Last updated on 2019-11-27
by Michael Dewey
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.
Preparing for meta-analysis
- The primary studies often use a range of
statistics to present their
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,
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).
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.
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).
provides a number of 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.
provides some methods for converting to effect sizes
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).
provides a variety of conversions based on Cohen's d.
converts between quantiles and means and standard deviations.
estimates means and standard deviations from
various order statistics.
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.
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
variance-covariance matrix for multivariate meta-analysis
metafuse uses a fused lasso to merge
covariate estimates across a number of independent datasets.
Fitting the model
Four packages provide the inverse variance weighted,
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
which provide meta-analysis of correlation
MAd which provides meta-analysis
of mean differences.
MAc and MAd provide
a range of graphics.
provides an extensive range of functions
for the meta-analysis of psychometric studies.
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.
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
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.
includes binomial-normal hierarchical models and can use weakly
informative priors for the heterogeneity and treatment effect parameters.
provides facilities using Stan for hierarchical
Bayesian models, graphical facilities are provided.
provides facilities using a
Bayesian approach and has graphical facilities.
uses Bayesian synthesis to generate priors from various
Some packages concentrate on providing
a specialised version of the core
meta-analysis function without providing
the range of ancillary
functions. These are:
which subsumes a very wide variety of models under the method
of confidence distributions and
also provides a graphical display,
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
- metagen provides a range of methods for
random effects models and also facilities
for extensive simulation studies of the
properties of those methods.
metaplus fits random effects
models relaxing the usual
assumption that the random effects have a normal
distribution by providing t or a mixture
fits random effects models to binary data using
a variety of methods for confidence intervals.
estimates exact confidence intervals in random effects
models using an efficient algorithm.
estimates exact confidence intervals in random effects
normal-normal models and also provides plots of them.
gives cluster-robust variance estimates.
implements prediction intervals for random effects meta-analysis.
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
proposes a metric for estimating the proportion of effects
above a cut-off of scientific importance
provides imputation methods for missing binary data.
An extensive range of graphical procedures is available.
Forest plots are provided in forestmodel
(using ggplot2), forestplot,
metansue, psychmeta, and rmeta.
Although the most basic plot can be produced
by any of them
they each provide their own choice of enhancements.
Funnel plots are provided in
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
funnel plots 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.
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.
Confidence intervals for the heterogeneity parameter
are provided in metafor,
metagen, and psychmeta.
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.
implements a Monte Carlo based test for heterogeneity.
Investigating small study bias
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.
provides sensitivity analysis of the effect of
The issue of whether small studies give different results
from large studies has been addressed by visual
examination of the funnel plots mentioned above.
- 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
An exploratory technique for detecting
an excess of statistically
significant studies is provided by PubBias.
metamisc provides funnel plots and tests for asymmetry.
provides methods using only the statistically significant studies,
methods for the special case of replication studies
and sample size determinations.
performs sensitivity analysis of the number of unpublished
studies needed to have a specified influence.
A recurrent issue in meta-analysis has been
the problem of unobserved studies.
Other study designs
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
SAMURAI performs a sensitivity
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
facilities for using the weight function model
of Vevea and Hedges.
Meta-analysis of significance values
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.
provides meta-analysis of Phase I dose-finding
implements meta-analysis of trials with difference in
restricted mean survival times
metap provides some facilities for
meta-analysis of significance values.
provides a smaller subset of methods.
TFisher provides Fisher's method using thresholding for
uses the method of harmonic mean of p-values which
is robust to correlation between the p-values.
provides a subset of methods and also a new method, ordmeta.
Some methods are also provided in some
of the genetics packages mentioned below.
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.
provides fixed effects and likelihood
based random effects model fitting procedures.
Both these packages include meta-regression,
metafor also provides for clustered and
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.
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.
provides a Bayesian analysis using
network meta-analysis of dose response
robumeta provides robust variance
estimation for clustered and hierarchical estimates.
has a function for multivariate m-a in the context
of atomic weights and estimating
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
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.
provides Bayesian inference analysis for bivariate meta-analysis
of diagnostic test studies and an extensive range of
CopulaREMADA uses a copula based mixed model
considers the case where the primary studies provide
analysis using multiple cut-offs.
Graphical methods are also provided.
HSROC provides HSROC with estimation
in a Bayesian framework. Graphical facilities are provided.
The case of imperfect reference standards is catered for.
uses the beta-binomial model to yield marginal mean sensitivity
and specificity. Graphical facilities are available.
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.
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.
uses generalised meta-analysis to handle the situation
where the studies do not all use the same regressors
uses the Bayesian approach of hierarchical meta-regression
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
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
This is provided in a Bayesian framework by
which acts as a front-end to BUGS
or JAGS, and pcnetmeta,
which uses JAGS.
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.
provides decision-invariant bias adjustment
thresholds and intervals the
smallest changes to the data that would result in a change of decision.
detects outliers in NMA using forward search
provides access to a database of network meta-analyses
There are a number of packages specialising
in genetic data:
combines case-control and family study data,
graphical facilities are provided,
uses a Bayesian approach to study cross-phenotype genetic
proposes a new statistical method to detect epistasis,
gap combines p-values,
getmstatistic quantifies systematic heterogeneity,
uses a Gaussian mixture copula model for high-throughput
provides methods for analysis of repeated measures
provides several methods for performing Mendelian randomisation
analyses with summarised data,
MetABEL provides meta-analysis of
genome wide SNP association results,
provides an extensive set of functions for genetic studies,
metaMA provides meta-analysis of
p-values or moderated
effect sizes to find differentially expressed genes,
performs meta-analysis for pathway enrichment,
MetaPCA provides meta-analysis in
the dimension reduction of genomic data,
metaRNASeq meta-analysis from multiple RNA
MetaSubtract uses leave-one-out methods to
validate meta-GWAS results,
MultiMeta for meta-analysis
of multivariate GWAS
results with graphics, designed to accept GEMMA format,
for the SKAT test,
provides a method for identifying gene-environment interactions
provides methods for aggregating lists of genes.
Extensive facilities for simulation are provided in
metagen including the ability to make use
of parallel processing.
facilities for simulation of psychometric data-sets.
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
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.
provides functions for meta-analysis of a single longitudinal and
a single time-to-event outcome from multiple studies using joint models
provides data-sets to accompany a French language book
on meta-analysis in the agricultural sciences.
generates a flowchart conforming to the PRISMA statement.