Calculation of fixed effect and random effects estimates (risk
ratio, odds ratio, risk difference, or arcsine difference) for
metaanalyses with binary outcome data. MantelHaenszel, inverse
variance, Peto method, and generalised linear mixed model (GLMM)
are available for pooling. For GLMMs, the
rma.glmm
function from R package
metafor (Viechtbauer 2010) is called internally.
metabin(event.e, n.e, event.c, n.c, studlab, data = NULL,
subset = NULL, exclude = NULL, method = ifelse(tau.common,
"Inverse", gs("method")), sm = ifelse(!is.na(charmatch(tolower(method),
c("peto", "glmm"), nomatch = NA)), "OR", gs("smbin")),
incr = gs("incr"), allincr = gs("allincr"),
addincr = gs("addincr"), allstudies = gs("allstudies"),
MH.exact = gs("MH.exact"), RR.cochrane = gs("RR.cochrane"),
model.glmm = "UM.FS", level = gs("level"),
level.comb = gs("level.comb"), comb.fixed = gs("comb.fixed"),
comb.random = gs("comb.random"), hakn = gs("hakn"),
method.tau = ifelse(!is.na(charmatch(tolower(method), "glmm", nomatch =
NA)), "ML", gs("method.tau")), tau.preset = NULL, TE.tau = NULL,
tau.common = gs("tau.common"), prediction = gs("prediction"),
level.predict = gs("level.predict"), method.bias = ifelse(sm == "OR",
"score", gs("method.bias")), backtransf = gs("backtransf"),
pscale = 1, title = gs("title"), complab = gs("complab"),
outclab = "", label.e = gs("label.e"), label.c = gs("label.c"),
label.left = gs("label.left"), label.right = gs("label.right"),
byvar, bylab, print.byvar = gs("print.byvar"),
byseparator = gs("byseparator"), print.CMH = gs("print.CMH"),
keepdata = gs("keepdata"), warn = gs("warn"), control = NULL, ...)
Number of events in experimental group.
Number of observations in experimental group.
Number of events in control group.
Number of observations in control group.
An optional vector with study labels.
An optional data frame containing the study information, i.e., event.e, n.e, event.c, and n.c.
An optional vector specifying a subset of studies to be used.
An optional vector specifying studies to exclude from metaanalysis, however, to include in printouts and forest plots.
A character string indicating which method is to be
used for pooling of studies. One of "Inverse"
,
"MH"
, "Peto"
, or "GLMM"
, can be abbreviated.
A character string indicating which summary measure
("RR"
, "OR"
, "RD"
, or "ASD"
) is to be
used for pooling of studies, see Details.
Could be either a numerical value which is added to
each cell frequency for studies with a zero cell count or the
character string "TACC"
which stands for treatment arm
continuity correction, see Details.
A logical indicating if incr
is added to each
cell frequency of all studies if at least one study has a zero
cell count. If FALSE (default), incr
is added only to each
cell frequency of studies with a zero cell count.
A logical indicating if incr
is added to each
cell frequency of all studies irrespective of zero cell counts.
A logical indicating if studies with zero or all
events in both groups are to be included in the metaanalysis
(applies only if sm
is equal to "RR"
or
"OR"
).
A logical indicating if incr
is not to be
added to all cell frequencies for studies with a zero cell count
to calculate the pooled estimate based on the MantelHaenszel
method.
A logical indicating if 2*incr
instead of
1*incr
is to be added to n.e
and n.c
in the
calculation of the risk ratio (i.e., sm="RR"
) for studies
with a zero cell. This is used in RevMan 5, the Cochrane
Collaboration's program for preparing and maintaining Cochrane
reviews.
A character string indicating which GLMM should
be used. One of "UM.FS"
, "UM.RS"
, "CM.EL"
,
and "CM.AL"
, see Details.
The level used to calculate confidence intervals for individual studies.
The level used to calculate confidence intervals for pooled estimates.
A logical indicating whether a fixed effect metaanalysis should be conducted.
A logical indicating whether a random effects metaanalysis should be conducted.
A logical indicating whether the method by Hartung and Knapp should be used to adjust test statistics and confidence intervals.
A character string indicating which method is
used to estimate the betweenstudy variance \(\tau^2\). Either
"DL"
, "PM"
, "REML"
, "ML"
,
"HS"
, "SJ"
, "HE"
, or "EB"
, can be
abbreviated.
Prespecified value for the squareroot of the betweenstudy variance \(\tau^2\).
Overall treatment effect used to estimate the betweenstudy variance \(\tau^2\).
A logical indicating whether tausquared should be the same across subgroups.
A logical indicating whether a prediction interval should be printed.
The level used to calculate prediction interval for a new study.
A character string indicating which test for
funnel plot asymmetry is to be used. Either "rank"
,
"linreg"
, "mm"
, "count"
, "score"
, or
"peters"
, can be abbreviated. See function
metabias
A logical indicating whether results for odds
ratio (sm="OR"
) and risk ratio (sm="RR"
) should be
back transformed in printouts and plots. If TRUE (default),
results will be presented as odds ratios and risk ratios;
otherwise log odds ratios and log risk ratios will be shown.
A numeric defining a scaling factor for printing of risk differences.
Title of metaanalysis / systematic review.
Comparison label.
Outcome label.
Label for experimental group.
Label for control group.
Graph label on left side of forest plot.
Graph label on right side of forest plot.
An optional vector containing grouping information
(must be of same length as event.e
).
A character string with a label for the grouping variable.
A logical indicating whether the name of the grouping variable should be printed in front of the group labels.
A character string defining the separator between label and levels of grouping variable.
A logical indicating whether result of the CochranMantelHaenszel test for overall effect should be printed.
A logical indicating whether original data (set) should be kept in meta object.
A logical indicating whether warnings should be printed
(e.g., if incr
is added to studies with zero cell
frequencies).
Additional arguments passed on to
rma.glmm
function.
An object of class c("metabin", "meta")
with corresponding
print
, summary
, and forest
functions. The
object is a list containing the following components:
As defined above.
As defined above.
As defined above.
As defined above.
As defined above.
As defined above.
As defined above.
As defined above.
Estimated treatment effect and standard error of individual studies.
Lower and upper confidence interval limits for individual studies.
zvalue and pvalue for test of treatment effect for individual studies.
Weight of individual studies (in fixed and random effects model).
Estimated overall treatment effect, e.g., log risk ratio or risk difference, and standard error (fixed effect model).
Lower and upper confidence interval limits (fixed effect model).
zvalue and pvalue for test of overall treatment effect (fixed effect model).
Estimated overall treatment effect, e.g., log risk ratio or risk difference, and standard error (random effects model).
Lower and upper confidence interval limits (random effects model).
zvalue or tvalue and corresponding pvalue for test of overall treatment effect (random effects model).
As defined above.
Standard error utilised for prediction interval.
Lower and upper limits of prediction interval.
Number of studies combined in metaanalysis.
Heterogeneity statistic Q.
Degrees of freedom for heterogeneity statistic.
Pvalue of heterogeneity test.
Heterogeneity statistic for likelihoodratio test
(only if method = "GLMM"
).
Degrees of freedom for likelihoodratio test
Pvalue of likelihoodratio test.
Squareroot of betweenstudy variance.
Standard error of squareroot of betweenstudy variance.
Scaling factor utilised internally to calculate common tausquared across subgroups.
CochranMantelHaenszel test statistic for overall effect.
Degrees of freedom for CochranMantelHaenszel test statistic.
Pvalue of CochranMantelHaenszel test.
Increment added to cells in the experimental and control group, respectively.
Logical flag indicating if any study included in metaanalysis has any zero cell frequencies.
Logical flag indicating if any study has zero cell frequencies in both treatment groups.
Degrees of freedom for test of treatment effect for
HartungKnapp method (only if hakn = TRUE
).
Number of studies combined in metaanalysis using MantelHaenszel method.
Levels of grouping variable  if byvar
is not
missing.
Estimated treatment effect and
standard error in subgroups (fixed effect model)  if
byvar
is not missing.
Lower and upper confidence interval limits in
subgroups (fixed effect model)  if byvar
is not missing.
zvalue and pvalue for test of
treatment effect in subgroups (fixed effect model)  if
byvar
is not missing.
Estimated treatment effect and standard error in
subgroups (random effects model)  if byvar
is not
missing.
Lower and upper confidence
interval limits in subgroups (random effects model)  if
byvar
is not missing.
zvalue or tvalue and
corresponding pvalue for test of treatment effect in subgroups
(random effects model)  if byvar
is not missing.
Weight of subgroups (in fixed and
random effects model)  if byvar
is not missing.
Degrees of freedom for test of treatment effect
for HartungKnapp method in subgroups  if byvar
is not
missing and hakn = TRUE
.
Harmonic mean of number of observations in
subgroups (for back transformation of FreemanTukey Double
arcsine transformation)  if byvar
is not missing.
Number of events in experimental group in
subgroups  if byvar
is not missing.
Number of observations in experimental group in
subgroups  if byvar
is not missing.
Number of events in control group in subgroups 
if byvar
is not missing.
Number of observations in control group in subgroups 
if byvar
is not missing.
Number of studies combined within subgroups  if
byvar
is not missing.
Number of all studies in subgroups  if byvar
is not missing.
Overall within subgroups heterogeneity statistic Q
(based on fixed effect model)  if byvar
is not missing.
Overall within subgroups heterogeneity statistic
Q (based on random effects model)  if byvar
is not
missing (only calculated if argument tau.common
is TRUE).
Degrees of freedom for test of overall within
subgroups heterogeneity  if byvar
is not missing.
Pvalue of within subgroups heterogeneity
statistic Q (based on fixed effect model)  if byvar
is
not missing.
Pvalue of within subgroups heterogeneity
statistic Q (based on random effects model)  if byvar
is
not missing.
Overall between subgroups heterogeneity statistic
Q (based on fixed effect model)  if byvar
is not
missing.
Overall between subgroups heterogeneity statistic
Q (based on random effects model)  if byvar
is not
missing.
Degrees of freedom for test of overall between
subgroups heterogeneity  if byvar
is not missing.
Pvalue of between subgroups heterogeneity
statistic Q (based on fixed effect model)  if byvar
is
not missing.
Pvalue of between subgroups heterogeneity
statistic Q (based on random effects model)  if byvar
is
not missing.
Squareroot of betweenstudy variance within subgroups
 if byvar
is not missing.
Scaling factor utilised internally to calculate common
tausquared across subgroups  if byvar
is not missing.
Heterogeneity statistic H within subgroups  if
byvar
is not missing.
Lower and upper confidence limti for
heterogeneity statistic H within subgroups  if byvar
is
not missing.
Heterogeneity statistic I2 within subgroups  if
byvar
is not missing.
Lower and upper confidence limit for
heterogeneity statistic I2 within subgroups  if byvar
is
not missing.
As defined above.
Original data (set) used in function call (if
keepdata = TRUE
).
Information on subset of original data used in
metaanalysis (if keepdata = TRUE
).
GLMM object generated by call of
rma.glmm
function (fixed effect model).
GLMM object generated by call of
rma.glmm
function (random effects model).
Function call.
Version of R package meta used to create object.
Version of R package metafor used for GLMMs.
Treatment estimates and standard errors are calculated for each study. The following measures of treatment effect are available:
Risk ratio (sm = "RR"
)
Odds ratio (sm = "OR"
)
Risk difference (sm = "RD"
)
Arcsine difference (sm = "ASD"
)
For several arguments defaults settings are utilised (assignments
using gs
function). These defaults can be changed
using the settings.meta
function.
Internally, both fixed effect and random effects models are
calculated regardless of values chosen for arguments
comb.fixed
and comb.random
. Accordingly, the estimate
for the random effects model can be extracted from component
TE.random
of an object of class "meta"
even if
argument comb.random = FALSE
. However, all functions in R
package meta will adequately consider the values for
comb.fixed
and comb.random
. E.g. function
print.meta
will not print results for the random
effects model if comb.random = FALSE
.
By default, both fixed effect and random effects models are
considered (see arguments comb.fixed
and
comb.random
). If method
is "MH"
(default), the
MantelHaenszel method is used to calculate the fixed effect
estimate; if method
is "Inverse"
, inverse variance
weighting is used for pooling; if method
is "Peto"
,
the Peto method is used for pooling. For the Peto method, Peto's
log odds ratio, i.e. (O  E) / V
and its standard error
sqrt(1 / V)
with O  E
and V
denoting
"Observed minus Expected" and "V", are utilised in the random
effects model. Accordingly, results of a random effects model using
sm = "Peto"
can be (slightly) different to results from a
random effects model using sm = "MH"
or sm =
"Inverse"
.
A distinctive and frequently overlooked advantage of binary
endpoints is that individual patient data (IPD) can be extracted
from a twobytwo table. Accordingly, statistical methods for IPD,
i.e., logistic regression and generalised linear mixed models, can
be utilised in a metaanalysis of binary outcomes (Stijnen et al.,
2010; Simmonds et al., 2016). These methods are available (argument
method = "GLMM"
) for the odds ratio as summary measure by
calling the rma.glmm
function from R package
metafor internally.
Four different GLMMs are available for
metaanalysis with binary outcomes using argument model.glmm
(which corresponds to argument model
in the
rma.glmm
function):
1.  Logistic regression model with fixed study effects (default) 
(model.glmm = "UM.FS" , i.e., Unconditional
Model  Fixed Study effects) 

2.  Mixedeffects logistic regression model with random study effects 
(model.glmm = "UM.RS" , i.e., Unconditional
Model  Random Study effects) 

3.  Generalised linear mixed model (conditional HypergeometricNormal) 
(model.glmm = "CM.EL" , i.e., Conditional
Model  Exact Likelihood) 

4.  Generalised linear mixed model (conditional BinomialNormal) 
Details on these four GLMMs as well as additional arguments which
can be provided using argument '…{}
' in metabin
are described in rma.glmm
where you can also
find information on the iterative algorithms used for estimation.
Note, regardless of which value is used for argument
model.glmm
, results for two different GLMMs are calculated:
fixed effect model (with fixed treatment effect) and random effects
model (with random treatment effects).
For studies with a zero cell count, by default, 0.5 is added to all
cell frequencies of these studies; if incr
is "TACC"
a treatment arm continuity correction is used instead (Sweeting et
al., 2004; Diamond et al., 2007). For odds ratio and risk ratio,
treatment estimates and standard errors are only calculated for
studies with zero or all events in both groups if allstudies
is TRUE
. This continuity correction is used both to
calculate individual study results with confidence limits and to
conduct metaanalysis based on the inverse variance method. For
Peto method and GLMMs no continuity correction is used. For the
MantelHaenszel method, by default (if MH.exact
is FALSE),
incr
is added to all cell frequencies of a study with a zero
cell count in the calculation of the pooled risk ratio or odds
ratio as well as the estimation of the variance of the pooled risk
difference, risk ratio or odds ratio. This approach is also used in
other software, e.g. RevMan 5 and the Stata procedure
metan. According to Fleiss (in Cooper & Hedges, 1994), there is no
need to add 0.5 to a cell frequency of zero to calculate the
MantelHaenszel estimate and he advocates the exact method
(MH.exact
= TRUE). Note, estimates based on exact
MantelHaenszel method or GLMM are not defined if the number of
events is zero in all studies either in the experimental or control
group.
Argument byvar
can be used to conduct subgroup analysis for
all methods but GLMMs. Instead use the metareg
function for GLMMs which can also be used for continuous
covariates.
A prediction interval for the treatment effect of a new study is
calculated (Higgins et al., 2009) if arguments prediction
and comb.random
are TRUE
.
R function update.meta
can be used to redo the
metaanalysis of an existing metabin object by only specifying
arguments which should be changed.
For the random effects, the method by Hartung and Knapp (2001) is
used to adjust test statistics and confidence intervals if argument
hakn = TRUE
. For GLMMs, a method similar to Knapp and
Hartung (2003) is implemented, see description of argument
tdist
in rma.glmm
.
The DerSimonianLaird estimate (1986) is used in the random effects
model if method.tau = "DL"
. The iterative PauleMandel
method (1982) to estimate the betweenstudy variance is used if
argument method.tau = "PM"
. Internally, R function
paulemandel
is called which is based on R function
mpaule.default from R package metRology from S.L.R. Ellison
<s.ellison at lgc.co.uk>.
If R package metafor (Viechtbauer 2010) is installed, the
following methods to estimate the betweenstudy variance
\(\tau^2\) (argument method.tau
) are also available:
Restricted maximumlikelihood estimator (method.tau =
"REML"
)
Maximumlikelihood estimator (method.tau = "ML"
)
HunterSchmidt estimator (method.tau = "HS"
)
SidikJonkman estimator (method.tau = "SJ"
)
Hedges estimator (method.tau = "HE"
)
Empirical Bayes estimator (method.tau = "EB"
)
For these methods the R function rma.uni
of R package
metafor is called internally. See help page of R function
rma.uni
for more details on these methods to estimate
betweenstudy variance.
Cooper H & Hedges LV (1994): The Handbook of Research Synthesis. Newbury Park, CA: Russell Sage Foundation
Diamond GA, Bax L, Kaul S (2007): Uncertain Effects of Rosiglitazone on the Risk for Myocardial Infarction and Cardiovascular Death. Annals of Internal Medicine, 147, 57881
DerSimonian R & Laird N (1986): Metaanalysis in clinical trials. Controlled Clinical Trials, 7, 17788
Fleiss JL (1993): The statistical basis of metaanalysis. Statistical Methods in Medical Research, 2, 12145
Greenland S & Robins JM (1985): Estimation of a common effect parameter from sparse followup data. Biometrics, 41, 5568
Hartung J & Knapp G (2001): A refined method for the metaanalysis of controlled clinical trials with binary outcome. Statistics in Medicine, 20, 387589
Higgins JPT, Thompson SG, Spiegelhalter DJ (2009): A reevaluation of randomeffects metaanalysis. Journal of the Royal Statistical Society: Series A, 172, 13759
Knapp G & Hartung J (2003): Improved tests for a random effects metaregression with a single covariate. Statistics in Medicine, 22, 2693710
Review Manager (RevMan) [Computer program]. Version 5.3. Copenhagen: The Nordic Cochrane Centre, The Cochrane Collaboration, 2014
Paule RC & Mandel J (1982): Consensus values and weighting factors. Journal of Research of the National Bureau of Standards, 87, 37785
Pettigrew HM, Gart JJ, Thomas DG (1986): The bias and higher cumulants of the logarithm of a binomial variate. Biometrika, 73, 42535
R<U+00FC>cker G, Schwarzer G, Carpenter JR (2008): Arcsine test for publication bias in metaanalyses with binary outcomes. Statistics in Medicine, 27, 74663
Simmonds MC, Higgins JP (2016): A general framework for the use of logistic regression models in metaanalysis. Statistical Methods in Medical Research, 25, 285877
StataCorp. 2011. Stata Statistical Software: Release 12. College Station, TX: StataCorp LP.
Stijnen T, Hamza TH, Ozdemir P (2010): Random effects metaanalysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29, 304667
Sweeting MJ, Sutton AJ, Lambert PC (2004): What to add to nothing? Use and avoidance of continuity corrections in metaanalysis of sparse data. Statistics in Medicine, 23, 135175
Viechtbauer W (2010): Conducting MetaAnalyses in R with the Metafor Package. Journal of Statistical Software, 36, 148
update.meta
, forest
,
funnel
, metabias
,
metacont
, metagen
,
metareg
, print.meta
# NOT RUN {
# Calculate odds ratio and confidence interval for a single study
#
metabin(10, 20, 15, 20, sm = "OR")
# Different results (due to handling of studies with double zeros)
#
metabin(0, 10, 0, 10, sm = "OR")
metabin(0, 10, 0, 10, sm = "OR", allstudies = TRUE)
# Use subset of Olkin (1995) to conduct metaanalysis based on
# inverse variance method (with risk ratio as summary measure)
#
data(Olkin95)
m1 < metabin(event.e, n.e, event.c, n.c,
data = Olkin95, subset = c(41, 47, 51, 59),
method = "Inverse")
summary(m1)
# Use different subset of Olkin (1995)
#
m2 < metabin(event.e, n.e, event.c, n.c,
data = Olkin95, subset = year < 1970,
method = "Inverse", studlab = author)
summary(m2)
forest(m2)
# Metaanalysis with odds ratio as summary measure
#
m3 < metabin(event.e, n.e, event.c, n.c,
data = Olkin95, subset = year < 1970,
sm = "OR", method = "Inverse", studlab = author)
# Same metaanalysis result using 'update.meta' function
m3 < update(m2, sm = "OR")
summary(m3)
# Metaanalysis based on MantelHaenszel method (with odds ratio as
# summary measure)
#
m4 < update(m3, method = "MH")
summary(m4)
# Metaanalysis based on Peto method (only available for odds ratio
# as summary measure)
#
m5 < update(m3, method = "Peto")
summary(m5)
# }
# NOT RUN {
# Metaanalysis using generalised linear mixed models (only if R
# packages 'metafor' and 'lme4' are available)
#
if (suppressMessages(require(metafor, quietly = TRUE, warn = FALSE)) &
require(lme4, quietly = TRUE)) {
# Logistic regression model with (k = 4) fixed study effects
# (default: model.glmm = "UM.FS")
#
m6 < metabin(event.e, n.e, event.c, n.c,
data = Olkin95, subset = year < 1970,
method = "GLMM")
# Same results:
m6 < update(m2, method = "GLMM")
summary(m6)
# Mixedeffects logistic regression model with random study effects
# (warning message printed due to argument 'nAGQ')
#
m7 < update(m6, model.glmm = "UM.RS")
#
# Use additional argument 'nAGQ' for internal call of 'rma.glmm'
# function
#
m7 < update(m6, model.glmm = "UM.RS", nAGQ = 1)
summary(m7)
# Generalised linear mixed model (conditional
# HypergeometricNormal) (R package 'BiasedUrn' must be available)
#
if (require(BiasedUrn, quietly = TRUE)) {
m8 < update(m6, model.glmm = "CM.EL")
summary(m8)
}
# Generalised linear mixed model (conditional BinomialNormal)
#
m9 < update(m6, model.glmm = "CM.AL")
summary(m9)
# Logistic regression model with (k = 70) fixed study effects
# (about 18 seconds with Intel Core i73667U, 2.0GHz)
#
m10 < metabin(event.e, n.e, event.c, n.c,
data = Olkin95, method = "GLMM")
summary(m10)
# Mixedeffects logistic regression model with random study effects
#  about 50 seconds with Intel Core i73667U, 2.0GHz
#  several warning messages, e.g. "failure to converge, ..."
#
summary(update(m10, model.glmm = "UM.RS"))
# Conditional HypergeometricNormal GLMM
#  long computation time (about 12 minutes with Intel Core
# i73667U, 2.0GHz)
#  estimation problems for this very large dataset:
# * warning that Choleski factorization of Hessian failed
# * confidence interval for treatment effect smaller in random
# effects model compared to fixed effect model
#
if (require(BiasedUrn, quietly = TRUE)) {
system.time(m11 < update(m10, model.glmm = "CM.EL"))
summary(m11)
}
# Generalised linear mixed model (conditional BinomialNormal)
# (less than 1 second with Intel Core i73667U, 2.0GHz)
#
summary(update(m10, model.glmm = "CM.AL"))
}
# }
# NOT RUN {
# }
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