Calculation of common and random effects estimates for meta-analyses
with binary outcome data.
The following measures of treatment effect are available (Rücker et
al., 2009):
Risk ratio (sm = "RR")
Odds ratio (sm = "OR")
Risk difference (sm = "RD")
Arcsine difference (sm = "ASD")
Diagnostic Odds ratio (sm = "DOR")
Vaccine efficacy or vaccine effectiveness (sm = "VE")
Note, mathematically, odds ratios and diagnostic odds ratios are
identical, however, the labels in printouts and figures
differ. Furthermore, log risk ratio (logRR) and log vaccine ratio
(logVR) are mathematical identical, however, back-transformed
results differ as vaccine efficacy or effectiveness is defined as
VE = 100 * (1 - RR).
A three-level random effects meta-analysis model (Van den Noortgate
et al., 2013) is utilized if argument cluster is used and at
least one cluster provides more than one estimate. Internally,
rma.mv is called to conduct the analysis and
weights.rma.mv with argument type =
"rowsum" is used to calculate random effects weights.
Default settings are utilised for several arguments (assignments
using gs function). These defaults can be changed for
the current R session using the settings.meta
function.
Furthermore, R function update.meta can be used to
rerun a meta-analysis with different settings.
Meta-analysis method
By default, both common effect (also called common effect) and
random effects models are considered (see arguments common
and random). If method is "MH" (default), the
Mantel-Haenszel method (Greenland & Robins, 1985; Robins et al.,
1986) is used to calculate the common 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 (Yusuf et al., 1985); if
method is "SSW", the sample size method is used for
pooling (Bakbergenuly et al., 2020).
While the Mantel-Haenszel and Peto method are defined under the
common 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 (DerSimonian and Laird,
1986). Accordingly, 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 common 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) |
| (model.glmm = "CM.AL", i.e., Conditional
Model - Approximate Likelihood) |
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:
common effect model (with fixed treatment effect) and random
effects model (with random treatment effects).
Continuity correction
Three approaches are available to apply a continuity correction:
Only studies with a zero cell count (method.incr =
"only0")
All studies if at least one study has a zero cell count
(method.incr = "if0all")
All studies irrespective of zero cell counts
(method.incr = "all")
By default, a continuity correction is only applied to studies with
a zero cell count (method.incr = "only0"). This method
showed the best performance for the odds ratio in a simulation
study under the random effects model (Weber et al., 2020).
The continuity correction method is used both to calculate
individual study results with confidence limits and to conduct
meta-analysis based on the inverse variance method. For the risk
difference, the method is only considered to calculate standard
errors and confidence limits. For Peto method and GLMMs no
continuity correction is used in the meta-analysis. Furthermore,
the continuity correction is ignored for individual studies for the
Peto method.
For studies with a zero cell count, by default, 0.5 (argument
incr) is added to all cell frequencies for the odds ratio or
only the number of events for the risk ratio (argument
RR.Cochrane = FALSE, default). The increment is added to all
cell frequencies for the risk ratio if argument RR.Cochrane =
TRUE. For the risk difference, incr is only added to all
cell frequencies to calculate the standard error. Finally, a
treatment arm continuity correction is used if incr = "TACC"
(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 = TRUE.
For the Mantel-Haenszel method, by default (if MH.exact is
FALSE), incr is added to 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.
Subgroup analysis
Argument subgroup 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.
Exclusion of studies from meta-analysis
Arguments subset and exclude can be used to exclude
studies from the meta-analysis. Studies are removed completely from
the meta-analysis using argument subset, while excluded
studies are shown in printouts and forest plots using argument
exclude (see Examples in metagen).
Meta-analysis results are the same for both arguments.
Presentation of meta-analysis results
Internally, both common effect and random effects models are
calculated regardless of values choosen for arguments
common and 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 random = FALSE. However, all functions in R
package meta will adequately consider the values for
common and random. E.g. function
print.meta will not print results for the random
effects model if random = FALSE.
A prediction interval will only be shown if prediction =
TRUE.