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 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
results. Convenience functions to convert these onto a common
metric are presented by:
which converts from various statistics to
d, g, r, z and the log odds ratio,
which converts to correlation coefficients,
which converts to mean differences,
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).
provides functions to read and work
with files output by RevMan 4 and 5.
Fitting the model
Four packages provide the inverse variance weighted, Mantel-Haenszel,
and Peto methods:
For binary data
provides the binomial-normal model.
For sparse binary data
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
which provide meta-analysis of correlation coefficients and
which provides meta-analysis of mean differences.
provide a range of graphics.
provides an extensive range of functions for the meta-analysis of
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.
provides a method with priors suggested by Higgins.
provides meta-analysis using
beta-binomial prior distributions.
Some packages concentrate on providing a specialised version of the core
meta-analysis function without providing the range of ancillary
functions. These are:
which uses a more sophisticated approach to the likelihood,
which as well as the method of moments provides
two likelihood-based methods, and
another improved method of obtaining confidence intervals.
provides a range of methods for
random effects models and also facilities
for extensive simulation studies of the
properties of those methods.
fits random effects models relaxing the usual
assumption that the random effects have a normal
distribution. Also provides some diagnostics.
An extensive range of graphical procedures is available.
Forest plots are provided in
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
In addition to the standard funnel plots
an enhanced funnel plot to assess the impact of extra evidence
is available in
extfunnel, and a funnel plot
for limit meta-analysis in
Radial (Galbraith) plots are provided in
L'Abbe plots are provided in
Baujat plots are provided in
An extensive series of plots of diagnostic statistics is
Confidence intervals for the heterogeneity parameter
are provided in
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.
both the non-parametric method suggested by Begg and Mazumdar
and a range of regression tests modelled after the approach of Egger.
An exploratory technique for detecting an excess of statistically
significant studies is provided by
A recurrent issue in meta-analysis has been
the problem of unobserved studies.
Meta-analysis of significance values
Rosenthal's fail safe n is provided by
provides it as well as two
more recent methods by Orwin and Rosenberg.
Duval's trim and fill method is provided by
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.
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.
performs a sensitivity analysis assuming
the number of unobserved studies is known,
perhaps from a trial registry, but not their outcome.
provides some facilities for
meta-analysis of significance values.
Some of these 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:
assumes the within study covariances
are known and as well as fixed effects 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,
also provides for clustered and
provides multivariate meta-analysis
using the method of moments for random effects
although not meta-regression,
is available from R-Forge and
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.
concentrates on the situation where individual
studies have information on the dose-response relationship.
provides robust variance estimation for
clustered and hierarchical estimates.
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 called Reitsma's
method, and the heirarchical summary receiver operating
characteristic (HSROC) method.
In many situations these are equivalent.
provides various descriptive statistics
and univariate methods (diagnostic odds ratio and Lehman
model) as well as the bivariate method due to Reitsma.
In addition meta-regression is provided.
A range of graphical methods is also available.
provides HSROC with estimation in a Bayesian framework.
Graphical methods are provided.
The case of imperfect reference standards is catered for.
provides a method for the Reitsma model
incuding the case of an imperfect reference standard
provides the method of Riley which estimates a common
within and between correlation. Graphical output is also provided.
provides Bayesian meta-analysis with a bivariate
random effects model (using JAGS to implement the MCMC method).
uses a copula based mixed model
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.
provides meta-regression (multiple
moderators are catered for).
Various packages rely on
provide meta-regression (meta,
MAd) and all three of these provide bubble plots.
also provide meta-regression.
provides meta-regression for multivariate meta-analysis
provides for the meta-regression of diagnostic test studies.
Individual participant data (IPD)
Where all studies can provide indiviual 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:
which uses information on aggregate
summary statistics and
a covariate of interest to assess whether a full IPD analysis
would have more power.
which is designed for ecological studies
enables estimation of an individual level
logistic regression from aggregate data or individual data.
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
which uses JAGS.
works in a frequentist
provide network graphs and
provides a heatmap for displaying inconsistency
There are a number of packages specialising in genetic data:
for startified GWAS meta-analysis
combines p-values using Fisher's method,
provides meta-analysis of microarray data,
provides meta-analysis of genome wide SNP association
provides microarray meta-analysis of for differentially
expressed dene detection,
provides meta-analysis of p-values or moderated
effect sizes to find differentially expressed genes,
provides meta-analysis in the dimension reduction
of genomic data,
provides objective quality control and
inclusion/exclusion criteria for genomic meta-analysis,
meta-analysis from multiple RNA
for meta-analysis of multivariate GWAS
results with graphics, designed to accept GEMMA format,
provide meta-analysis for the SKAT test.
provides single case meta-analysis. It is part of a suite of packages
dedicated to single-case designs.
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
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 an interface via the Rcmdr GUI
to do the heavy lifting and
provides a shiny