Methods for multiviarate randomeffects metaanalysis
Description
Methods for multiviarate randomeffects metaanalysis
Usage
1 
Arguments
data 
dataset 
rhow 
withinstudy correlation 
type 
either "continuous" or "binary", indicating the type of outcomes. 
k 
integer indicating the number of outcomes 
method 
either "nn.reml", "nn.cl", "nn.mom", "nn.rs", "bb.cl", "bn.cl", "tb.cl" or "tn.cl", indicating the estimation method. 
Details
Inference on the multivariate randomeffects metaanalysis for both continuous and binary outcomes
The function can be used in metaanalyses with continous outcomes and binary outcomes (e.g., mean differences, diagnostic test results in diagnostic accuracy studies, the exposure status of both cases and controls in casecontrol studies and so on).
Different estimators with and without the knowledge of withinstudy correlations are implemented in this function. The estimation methods include

Restricted maximum likelihood for MMA with continuous outcomes(nn.reml)

Composite likelihood method for MMA with continuous outcomes (nn.cl)

Moment of method for MMA with continuous outocmes (nn.mom)

Improved method for Riley model for MMA with continuous outcomes (nn.rs)

Marginal bivariate normal model for MMA with binary outcomes (bn.cl)

Marginal betabinomial model for MMA with binary outcomes(bb.cl)

Hybrid model for disease prevalence along with sensitivity and specificity for diagnostic test accuracy (tb.cl)

Trivariate model for multivariate metaanalysis of diagnostic test accuracy(tn.cl)
Value
An object of class "mmeta"
. The object is a list containing the following components:
beta 
estimated coefficients of the model. 
beta.cov 
covariance matrix of the coefficients. 
Multivariate randomeffects meta analysis
We consider a metaanalysis with m studies where two outcomes in each study are of interest. For the ith study, denote Y_{ij} and s_{ij} the summary measure for the jth outcome of interest and associated standard error respectively, both assumed known, i=1, …, m, and j=1,2. Each summary measure Y_{ij} is an estimate of the true effect size θ_{ij}. To account for heterogeneity in effect size across studies, we assume θ_{ij} to be independently drawn from a common distribution with overall effect size β_j and between study variance τ_j^2, j=1,2. Under normal distribution assumption for Y_{ij} and θ_{ij}, the general bivariate randomeffects metaanalysis can be written as
≤ft(\begin{array}{c}Y_{i1} \\ Y_{i2} \end{array} \right)\sim N≤ft( ≤ft(\begin{array}{c}θ_{i1} \\ θ_{i2} \end{array} \right), {{Δ_i}}\right),\quad {{Δ_i}}=≤ft(\begin{array}{cc} s_{i1}^2 & s_{i1}s_{i2}ρ_{\textrm{W}i}\\ s_{i1}s_{i2}ρ_{\textrm{W}i} & s_{i2}^2 \end{array} \right),
≤ft(\begin{array}{c}θ_{i1} \\ θ_{i2} \end{array} \right)\sim N≤ft(≤ft(\begin{array}{c}β_{1} \\ β_{2} \end{array} \right), {{Ω}}\right),\quad {{Ω}}=≤ft(\begin{array}{cc} τ_{1}^2 & τ_{1}τ_{2}ρ_{\textrm{B}}\\ τ_{1}τ_{2}ρ_{\textrm{B}} & τ_{2}^2 \end{array} \right),
where Δ_i and Ω are the respective withinstudy and betweenstudy covariance matrices, and ρ_{\textrm{W}i} and ρ_{\textrm{B}} are the respective withinstudy and betweenstudy correlations.
Restricted maximum likelihood for MMA
When the withinstudy correlations are known, inference on the overall effect sizes β_1 and β_2 or their comparative measures (e.g., β_1β_2) can be based on the marginal distribution of ≤ft(Y_{i1}, Y_{i2}\right)
≤ft(\begin{array}{c}Y_{i1} \\ Y_{i2} \end{array} \right)\sim N≤ft(≤ft(\begin{array}{c}β_{1} \\ β_{2} \end{array} \right), \bf{V_i}\right), \bf{V_i}=Δ_i+Ω=≤ft(\begin{array}{cc} s_{i1}^2+τ_{1}^2 & s_{i1}s_{i2}ρ_{wi}+τ_{1}τ_{2}ρ_{\textrm{B}}\\ s_{i1}s_{i2}ρ_{wi}+τ_{1}τ_{2}ρ_{\textrm{B}} & s_{i2}^2+τ_{2}^2 \end{array} \right).
For simplicity of notation, denote \bf{Y_i}=(Y_{i1}, Y_{i2})^{T}, {{β}}=(β_1, β_2)^T, η_1=(β_1,τ_1^2)^{T} and η_2=(β_2,τ_2^2)^{T}. The restricted likelihood of (η_1, η_2, ρ_{\textrm{B}}) can be written as
\log L({{η}}_1, {{η}}_2, ρ_{\textrm{B}}) ={1\over 2} ≤ft[\log ≤ft( \Big{}∑_{i=1}^m \bf{V_i}^{1}\Big{}\right)+ ∑_{i=1}^m≤ft\{ \log \bf{V_i} + (\bf{Y_i}{{β}})^T \bf{V_i}^{1} (\bf{Y_i}{{β}}) \right\}\right].
The parameters (η_1, η_2, ρ_{\textrm{B}}) can be estimated by the restricted maximum likelihood (REML) approach as described in Van Houwelingen et al. (2002). The REML method for
MMA is specified via method
argument (method="nn.reml"
).
The standard inference procedures, such as the maximum likelihood or maximum restricted likelihood inference, require the withinstudy correlations, which are usually unavailable.
In case withinstudy correlations are unknown, then one can leave the ρ_w argument unspecified, and specify a method that does not require the withinstudy correlations via method
argument.
Composite likelihood method for MMA with continuous outcomes
Chen et al. (2014) proposed a pseudolikelihood method for MMA with unknown withinstudy correlation.
The pseudolikelihood method does not require withinstudy correlations, and is not prone to singular covariance matrix problem.
In addition, it can properly estimate the covariance between pooled estimates for different outcomes,
which enables valid inference on functions of pooled estimates, and can be applied to metaanalysis where some studies have outcomes MCAR.
This composite likelihood method for MMA is specified via method
argument (method="nn.cl"
).
Moment of method for MMA with continuous outocmes
Chen et al. (2015) proposed a simple noniterative method that can be used for the analysis of multivariate metaanalysis datasets
that has no convergence problems and does not require the use of withinstudy correlations.
The strategy is to use standard univariate methods for the marginal effects but also provides valid joint inference for multiple parameters.
This method method can directly handle missing outcomes under missing completely at random assumption.
This moment of method for MMA is specified via method
argument (method="nn.mom"
)
Improved method for Riley model for MMA with continuous outcomes
Riley et al.(2008) proposed a working model and an overall synthesis correlation parameter to account for the marginal correlation between outcomes,
where the only data needed are those required for a separate univariate randomeffects metaanalysis. As withinstudy correlations are not required,
the Riley method is applicable to a wide variety of evidence synthesis situations.
However, the standard variance estimator of the Riley method is not entirely correct under many important settings.
As a consequence, the coverage of a function of pooled estimates may not reach the nominal level even when the number of studies in the multivariate metaanalysis is large.
Hong et al. (2015) improved the Riley method by proposing a robust variance estimator,
which is asymptotically correct even when the model is misspecified (i.e., when the likelihood function is incorrect).
The improved method for Riley model MMA is specified via method
argument (method="nn.rs"
)
Marginal bivariate normal model for MMA with binary outcomes
Diagnostic systematic review is a vital step in the evaluation of diagnostic technologies. In many applications, it invovles pooling paris of sensitivity and specificity
of a dichotomized diagnostic test from multiple studies.
Chen et al. (2014) proposed a composite likelihood method for bivariate metaanalysis in diagnostic systematic reviews.
The idea of marginal bivariate normal model for MMA with binary outcomes is to construct a composite likelihood (CL) funciton by using an independent working assumption between sensitivity and specificity.
There are three immediate advantages of using this CL method. First, the nonconvergence or non positive definite covariance matrix problem is resolved since there is no correlation parameter involved in the CL.
Secondly, because the twodimensional integration involved in the standard likelihood is substituted by onedimensional integrals, the approximation errors are substantially reduced. Thirdly, the inference based on the CL only relies on the marginal normality of logit sensitivity and specificity.
Hence the proposed method can be more robust than the standard likelihood inference to misspecifications of the joint distribution assumption.
This method is specified via method
argument (method="bn.cl"
)
Marginal betabinomial model for MMA with binary outcomes
When conducting a metaanalysis of studies with bivariate binary outcomes, challenges arise when the withinstudy correlation and betweenstudy heterogeneity should be taken into account.
Chen et al. (2015) proposed a marginal betabinomial model for the metaanalysis of studies with binary outcomes.
This model is based on the composite likelihood approach, and has several attractive features compared to the existing models such as bivariate generalized linear mixed model (Chu and Cole, 2006) and Sarmanov betabinomial model (Chen et al., 2012).
The advantages of the proposed marginal model include modeling the probabilities in the original scale, not requiring any transformation of probabilities or any link function, having closedform expression of likelihood function, and no constraints on the correlation parameter.
More importantly, since the marginal betabinomial model is only based on the marginal distributions,
it does not suffer from potential misspecification of the joint distribution of bivariate studyspecific probabilities.
Such misspecification is difficult to detect and can lead to biased inference using currents methods.
This method is specified via method
argument (method="bb.cl"
)
Hybrid model for disease prevalence along with sensitivity and specificity for diagnostic test accuracy
Metaanalysis of diagnostic test accuracy often involves mixture of casecontrol and cohort studies.
The existing bivariate random effects models, which jointly model bivariate accuracy indices (e.g., sensitivity and specificity),
do not differentiate cohort studies from casecontrol studies, and thus do not utilize the prevalence information contained in the cohort studies.
The trivariate generalized linear mixed models are only applicable to cohort studies, and more importantly, they assume the common correlation structure across studies, and the trivariate normality on disease prevalence, test sensitivity and specificity after transformation by some prespecified link functions.
In practice, very few studies provide justifications of these assumptions, and sometimes these assumptions are violated.
Chen et al. (2015) evaluated the performance of the commonly used random effects model under violations of these assumptions and propose a simple and robust method to fully utilize
the information contained in casecontrol and cohort studies.
The proposed method avoids making the aforementioned assumptions and can provide valid joint inferences for any functions of overall summary measures of diagnostic accuracy.
This method is specified via method
argument (method="tb.cl"
)
Trivariate model for multivariate metaanalysis of diagnostic test accuracy
The standard methods for evaluating diagnostic accuracy only focus on sensitivity and specificity and ignore the information on disease prevalence contained in cohort studies.
Consequently, such methods cannot provide estimates of measures related to disease prevalence, such as population averaged or overall positive and negative predictive values,
which reflect the clinical utility of a diagnostic test.
Chen et al. (2014) proposed a hybrid approach that jointly models the disease prevalence along with the diagnostic test sensitivity and specificity in cohort studies,
and the sensitivity and specificity in casecontrol studies. In order to overcome the potential computational difficulties in the standard full likelihood inference of the proposed hybrid model,
an alternative inference procedure was proposed based on the composite likelihood. Such composite likelihood based inference does not suffer computational problems and maintains high relative efficiency.
In addition, it is more robust to model misspecifications compared to the standard full likelihood inference.
This method is specified via method
argument (method="tn.cl"
)
Author(s)
Yong Chen, Yulun Liu
References
Chen, Y., Hong, C. and Riley, R. D. (2015). An alternative pseudolikelihood method for multivariate randomeffects metaanalysis. Statistics in medicine, 34(3), 361380.
Chen, Y., Hong, C., Ning, Y. and Su, X. (2015). Metaanalysis of studies with bivariate binary outcomes: a marginal betabinomial model approach, Statistics in Medicine (in press).
Hong, C., Riley, R. D. and Chen, Y. (2015). An improved method for multivariate randomeffects metaanalysis (in preparation).
Chen, Y., Liu, Y., Ning, J., Nie, L., Zhu, H. and Chu, H. (2014). A composite likelihood method for bivariate metaanalysis in diagnostic systematic reviews. Statistical methods in medical research (in press).
Chen, Y., Cai, Y., Hong, C. and Jackson, D. (2015). Inference for correlated effect sizes using multiple univariate metaanalyses, Statistics in Medicine (provisional acceptance).
Chen, Y., Liu, Y., Ning, J., Cormier J. and Chu H. (2014). A hybrid model for combining casecontrol and cohort studies in systematic reviews of diagnostic tests, Journal of the Royal Statistical Society: Series C (Applied Statistics) 64.3 (2015): 469489.
Chen, Y., Liu, Y., Chu, H., Lee, M. and Schmid C. (2015). A simple and robust method for multivariate metaanalysis of diagnostic test accuracy, Statistics in Medicine (under revision).
Examples
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