regarding heterogeneity across studies would be necessary in the context of metaanalysis for the rareevent outcomes. A quantification of heterogeneity under the betabinomial model and a model extension for incorporating studylevel covariates could be essential in practical use of the method. Outcome reporting bias is another important issue to be discussed in the bivariate metaanalysis. It should be cared that the outcome reporting bias for safety outcomes might differ from efficacy outcomes, which have a possible tendency of not reporting clinically relevant adverse events with a few observations (Bennetts et al. 2017). Further research of extending the method to allow for a mixed treatment comparison would be welcome.
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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