Calculation of fixed effect and random effects estimates (risk
ratio, odds ratio, risk difference, or arcsine difference) for
meta-analyses with binary outcome data. Mantel-Haenszel, 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 meta-analysis, 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 meta-analysis
(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 Mantel-Haenszel
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 meta-analysis should be conducted.
A logical indicating whether a random effects meta-analysis 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 between-study variance \(\tau^2\). Either
"DL", "PM", "REML", "ML",
"HS", "SJ", "HE", or "EB", can be
abbreviated.
Prespecified value for the square-root of the between-study variance \(\tau^2\).
Overall treatment effect used to estimate the between-study variance \(\tau^2\).
A logical indicating whether tau-squared 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 meta-analysis / 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 Cochran-Mantel-Haenszel 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.
z-value and p-value 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).
z-value and p-value 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).
z-value or t-value and corresponding p-value 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 meta-analysis.
Heterogeneity statistic Q.
Degrees of freedom for heterogeneity statistic.
P-value of heterogeneity test.
Heterogeneity statistic for likelihood-ratio test
(only if method = "GLMM").
Degrees of freedom for likelihood-ratio test
P-value of likelihood-ratio test.
Square-root of between-study variance.
Standard error of between-study variance.
Scaling factor utilised internally to calculate common tau-squared across subgroups.
Cochran-Mantel-Haenszel test statistic for overall effect.
Degrees of freedom for Cochran-Mantel-Haenszel test statistic.
P-value of Cochran-Mantel-Haenszel test.
Increment added to cells in the experimental and control group, respectively.
Logical flag indicating if any study included in meta-analysis 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
Hartung-Knapp method (only if hakn = TRUE).
Number of studies combined in meta-analysis using Mantel-Haenszel 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.
z-value and p-value 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.
z-value or t-value and
corresponding p-value 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 Hartung-Knapp method in subgroups - if byvar is not
missing and hakn = TRUE.
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.
P-value of within subgroups heterogeneity
statistic Q (based on fixed effect model) - if byvar is
not missing.
P-value 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.
P-value of between subgroups heterogeneity
statistic Q (based on fixed effect model) - if byvar is
not missing.
P-value of between subgroups heterogeneity
statistic Q (based on random effects model) - if byvar is
not missing.
Square-root of between-study variance within subgroups
- if byvar is not missing.
Scaling factor utilised internally to calculate common
tau-squared 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
meta-analysis (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 (R<U+00FC>cker et al., 2009):
Risk ratio (sm = "RR")
Odds ratio (sm = "OR")
Risk difference (sm = "RD")
Arcsine difference (sm = "ASD")
By default, both fixed effect and random effects models are
considered (see arguments comb.fixed and
comb.random). If method is "MH" (default), the
Mantel-Haenszel method (Greenland & Robins, 1985; Robins et al.,
1986) is used to calculate the fixed effect estimate; if
method is "Inverse", inverse variance weighting is
used for pooling (Fleiss, 1993); if method is "Peto",
the Peto method is used for pooling (Yussuf et al., 1985).
While the Mantel-Haenszel and Peto method are defined under the
fixed effect model, random effects variants based on these methods
are also implemented in metabin. Following RevMan 5, the
Mantel-Haenszel estimator is used in the calculation of the
between-study heterogeneity statistic Q which is used in the
DerSimonian-Laird estimator. Accordlingly, the results for the
random effects meta-analysis using the Mantel-Haenszel or inverse
variance method are typically very similar. 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 its variance, are utilised in the
random effects model. Accordingly, results of a random effects
model using sm = "Peto" can be 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 two-by-two table. Accordingly, statistical methods for IPD,
i.e., logistic regression and generalised linear mixed models, can
be utilised in a meta-analysis 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
meta-analysis 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. | Mixed-effects logistic regression model with random study effects |
(model.glmm = "UM.RS", i.e., Unconditional
Model - Random Study effects) |
|
| 3. | Generalised linear mixed model (conditional Hypergeometric-Normal) |
(model.glmm = "CM.EL", i.e., Conditional
Model - Exact Likelihood) |
|
| 4. | Generalised linear mixed model (conditional Binomial-Normal) |
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 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.
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 meta-analysis based on the inverse variance method. For
Peto method and GLMMs no continuity correction is used. For the
Mantel-Haenszel 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
Mantel-Haenszel estimate and he advocates the exact method
(MH.exact = TRUE). Note, estimates based on exact
Mantel-Haenszel 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
a categorical covariate. The metareg function can be
used instead for more than one categorical covariate or continuous
covariates.
A prediction interval for the proportion in a new study (Higgins et
al., 2009) is calculated if arguments prediction and
comb.random are TRUE. Note, the definition of
prediction intervals varies in the literature. This function
implements equation (12) of Higgins et al., (2009) which proposed a
t distribution with K-2 degrees of freedom where
K corresponds to the number of studies in the meta-analysis.
R function update.meta can be used to redo the
meta-analysis 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 following methods to estimate the between-study variance
\(\tau^2\) (argument method.tau) are available for the
inverse variance method:
DerSimonian-Laird estimator (method.tau = "DL")
Paule-Mandel estimator (method.tau = "PM")
Restricted maximum-likelihood estimator (method.tau =
"REML")
Maximum-likelihood estimator (method.tau = "ML")
Hunter-Schmidt estimator (method.tau = "HS")
Sidik-Jonkman estimator (method.tau = "SJ")
Hedges estimator (method.tau = "HE")
Empirical Bayes estimator (method.tau = "EB")
See metagen for more information on these
estimators. Note, the maximum-likelihood method is utilized for
GLMMs.
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, 578--81
DerSimonian R & Laird N (1986): Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177--88
Fleiss JL (1993): The statistical basis of meta-analysis. Statistical Methods in Medical Research, 2, 121--45
Greenland S & Robins JM (1985): Estimation of a common effect parameter from sparse follow-up data. Biometrics, 41, 55--68
Hartung J & Knapp G (2001): A refined method for the meta-analysis of controlled clinical trials with binary outcome. Statistics in Medicine, 20, 3875--89
Higgins JPT, Thompson SG, Spiegelhalter DJ (2009): A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A, 172, 137--59
Knapp G & Hartung J (2003): Improved tests for a random effects meta-regression with a single covariate. Statistics in Medicine, 22, 2693--710
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, 377--85
Pettigrew HM, Gart JJ, Thomas DG (1986): The bias and higher cumulants of the logarithm of a binomial variate. Biometrika, 73, 425--35
Robins J, Breslow N, Greenland S (1986): Estimators of the Mantel-Haenszel Variance Consistent in Both Sparse Data and Large-Strata Limiting Models. Biometrics, 42, 311--23
R<U+00FC>cker G, Schwarzer G, Carpenter J, Olkin I (2009): Why add anything to nothing? The arcsine difference as a measure of treatment effect in meta-analysis with zero cells. Statistics in Medicine, 28, 721--38
Simmonds MC, Higgins JP (2016): A general framework for the use of logistic regression models in meta-analysis. Statistical Methods in Medical Research, 25, 2858--77
StataCorp. 2011. Stata Statistical Software: Release 12. College Station, TX: StataCorp LP.
Stijnen T, Hamza TH, Ozdemir P (2010): Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29, 3046--67
Sweeting MJ, Sutton AJ, Lambert PC (2004): What to add to nothing? Use and avoidance of continuity corrections in meta-analysis of sparse data. Statistics in Medicine, 23, 1351--75
Viechtbauer W (2010): Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36, 1--48
Yusuf S, Peto R, Lewis J, Collins R, Sleight P (1985): Beta blockade during and after myocardial infarction: An overview of the randomized trials. Progress in Cardiovascular Diseases, 27, 335--71
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 meta-analysis 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)
# Meta-analysis 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 meta-analysis result using 'update.meta' function
m3 <- update(m2, sm = "OR")
summary(m3)
# Meta-analysis based on Mantel-Haenszel method (with odds ratio as
# summary measure)
#
m4 <- update(m3, method = "MH")
summary(m4)
# Meta-analysis based on Peto method (only available for odds ratio
# as summary measure)
#
m5 <- update(m3, method = "Peto")
summary(m5)
# }
# NOT RUN {
# Meta-analysis 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)
# Mixed-effects 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
# Hypergeometric-Normal) (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 Binomial-Normal)
#
m9 <- update(m6, model.glmm = "CM.AL")
summary(m9)
# Logistic regression model with (k = 70) fixed study effects
# (about 18 seconds with Intel Core i7-3667U, 2.0GHz)
#
m10 <- metabin(event.e, n.e, event.c, n.c,
data = Olkin95, method = "GLMM")
summary(m10)
# Mixed-effects logistic regression model with random study effects
# - about 50 seconds with Intel Core i7-3667U, 2.0GHz
# - several warning messages, e.g. "failure to converge, ..."
#
summary(update(m10, model.glmm = "UM.RS"))
# Conditional Hypergeometric-Normal GLMM
# - long computation time (about 12 minutes with Intel Core
# i7-3667U, 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 Binomial-Normal)
# (less than 1 second with Intel Core i7-3667U, 2.0GHz)
#
summary(update(m10, model.glmm = "CM.AL"))
}
# }
# NOT RUN {
# }
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