This task view covers packages which include facilities for meta-analysis of summary statistics from primary studies. The task view does not consider the meta-analysis of individual participant data (IPD) which can be handled by any of the standard linear modelling functions but it does include some packages which offer special facilities for IPD.

The standard meta-analysis model is a form of weighted least squares and so any of the wide range of R packages providing weighted least squares would in principle be able to fit the model. The advantage of using a specialised package is that (a) it takes care of the small tweaks necessary (b) it provides a range of ancillary functions for displaying and investigating the model. Where the model is referred to below it is this model which is meant.

Where summary statistics are not available a meta-analysis of significance levels is possible. This is not completely unconnected with the problem of adjustment for multiple comparisons but the packages below which offer this, chiefly in the context of genetic data, also offer additional functionality.

Univariate meta-analysis

Preparing for meta-analysis

• The primary studies often use a range of statistics to present their results. Convenience functions to convert these onto a common metric are presented by: compute.es which converts from various statistics to d, g, r, z and the log odds ratio, MAc which converts to correlation coefficients, MAd which converts to mean differences, and metafor which converts to effect sizes an extensive set of measures for comparative studies (such as binary data, person years, mean differences and ratios and so on), for studies of association (a wide range of correlation types), for non-comparative studies (proportions, incidence rates, and mean change). It also provides for a measure used in psychometrics (Cronbach's alpha). esc provides a range of effect size calculations with partial overlap with metafor but with some extras, noticeably for converting test statistics, also includes a convenience function for collating its output for input to another package like metafor or producing a CSV file. effsize contains functions to compute effect sizes mean difference (Cohen's d and Hedges g), dominance matrices (Cliff's Delta) and stochastic superiority (Vargha-Delaney A). effectsize provides a large number of different effect sizes and converts between them. psychmeta provides extensive facilties for converting effect sizes and for correcting for a variety of restrictions and measurement errors. metansue provides some methods for converting to effect sizes es.dif from raw data computes Cohen's d, Hedges' d, biased/unbiased c (an effect size between a mean and a constant) and e (an effect size between means without assuming the variance equality). MOTE provides a variety of conversions based on Cohen's d. estmeansd converts between quantiles and means and standard deviations. metaBLUE estimates means and standard deviations from various order statistics. SingleCaseES provides basic effect sizes for single-case designs, both parametric and non-overlap.
• meta provides functions to read and work with files output by RevMan 4 and 5.
• metagear provides many tools for the systematic review process including screening articles, downloading the articles, generating a PRISMA diagram, and some tools for effect sizes. revtools provides tools for downloading from bibliographic databases and uses machine learning methods to process them.
• metavcov computes the variance-covariance matrix for multivariate meta-analysis when correlations between outcomes can be provided but not between treatment effects, and clubSandwich imputes variance-covariance matrix for multivariate meta-analysis
• metafuse uses a fused lasso to merge covariate estimates across a number of independent datasets.
• metapower provides power analysis for meta-analysis and meta-regression

Fitting the model

• Four packages provide the inverse variance weighted, Mantel-Haenszel, and Peto methods: epiR, meta, metafor, and rmeta.
• For binary data metafor provides the binomial-normal model.
• For sparse binary data exactmeta provides an exact method which does not involve continuity corrections.
• Packages which work with specific effect sizes may be more congenial to workers in some areas of science and include MAc and metacor which provide meta-analysis of correlation coefficients and MAd which provides meta-analysis of mean differences. MAc and MAd provide a range of graphics. psychometric provides an extensive range of functions for the meta-analysis of psychometric studies. mixmeta provides an integrated interface to standard meta-analysis and extensions like multivariate and dose-response.
• psychmeta implements the Hunter-Schmidt method including corrections for reliability and range-restriction issues
• Bayesian approaches are contained in various packages. bspmma which provides two different models: a non-parametric and a semi-parametric. Graphical display of the results is provided. metamisc provides a method with priors suggested by Higgins. mmeta provides meta-analysis using beta-binomial prior distributions. A Bayesian approach is also provided by bmeta which provides forest plots via forestplot and diagnostic graphical output. bayesmeta includes shrinkage estimates, posterior predictive p-values and forest plots via either metafor or forestplot. Diagnostic graphical output is available. MetaStan includes binomial-normal hierarchical models and can use weakly informative priors for the heterogeneity and treatment effect parameters. baggr provides facilities using Stan for hierarchical Bayesian models, graphical facilities are provided. BayesCombo provides facilities using a Bayesian approach and has graphical facilities. RBesT uses Bayesian synthesis to generate priors from various sources.
• Some packages concentrate on providing a specialised version of the core meta-analysis function without providing the range of ancillary functions. These are: metaLik which uses a more sophisticated approach to the likelihood, metamisc which as well as the method of moments provides two likelihood-based methods, and metatest which provides another improved method of obtaining confidence intervals. metaBMA has a Bayesian approach using model averaging, a variety of priors are provided and it is possible for the user to define new ones.
• metaplus fits random effects models relaxing the usual assumption that the random effects have a normal distribution by providing t or a mixture of normals.
• ratesci fits random effects models to binary data using a variety of methods for confidence intervals.
• RandMeta estimates exact confidence intervals in random effects models using an efficient algorithm.
• rma.exact estimates exact confidence intervals in random effects normal-normal models and also provides plots of them.
• clubSandwich gives cluster-robust variance estimates.
• pimeta provides a range of methods for prediction interval estimation from random effects models and ahs graphical facilities.
• metamedian implements several methods to meta-analyze one-group or two-group studies that report the median of the outcome. These methods estimate the pooled median in the one-group context and the pooled raw difference of medians across groups in the two-group context
• MetaUtility proposes a metric for estimating the proportion of effects above a cut-off of scientific importance
• metasens provides imputation methods for missing binary data.
• metagam provides a framework for meta-analysis of generalised additive models including the case where individual paticipant data cannot be shared across locations.
• metawho implements a method for combining within study interactions
• metarep provides replicability analyses after a conventional analysis

Graphical methods

An extensive range of graphical procedures is available.

• Forest plots are provided in forestmodel (using ggplot2), forestplot, meta, metafor, metansue, psychmeta, and rmeta. Although the most basic plot can be produced by any of them they each provide their own choice of enhancements. metaviz provides a range of enhancements.
• Funnel plots are provided in meta, metafor, metansue, psychometric rmeta and weightr. In addition to the standard funnel plots an enhanced funnel plot to assess the impact of extra evidence is available in extfunnel, a funnel plot for limit meta-analysis in metasens, and metaviz provides an extensive range of enhanced funnel plots and also facilities for their use in the context of visual inference.
• Radial (Galbraith) plots are provided in meta and metafor.
• L'Abbe plots are provided in meta and metafor.
• Baujat plots are provided in meta and metafor.
• metaplotr provides a crosshair plot
• MetaAnalyser provides an interactive visualisation of the results of a meta-analysis.
• metaviz provides rainforestplots, an enhanced version of forest plots. It accepts input from metafor.

Investigating heterogeneity

• Confidence intervals for the heterogeneity parameter are provided in metafor and psychmeta.
• altmeta presents a variety of alternative methods for measuring and testing heterogeneity with a focus on robustness to outlying studies.
• metaforest investigates heterogeneity using random forests. Note that it has nothing to do with forest plots.
• mc.heterogeneity implements a Monte Carlo based test for heterogeneity.

Model criticism

• An extensive series of plots of diagnostic statistics is provided in metafor.
• metaplus provides outlier diagnostics.
• psychmeta provides leave-one-out methods.
• ConfoundedMeta conducts a sensitivity analysis to estimate the proportion of studies with true effect sizes above a threshold.
• EValue provides sensitivity analysis of the effect of unmeasured confounders

Investigating small study bias

The issue of whether small studies give different results from large studies has been addressed by visual examination of the funnel plots mentioned above. In addition:

• meta and metafor provide both the non-parametric method suggested by Begg and Mazumdar and a range of regression tests modelled after the approach of Egger.
• xmeta provides a method in the context of multivariate meta-analysis.
• An exploratory technique for detecting an excess of statistically significant studies is provided by PubBias.
• metamisc provides funnel plots and tests for asymmetry.
• puniform provides methods using only the statistically significant studies, methods for the special case of replication studies and sample size determinations.
• PublicationBias performs sensitivity analysis of the number of unpublished studies needed to have a specified influence.

Unobserved studies

A recurrent issue in meta-analysis has been the problem of unobserved studies.

• Rosenthal's fail safe n is provided by MAc and MAd. metafor provides it as well as two more recent methods by Orwin and Rosenberg.
• Duval's trim and fill method is provided by meta and metafor.
• metasens provides Copas's selection model and also the method of limit meta-analysis (a regression based approach for dealing with small study effects) due to Rücker et al.
• selectMeta provides various selection models: the parametric model of Iyengar and Greenhouse, the non-parametric model of Dear and Begg, and proposes a new non-parametric method imposing a monotonicity constraint.
• SAMURAI performs a sensitivity analysis assuming the number of unobserved studies is known, perhaps from a trial registry, but not their outcome.
• The metansue package allows the inclusion by multiple imputation of studies known only to have a non-significant result.
• weightr provides facilities for using the weight function model of Vevea and Hedges.
• publipha estimates models accounting for publication bias or p-hacking using a Bayesian framework

Other study designs

• SCMA provides single case meta-analysis. It is part of a suite of packages dedicated to single-case designs.
• joint.Cox provides facilities for the meta-analysis of studies of joint time-to-event and disease progression.
• metamisc provides for meta-analysis of prognostic studies using the c statistic or the O/E ratio. Some plots are provided.
• dfmeta provides meta-analysis of Phase I dose-finding clinical trials
• metaRMST implements meta-analysis of trials with difference in restricted mean survival times

Meta-analysis of significance values

• Fisher's method and Lancaster's are available in aggregation, metap, metapro and poolr.
• Stouffer's method, Tippett's and Wilkinson's are available in metap and poolr.
• Edgington's method, inverse-t, logit, mean of p, and mean of z are all available in metap.

In all cases poolr considers correlated p-values in addition to independent. The others above do not.

• TFisher provides Fisher's method using both hard and soft thresholding for the p-values. There is a wrapper in metap for the hard threshold case.
• harmonicmeanp uses the method of harmonic mean of p-values which is robust to correlation between the p-values.
• metapro provides a new method, ordmeta.
• metap provides simple graphics.

Some methods are also provided in some of the genetics packages mentioned below.

Multivariate meta-analysis

Standard methods outlined above assume that the effect sizes are independent. This assumption may be violated in a number of ways: within each primary study multiple treatments may be compared to the same control, each primary study may report multiple endpoints, or primary studies may be clustered for instance because they come from the same country or the same research team. In these situations where the outcome is multivariate:

• mvmeta assumes the within study covariances are known and provides a variety of options for fitting random effects. metafor provides fixed effects and likelihood based random effects model fitting procedures. Both these packages include meta-regression, metafor also provides for clustered and hierarchical models.
• mvtmeta provides multivariate meta-analysis using the method of moments for random effects although not meta-regression,
• metaSEM provides multivariate (and univariate) meta-analysis and meta-regression by embedding it in the structural equation framework and using OpenMx for the structural equation modelling. It can provide a three-level meta-analysis taking account of clustering and allowing for level 2 and level 3 heterogeneity. It also provides via a two-stage approach meta-analysis of correlation or covariance matrices.
• xmeta provides various functions for multivariate meta-analysis and also for detecting publication bias.
• dosresmeta concentrates on the situation where individual studies have information on the dose-response relationship. MBNMAdose provides a Bayesian analysis using network meta-analysis of dose response studies.
• robumeta provides robust variance estimation for clustered and hierarchical estimates.
• CIAAWconsensus has a function for multivariate m-a in the context of atomic weights and estimating isotope ratios.

Meta-analysis of studies of diagnostic tests

A special case of multivariate meta-analysis is the case of summarising studies of diagnostic tests. This gives rise to a bivariate, binary meta-analysis with the within-study correlation assumed zero although the between-study correlation is estimated. This is an active area of research and a variety of methods are available including what is referred to here as Reitsma's method, and the hierarchical summary receiver operating characteristic (HSROC) method. In many situations these are equivalent.

• mada provides various descriptive statistics and univariate methods (diagnostic odds ratio and Lehman model) as well as the bivariate method due to Reitsma. Meta-regression is provided. Graphical facilities are also available.
• Metatron provides a method for the Reitsma model incuding the case of an imperfect reference standard.
• metamisc provides the method of Riley which estimates a common within and between correlation. Graphical output is also provided.
• bamdit provides Bayesian meta-analysis with a bivariate random effects model (using JAGS to implement the MCMC method). Graphical methods are provided.
• meta4diag provides Bayesian inference analysis for bivariate meta-analysis of diagnostic test studies and an extensive range of graphical methods.
• CopulaREMADA uses a copula based mixed model
• diagmeta considers the case where the primary studies provide analysis using multiple cut-offs. Graphical methods are also provided.
• CopulaDTA uses the beta-binomial model to yield marginal mean sensitivity and specificity. Graphical facilities are available.
• NMADiagT provides network meta-analysis of diagnostic tests in a Bayesian framework using Stan as the engine, graphical output is provided.

Meta-regression

Where suitable moderator variables are available they may be included using meta-regression. All these packages are mentioned above, this just draws that information together.

• metafor provides meta-regression (multiple moderators are catered for). Various packages rely on metafor to provide meta-regression (meta, MAc, and MAd) and all three of these provide bubble plots. psychmeta also uses metafor.
• bmeta, metaLik, metansue, metaSEM, and metatest also provide meta-regression.
• mvmeta provides meta-regression for multivariate meta-analysis as do metafor and metaSEM.
• mada provides for the meta-regression of diagnostic test studies.
• GENMETA uses generalised meta-analysis to handle the situation where the studies do not all use the same regressors
• jarbes uses the Bayesian approach of hierarchical meta-regression
• metacart uses classification and regression trees to identify interactions between moderators

Individual participant data (IPD)

Where all studies can provide individual participant data then software for analysis of multi-centre trials or multi-centre cohort studies should prove adequate and is outside the scope of this task view. Other packages which provide facilities related to IPD are:

• ipdmeta which uses information on aggregate summary statistics and a covariate of interest to assess whether a full IPD analysis would have more power.
• ecoreg which is designed for ecological studies enables estimation of an individual level logistic regression from aggregate data or individual data.

Network meta-analysis

Also known as multiple treatment comparison. This is a very active area of research and development. Note that some of the packages mentioned above under multivariate meta-analysis can also be used for network meta-analysis with appropriate setup.

This is provided in a Bayesian framework by pcnetmeta, which uses JAGS. It provides a number of data-sets. nmaINLA uses integrated nested Laplace approximations as an alternative to MCMC. It provides a number of data-sets. netmeta works in a frequentist framework. Both pcnetmeta and netmeta provide network graphs and netmeta provides a heatmap for displaying inconsistency and heterogeneity. nmathresh provides decision-invariant bias adjustment thresholds and intervals the smallest changes to the data that would result in a change of decision. NMAoutlier detects outliers in NMA using forward search, NMADiagT provides network meta-analysis of diagnostic tests in a Bayesian framework using Stan as the engine, graphical output is provided.

nmadb provides access to a database of network meta-analyses

Genetics

There are a number of packages specialising in genetic data: catmap combines case-control and family study data, graphical facilities are provided, CPBayes uses a Bayesian approach to study cross-phenotype genetic associations, etma proposes a new statistical method to detect epistasis, gap combines p-values, getmstatistic quantifies systematic heterogeneity, getspres uses standardised predictive random effects to explore heterogeneity in genetic association meta-analyses, GMCM uses a Gaussian mixture copula model for high-throughput experiments, MBNMAtime provides methods for analysis of repeated measures network meta-analysis, MendelianRandomization provides several methods for performing Mendelian randomisation analyses with summarised data, MetABEL provides meta-analysis of genome wide SNP association results, MetaIntegrator provides meta-analysis of gene expression data, metaMA provides meta-analysis of p-values or moderated effect sizes to find differentially expressed genes, MetaPath performs meta-analysis for pathway enrichment, MetaPCA provides meta-analysis in the dimension reduction of genomic data, metaRNASeq meta-analysis from multiple RNA sequencing experiments, MetaSubtract uses leave-one-out methods to validate meta-GWAS results, MetaSKAT provides meta-analysis for the SKAT test, ofGEM provides a method for identifying gene-environment interactions using meta-filtering, RobustRankAggreg provides methods for aggregating lists of genes, SPAtest combines association results.

Interfaces

• Plug-ins for Rcmdr are provided by: RcmdrPlugin.EZR which uses meta and metatest, RcmdrPlugin.MA which uses MAd and metafor, and RcmdrPlugin.RMTCJags for network meta-analysis using BUGS code.
• MAVIS provides a Shiny interface using metafor, MAc, MAd, and weightr, miniMeta provides a Shiny interface to meta.

Simulation

psychmeta provides facilities for simulation of psychometric data-sets.

Others

CRTSize provides meta-analysis as part of a package primarily dedicated to the determination of sample size in cluster randomised trials in particular by simulating adding a new study to the meta-analysis.

CAMAN offers the possibility of using finite semiparametric mixtures as an alternative to the random effects model where there is heterogeneity. Covariates can be included to provide meta-regression.

joineRmeta provides functions for meta-analysis of a single longitudinal and a single time-to-event outcome from multiple studies using joint models

KenSyn provides data-sets to accompany a French language book on meta-analysis in the agricultural sciences.

PRISMAstatement generates a flowchart conforming to the PRISMA statement.