1 Introduction
In evidencebased medicine, metaanalysis has been an essential tool for quantitatively summarizing multiple studies and producing integrated evidence. In general, the treatment effects from different sources of evidence are heterogeneous due to various factors such as study designs, participating subjects, and treatment administration. Therefore, such heterogeneity should be adequately addressed when combining evidence sources, otherwise statistical errors may be seriously underestimated and possibly result in misleading conclusions (Higgins and Green, 2011). The randomeffects model has been widely used to address these heterogeneities in most medical metaanalyses. The applications cover various types of systematic reviews, for example, conventional univariate metaanalysis (DerSimonian and Laird, 1986; Whitehead and Whitehead, 1991), metaanalysis of diagnostic test accuracy (Reitsuma et al., 2005), network metaanalysis for comparing the effectiveness of multiple treatments (Salanti, 2012), and individual participant metaanalysis (Riley et al., 2010).
However, in random effect metaanalysis, most existing standard inference methods for average treatment effect parameters underestimate statistical errors under realistic situations of medical metaanalysis, for example, the coverage probabilities of standard inference methods are usually smaller than the nominal confidence levels, even when the model is completely specified (Brockwell and Gordon, 2001; 2007). This may lead to highly overconfident conclusions. The main reason for this notable problem is that randomeffects models typically include heterogeneity variancecovariance parameters other than the average treatment effect, and most inference methods depend on large sample approximations for the number of trials to be synthesized, which is usually small or moderate in medical metaanalysis. In other words, the variability of the estimation of nuisance parameters is possibly ignored and the total statistical error can be underestimated when constructing confidence intervals. Recently, several confidence intervals that aim to improve the undercoverage property have been developed, for example, by Brockwell and Gordon (2007), Henmi and Copas (2010), Jackson and Bowden (2009), Jackson and Riley (2014), Noma (2011), Noma et al. (2017), Guolo (2012), and Sidik and Jonkman (2002). Although coverage properties are relatively improved under realistic situations, the validity of these methods is substantially guaranteed using large sample approximations. Moreover, most of these methods were developed in the context of traditional direct pairwise comparisons; therefore, the methods have limited applicability in recent, more advanced types of metaanalysis that use the complicated multivariate models noted above.
In this paper, we develop a unified method for constructing accurate confidence intervals for random effect models so that their coverage probabilities would equal to the nominal level regardless of the number of studies. To effectively circumvent the effects of nuisance parameters, we consider the likelihood ratio test (LRT) for the average treatment effect, and we define its
value based on the conditional distribution given the maximum likelihood estimator of the nuisance parameters rather than the unconditional distribution of the test statistic. For computing the
value, we adopt the Monte Carlo conditioning technique proposed by Lindqvist and Taraldsen (2005), and the confidence interval can be derived by inverting the LRT. As a result, the derived confidence intervals are shown to have accurate coverage probabilities and the proposed method can be generally applied to various types of metaanalysis involving complicated multivariate random effect models.In Section 2, we first provide an algorithm for computing the value of the LRT and derive an accurate confidence interval under general statistical models. In Section 3, the proposed method is applied to the simplest univariate randomeffects metaanalysis for direct pairwise comparisons. We conduct simulations for evaluating the empirical performances of the proposed confidence interval, and illustrate our method using the wellknown metaanalysis of intravenous magnesium for suspected acute myocardial infarction (Teo et al., 1991). In Sections 4 and 5, we discuss the applications to metaanalysis for diagnostic test accuracy and network metaanalysis, respectively, with illustrations using real datasets. Discussion is provided in Section 6.
2 Algorithm for confidence interval
We suppose are independent and each has the density or probability mass function with parameter of interest and nuisance parameter . For example, in the univariate metaanalysis described in Section 3, and correspond to the number of studies and estimated treatment effect in the th study, respectively, and we use the model with known to estimate the average treatment effect , so that and in this case. In general, , and could be multivariate, but we assume in this section that all of them are onedimensional in order to make our presentation simpler. Multivariate cases are considered in Sections 4 and 5
. The likelihood ratio test (LRT) statistic for testing null hypothesis
iswhere , and is the loglikelihood function. Under some regularity conditions, the asymptotic distribution of under is as . However, when the sample size is not large as is often the case in metaanalysis, the approximation is not accurate enough. The main reason is that there is an unknown nuisance parameter and its estimation error is not ignorable when is not large. To overcome this problem, we calculate the value of the statistic based on the conditional distribution , where is the maximum likelihood estimator of under . For computing the value, we adopt the Monte Carlo conditioning developed in Lindqvist and Taraldsen (2005).
To describe the general methodology, we further assume that can be expressed as for some function
and random variable
whose distribution is completely known. For example, when , it holds that with . Now, the maximum likelihood estimator under satisfies the following equation:where is the partial derivative of the likelihood function with respect to the nuisance parameter . Under , can be expressed as ; thereby, the above equation can be rewritten as
We define as the solution of the above equation with respect to . Using the result in Lindqvist and Taraldsen (2005), the value can be expressed as
(1) 
where the expectation is taken with respect to the distribution of , and
(2) 
with some function of . As noted in Lindqvist and Taraldsen (2005), the choice of controls the efficiency of the Monte Carlo approximation in (1). However, the detailed discussion of this issue would extend of the scope of this paper; thus, we consider in this paper only . The algorithm for computing the value for testing H is given as follows.
Algorithm 1.
(Monte Carlo method for the value of LRT)

Compute the LRT and the constrained maximum likelihood estimator of under .

For with large , generate a random sample, , and compute , and from (2).

The Monte Carlo approximation of the value is given by
(3)
Using the value of the LRT of , the confidence interval of with nominal level can be constructed as the set of such that the value of the LRT of is larger than . Although the confidence limits cannot be expressed in closed form, they can be computed by simple numerical methods, for example the bisectional method that repeatedly bisects an interval and selects a subinterval in which a root exists until the process converges numerically, see Section 2 in Burden and Faires (2010). When or
are multivariate (vectorvalued) parameters, this method can be easily extended to the case. When
is multivariate, which is typical in many applications, the absolute value symbol in the weight (2) should be recognized as determinant since is a matrix. On the other hand, when is multivariate, we need to construct a confidence region (CR) rather than a confidence interval. In this case, the bisectional method cannot be directly applied, and methods for CRs would depend on each setting. In Section 4, we present a diagnostic metaanalysis in which a CR is traditionally used, and provide a feasible algorithm to compute a CR.3 Univariate randomeffects metaanalysis
3.1 The randomeffects model
The univariate randomeffects model has been widely used in metaanalysis due to its parametric simplicity. However, the accuracy of the inference is poor when the number of studies is small. We consider solving this problem using the LRT and confidence interval introduced in Section 2. We assume that there are clinical trials and that are the estimated treatment effects. We consider the randomeffect model:
(4) 
where is the true effect size of the th study, and is the average treatment effect. Here and are independent error terms within and across studies, respectively, assumed to be distributed as and . The withinstudies variances s are usually assumed to be known and fixed to their valid estimates calculated from each study. On the other hand, the across variance is an unknown parameter representing the heterogeneity between studies. Under these settings, Hardy and Thompson (1996) considered the likelihoodbased approach for estimating the average treatment effect .
3.2 Exact confidence intervals of model parameters
We first consider a confidence interval of by using Algorithm 1 in the previous section. To begin with, we consider the null hypothesis with nuisance parameter . Since under the model (4), the LRT statistic can be defined as
where
(5) 
The minimization can be achieved in standard ways such as the iterative method in Hardy and Thompson (1996).
From (5), the constrained maximum likelihood estimator of under H satisfies the following equation:
Let be random variables that which independently follow , then can be expressed as under H. Substituting the expression for in the above equation, we have
where . The above equation can be easily solved with respect to and the solution is given by
Regarding the weight (2), using the implicit function theorem, it holds that
and thereby we can compute the value of the LRT for H from Algorithm 1, and the confidence interval of by inverting the LRT.
A confidence interval of can be derived as well. By a similar derivation to that for , the values of the LRT of H can be computed from Algorithm 1 with
so that the confidence interval of can be similarly constructed.
3.3 Simulation study of coverage accuracy
We evaluated the finite sample performances of the proposed confidence interval of via Monte Carlo (MC) algorithm together with existing methods widely used in practice. We considered the restricted maximum likelihood (REML) method, the DirSimonian and Laird (DL) method (DirSimonian and Laird, 1986), and the likelihood ratio (LR) method (Hardy and Thompson, 1996). When implementing the MC method, we used Monte Carlo samples to compute the value. We fixed the true average treatment effect at , and the heterogeneity variance at and . Following Rockwell and Gordon (2001), withinstudy variances
s were generated from a scaled chisquared distribution with
degree of freedom, multiplied by , and truncated to lie within the interval . We changed the number of studies over and , and set the nominal level to . Based on simulation runs, we calculated the coverage probabilities (CP) and average lengths (AL) of the four confidence intervals. The results, shown in Table 1, indicate that the confidence intervals from the three methods other than MC are too liberal to achieve the appropriate nominal level even when . On the other hand, the MC method produces reasonable confidence intervals with appropriate coverage probabilities even when is .CP  AL  

3  5  7  9  3  5  7  9  
MC  0.956  0.957  0.948  0.951  2.079  1.156  0.869  0.714  
REML  0.809  0.848  0.852  0.870  1.004  0.729  0.620  0.548  
DL  0.815  0.849  0.855  0.869  1.007  0.729  0.621  0.547  
LR  0.868  0.876  0.885  0.899  1.126  0.807  0.677  0.592  
MC  0.945  0.953  0.943  0.947  2.459  1.400  1.063  0.878  
REML  0.795  0.839  0.857  0.880  1.195  0.905  0.787  0.699  
DL  0.800  0.845  0.855  0.873  1.197  0.905  0.784  0.696  
LR  0.833  0.879  0.887  0.901  1.340  0.999  0.849  0.748 
3.4 Example: treatment of suspected acute myocardial infarction
Here we applied the proposed method to a metaanalysis of the treatment of suspected acute myocardial infarction with intravenous magnesium (Teo et al., 1991), which is wellknown as it yielded conflicting results between metaanalyses and large clinical trials (LeLorier et al., 1997). For the dataset, we constructed a confidence interval of the average treatment effect of intravenous magnesium using the proposed MC method (with Monte Carlo samples in evaluating the value) as well as the REML, DL and LR methods considered in the previous section. Moreover, we also applied Peto’s fixed effect (PFE) method (Yusuf, 1985). The results are given in Table 2. The confidence intervals from the DL, LR, REML, and PFE methods were narrower than that of the MC method since the first four methods cannot adequately account for the additional variability in estimating the between study variance . Therefore, the confidence intervals from the first four methods did not cover , which does not change the interpretation of the results. On the other hand, the proposed MC method produced a longer confidence interval than the other four methods while also covering , that is, the corresponding test for was not significant with a significant level.
Method  Estimate  CI 

MC  0.449  (0.149, 1.103) 
DL  0.448  (0.233, 0.861) 
LR  0.449  (0.191, 0.903) 
REML  0.437  (0.213, 0.895) 
PFE  0.471  (0.280, 0.791) 
4 Bivariate Metaanalysis of Diagnostic Test Accuracy
4.1 Bivariate randomeffects model
There has been increasing interest in systematic reviews and metaanalyses of data from diagnostic accuracy studies. For this purpose, a bivariate randomeffect model (Reitsma et al., 2005 and Harbord et al., 2007) is widely used. Following Reitsma et al. (2005), we define and
as the logittransformed true sensitivity and specificity, respectively, in the
th study. The bivariate model assumes thatfollows a bivariate normal distribution:
(6) 
where and are the average logittransformed sensitivity and specificity, and and
are standard deviations of
and , respectively. Here the parameter allows correlation between and . The unknown parameters are and . Let and be the observed logittransformed sensitivity and specificity, and we assume that(7) 
For summarizing the results of the metaanalysis, the CR of would be more valuable than separate confidence intervals since sensitivity and specificity might be highly correlated. Reitsma et al. (2005) suggested the joint CR for as the interior points of the ellipse defined as
(8) 
where and are the maximum likelihood estimates of and , and
are estimated standard errors of
and , respectively, and is the square root of the upper point of the distribution with degrees of freedom. The joint CR (8) is approximately valid; specifically, the coverage error converges to as the number of studies goes to infinity. However, when is not sufficiently large, the coverage error is not negligible, and the region (8) undercovers the true .4.2 Exact CR of sensitivity and specificity
We consider a CR of under the models (6) and (7) based on the Monte Carlo method given in Section 2. Let be a vector of nuisance parameters, and write rather than to clarify the dependence of . From (6) and (7), the LRT statistic of the null hypothesis H is given by
where
with .
Under H, the constrained maximum likelihood estimator satisfies the following equations:
where
Under H, the observed data can be expressed as with and being the Cholesky decomposition of , that is, . Then the above equation can be rewritten as
and we define as the solution of the above equation with respect to . The solution can be numerically obtained by minimizing the sum of squared values of three equations with respect to . Moreover, concerning the weight (2), we may use the numerical derivative given . Hence, we can compute the value of the LRT of H from Algorithm 1.
From the LRT of H, the CR of can be defined as , where denotes the value of the test statistic . Since is twodimensional in this case, the computing boundary is not straightforward. The most feasible procedure is to approximate the boundary with a sufficiently large numbers of points. To this end, we first divide the interval by points . For each , we compute satisfying
which can be carried out via numerical methods (e.g. the bisectional method).
4.3 Example: screening test accuracy for alcohol problems
Here we provide a reanalysis of the dataset given in Kriston et al. (2008), including studies regarding a short screening test for alcohol problems. Following Reitsuma et al. (2005), we used logittransformed values of sensitivity and specificity, denoted by and , respectively, and associated standard errors and . For the bivariate summary data, we fitted the bivariate models (6) and (7), and computed CRs of based on the approximated CR of the form (8) given in Reitsma et al. (2005). Moreover, we computed the proposed CR with Monte Carlo samples for calculating values of the LRT, and evaluation points that were smoothed by a 7point moving average for the CR boundary. Following Reitsuma et al. (2005), the obtained two CRs of were transformed to the scale , where and are the sensitivity and false positive rate, respectively. The obtained two CRs are presented in Figure 1 with a plot of the observed data, summary points , and the summary receiver operating curve. The approximate CR is smaller than the proposed CR, which may indicate that the approximation method underestimates the variability of estimating nuisance variance parameters.
5 Network Metaanalysis
5.1 Multivariate randomeffects model
Suppose there are treatments in contract to a reference treatment, and let be an estimator of relative treatment effect for the th treatment in the th study. We consider the following multivariate randomeffects model:
(9) 
where , is a vector of true treatment effects in the th study, is a vector of average treatment effects, and is the withinstudy variancecovariance matrix. Here we focus on the model (9) known as the contrastbased model (Salanti et al., 2008; Dias and Ades, 2016), which is commonly used in practice.
In network metaanalysis, each study contains only treatments ( usually ranges from 2 to 5); thereby, several components in cannot be defined. When the corresponding treatments are not involved in the th study, the corresponding components in and are shrunk to the subvector and submatrix, respectively, in the model (9). Moreover, when the references treatment is not involved in the th study, we can adopt the data argumentation approach of White et al. (2012), in which a quasismall data set is added to the reference arm, e.g. 0.001 events for 0.01 patients. To clarify the setting in which and are shrunk to the subvector and submatrix, respectively, we introduce an index , , representing the treatment estimates that are available in the th study, and define the dimensional vector of ’s, excluding the th element that is equal to . Moreover, we define , and and are the shrunken dimensional vector and matrix of and , respectively. The model (9) can be rewritten as
(10) 
Regarding the structure of between study variance , since there are rarely enough studies to identify the unstructured model of , the compound symmetry structure is used in most cases (White, 2015), where is a matrix with all diagonal elements equal to 1 and all offdiagonal elements equal to .
We define , , , and with independently for each . The hierarchical model (10) can be expressed as the following randomeffects model:
(11) 
where with . The unknown parameters in (11) are and . The loglikelihood of the model (11) is given by
where and denotes the Kronecker product. Note that is an dimensional vector and is the total number of comparisons.
5.2 Confidence interval of the average treatment effect
In network metaanalysis, we are interested in not only the average treatment effects in contrast to the reference treatment, but also the treatment differences . To handle these issues in a unified manner, we focus on the linear combination with known vector . Define a fullrank matrix such that the first element of is . The parameter is equivalent to when we use instead of in the model (11), so that it is sufficient to consider a confidence interval of .
Define and to be and matrices such that , and . The model (11) can be rewritten as
(12) 
We first consider testing of H, noting that and are nuisance parameters. The LRT statistics can be defined as
where
Under H, the constrained maximum likelihood estimator and satisfy the following equations:
(13) 
where and . Under H, it holds that for and is the Cholesky decomposition of such that . The equation (13) can be rewritten as
(14) 
with . The solution of the first equation with respect to is given by
By replacing with in the second equation in (14), we obtain the following equation for :
where , and we define be the solution of the above equation with respect to . Hence, the solutions of (14) with respect to and are given by and , respectively. Concerning the weight function , from the implicit function theorem, it follows that
where . From (14), we have
On the other hand, because derivation of analytical expressions of the partial derivatives with respect to or requires tedious algebraic calculation, we can use numerical derivatives instead. Therefore, we can carry out Algorithm 1 in Section 2 to compute the value of LRT, and the confidence interval can be obtained as well by inverting the LRT.
5.3 Example: Schizophrenia data
Ades et al. (2010) carried out a network metaanalysis of antipsychotic medication for prevention of relapse of schizophrenia; this analysis includes 15 trials comparing eight treatments with placebo. In each trial, the outcomes available were the four outcome states at the end of followup: relapse, discontinuation of treatment due to intolerable side effects and other reasons, not reaching any of the three endpoints, and still in remission. We here considered the last outcome and adopted the odds ratio as the treatment effect measure.
We applied the multivariate randomeffects model (11) with a compound symmetry structure of . The reference treatment was set to “Placebo”. The estimates of betweenstudies standard deviation were 0.28 for the ML and 0.52 for the REML estimation methods, respectively, which shows that there is substantial heterogeneity between studies.
In Table 3 we present the results of three confidence intervals based on the MC method, the LRbased method with value calculated by the asymptotic distribution, and the REML method. The number of Monte Carlo samples in applying the MC method was consistently set to . In this analysis, the confidence intervals of the MC method were much wider than those of the LR method, and indicated larger uncertainty in the inference of average treatment effects. On the other hand, the confidence intervals of the REML method were wider than those of the MC method in the last three treatments although the REML method produced narrower intervals than the MC method in the first five treatments; this may be due to the difference between the ML and REML estimates of between study standard deviation (0.28 for ML and 0.52 for REML).
Placebo v.s.  ML  MC  LR  REML  REML 

Olanzapine  4.91  (2.11, 9.85)  (2.67, 8.55)  4.52  (2.15, 9.50) 
Amisulpride  3.38  (1.19, 9.54)  (1.56, 7.30)  3.36  (1.22, 9.31) 
Zotepine  2.66  (0.67, 13.10)  (0.91, 7.74)  2.66  (0.68, 10.32) 
Aripiprazole  2.07  (0.66, 6.52)  (0.93, 4.58)  2.07  (0.67, 6.38) 
Ziprasidone  5.03  (1.84, 14.14)  (2.32, 10.73)  4.90  (1.81, 13.26) 
Paliperidone  2.08  (0.65, 6.66)  (0.90, 4.81)  2.08  (0.65, 6.67) 
Haloperidol  2.65  (1.13, 5.74)  (1.26, 5.12)  2.36  (0.94, 5.95) 
Risperidone  5.46  (2.52, 12.88)  (2.40, 11.82)  5.05  (1.74, 14.64) 
6 Discussions
We developed a unified method for constructing confidence intervals of the average treatment effects in randomeffects metaanalysis. The proposed confidence intervals are based on the LRT, and we proposed a Monte Carlo method to compute its value. In terms of specific applications, we discussed three types of metaanalysis, univariate metaanalysis, diagnostic metaanalysis, and network metaanalysis, and demonstrated the usefulness of the proposed method. The developed inference method would be adapted to a variety of applications, e.g., the multivariate individual participant data metaanalysis (Burke et al., 2016). A limitation of the proposed methods might be rigorous justification of the exactness of the proposed inference methods. Although the maximum likelihood estimator of the variance parameters are sufficient statistics in the case that all the withinstudy variances are the same, this property might not hold rigorously, under general conditions. Although there were no theoretical proofs concerning the sufficiency of this estimator under general conditions, but we could clearly demonstrate the proposed methods could provide almost exact confidence intervals in the simulation studies. We guess the sufficiency property would be hold under more general settings, and the theoretical justifications would be one of relevant further issues.
In addition, the numerical results from our simulations and the illustrative examples suggest that statistical methods in the randomeffects models should be selected carefully in practice. Historically, there have been many discrepant results between metaanalyses and subsequent large randomized clinical trials (LeLorier et al, 1997), and in these cases the metaanalyses have typically tended to provide false results as in the magnesium example in Section 3.4. Many systematic biases, for example, publication bias (Begg and Berlin, 1988; Easterbrook et al., 1991), might be important sources of these discrepancies, but we should also be aware of the risk of providing overconfident and misleading interpretations caused by the statistical methods based on large sample approximations. Considering these risks, accurate inference methods would be preferred in practice. Although there have not been any accurate inference methods that can be broadly applied in randomeffects metaanalyses, our approach in this article may provide an explicit solution to this relevant problem.
Finally, methodological research on extensions of randomeffects metaanalyses to more complicated statistical models are still in progress (e.g., multivariate network metaanalyses, Riley et al., 2017), and the small sample problems generally exist in most of these applications. Our methods are applicable to these complicated models as well as more advanced approaches that might arise in future research. The developed methods should be effective tools as a unified methodological framework to obtain accurate solutions in medical evidence synthesis.
Acknowledgement This research was partially supported by JST CREST (grant number: JPMJCR1412) and JSPS KAKENHI (grant number: 17K19808, 15K15954, 18K12757).
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