Calculation of fixed and random effects estimates for meta-analyses with correlations; inverse variance weighting is used for pooling.
metacor(
cor,
n,
studlab,
data = NULL,
subset = NULL,
exclude = NULL,
sm = gs("smcor"),
level = gs("level"),
level.comb = gs("level.comb"),
comb.fixed = gs("comb.fixed"),
comb.random = gs("comb.random"),
hakn = gs("hakn"),
method.tau = gs("method.tau"),
method.tau.ci = if (method.tau == "DL") "J" else "QP",
tau.preset = NULL,
TE.tau = NULL,
tau.common = gs("tau.common"),
prediction = gs("prediction"),
level.predict = gs("level.predict"),
null.effect = 0,
method.bias = gs("method.bias"),
backtransf = gs("backtransf"),
title = gs("title"),
complab = gs("complab"),
outclab = "",
byvar,
bylab,
print.byvar = gs("print.byvar"),
byseparator = gs("byseparator"),
keepdata = gs("keepdata"),
control = NULL
)Correlation.
Number of observations.
An optional vector with study labels.
An optional data frame containing the study information, i.e., cor and n.
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 summary measure
("ZCOR" or "COR") is to be used for pooling of
studies.
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 \(\tau^2\) and its
square root \(\tau\). Either "DL", "PM",
"REML", "ML", "HS", "SJ",
"HE", or "EB", can be abbreviated.
A character string indicating which method is
used to estimate the confidence interval of \(\tau^2\) and
\(\tau\). Either "QP", "BJ", or "J", or
"", can be abbreviated.
Prespecified value for the square root of the between-study variance \(\tau^2\).
Overall effect used to estimate the between-study variance tau-squared.
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 numeric value specifying the effect under the null hypothesis.
A character string indicating which test is to
be used. Either "rank", "linreg", or "mm",
can be abbreviated. See function metabias
A logical indicating whether results for Fisher's
z transformed correlations (sm = "ZCOR") should be back
transformed in printouts and plots. If TRUE (default), results
will be presented as correlations; otherwise Fisher's z
transformed correlations will be shown.
Title of meta-analysis / systematic review.
Comparison label.
Outcome label.
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 original data (set) should be kept in meta object.
An optional list to control the iterative process to
estimate the between-study variance \(\tau^2\). This argument
is passed on to rma.uni.
An object of class c("metacor", "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.
Either Fisher's z transformation of correlations
(sm = "ZCOR") or correlations (sm="COR") for
individual studies.
Lower and upper confidence interval limits for individual studies.
z-value and p-value for test of effect in individual studies.
Weight of individual studies (in fixed and random effects model).
Estimated overall effect (Fisher's z transformation of correlation or correlation) and standard error (fixed effect model).
Lower and upper confidence interval limits (fixed effect model).
z-value and p-value for test of overall effect (fixed effect model).
Estimated overall effect (Fisher's z transformation of correlation or correlation) 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 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.
Between-study variance \(\tau^2\).
Standard error of \(\tau^2\).
Lower and upper limit of confidence interval for \(\tau^2\).
Square-root of between-study variance \(\tau\).
Lower and upper limit of confidence interval for \(\tau\).
Heterogeneity statistic H.
Lower and upper confidence limit for heterogeneity statistic H.
Heterogeneity statistic I\(^2\).
Lower and upper confidence limit for heterogeneity statistic I\(^2\).
Heterogeneity statistic R\(_b\).
Lower and upper confidence limit for heterogeneity statistic R\(_b\).
Degrees of freedom for test of effect for
Hartung-Knapp method (only if hakn = TRUE).
Pooling method: "Inverse".
Levels of grouping variable - if byvar is not
missing.
Estimated 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
effect in subgroups (fixed effect model) - if byvar is not
missing.
Estimated 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 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 effect for
Hartung-Knapp method in subgroups - if byvar is not
missing and hakn = TRUE.
Number of observations in experimental 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.
Heterogeneity statistic H within subgroups - if
byvar is not missing.
Lower and upper confidence limit for
heterogeneity statistic H within subgroups - if byvar is
not missing.
Heterogeneity statistic I\(^2\) within subgroups - if
byvar is not missing.
Lower and upper confidence limit for
heterogeneity statistic I\(^2\) 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).
Function call.
Version of R package meta used to create object.
Fixed effect and random effects meta-analysis of correlations based
either on Fisher's z transformation of correlations (sm =
"ZCOR") or direct combination of (untransformed) correlations
(sm = "COR") (see Cooper et al., p264-5 and p273-4). Only
few statisticians would advocate the use of untransformed
correlations unless sample sizes are very large (see Cooper et al.,
p265). The artificial example given below shows that the smallest
study gets the largest weight if correlations are combined directly
because the correlation is closest to 1.
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.
The following methods to estimate the between-study variance \(\tau^2\) are available:
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")
metagen for more information on these
estimators.The following methods to calculate a confidence interval for \(\tau^2\) and \(\tau\) are available.
| Argument | Method |
method.tau.ci = "J" |
Method by Jackson |
method.tau.ci = "BJ" |
Method by Biggerstaff and Jackson |
metagen for more information on these methods. No
confidence intervals for \(\tau^2\) and \(\tau\) are calculated
if method.tau.ci = "".Hartung and Knapp (2001) and Knapp and Hartung (2003) proposed an
alternative method for random effects meta-analysis based on a
refined variance estimator for the treatment estimate. Simulation
studies (Hartung and Knapp, 2001; IntHout et al., 2014; Langan et
al., 2019) show improved coverage probabilities compared to the
classic random effects method. However, in rare settings with very
homogeneous treatment estimates, the Hartung-Knapp method can be
anti-conservative (Wiksten et al., 2016). The Hartung-Knapp method
is used if argument hakn = TRUE.
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.
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.
Internally, both fixed effect and random effects models are
calculated regardless of values choosen 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. functions
print.meta and forest.meta will not
print results for the random effects model if comb.random =
FALSE.
Cooper H, Hedges LV, Valentine JC (2009): The Handbook of Research Synthesis and Meta-Analysis, 2nd Edition. New York: Russell Sage Foundation
DerSimonian R & Laird N (1986): Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177--88
Hartung J & Knapp G (2001): On tests of the overall treatment effect in meta-analysis with normally distributed responses. Statistics in Medicine, 20, 1771--82
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
IntHout J, Ioannidis JPA, Borm GF (2014): The Hartung-Knapp-Sidik-Jonkman method for random effects meta-analysis is straightforward and considerably outperforms the standard DerSimonian-Laird method. BMC Medical Research Methodology, 14, 25
Knapp G & Hartung J (2003): Improved tests for a random effects meta-regression with a single covariate. Statistics in Medicine, 22, 2693--710
Langan D, Higgins JPT, Jackson D, Bowden J, Veroniki AA, Kontopantelis E, et al. (2019): A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses. Research Synthesis Methods, 10, 83--98
Viechtbauer W (2010): Conducting Meta-Analyses in R with the Metafor Package. Journal of Statistical Software, 36, 1--48
Wiksten A, R<U+00FC>cker G, Schwarzer G (2016): Hartung-Knapp method is not always conservative compared with fixed-effect meta-analysis. Statistics in Medicine, 35, 2503--15
# NOT RUN {
m1 <- metacor(c(0.85, 0.7, 0.95), c(20, 40, 10))
# Print correlations (back transformed from Fisher's z
# transformation)
#
m1
# Print Fisher's z transformed correlations
#
print(m1, backtransf = FALSE)
# Forest plot with back transformed correlations
#
forest(m1)
# Forest plot with Fisher's z transformed correlations
#
forest(m1, backtransf = FALSE)
m2 <- update(m1, sm = "cor")
m2
# Identical forest plots (as back transformation is the identity
# transformation)
# forest(m2)
# forest(m2, backtransf = FALSE)
# }
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