Calculation of fixed and random effects estimates for meta-analyses with continuous outcome data; inverse variance weighting is used for pooling.
metacont(
n.e,
mean.e,
sd.e,
n.c,
mean.c,
sd.c,
studlab,
data = NULL,
subset = NULL,
exclude = NULL,
sm = gs("smcont"),
pooledvar = gs("pooledvar"),
method.smd = gs("method.smd"),
sd.glass = gs("sd.glass"),
exact.smd = gs("exact.smd"),
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"),
method.bias = gs("method.bias"),
backtransf = gs("backtransf"),
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"),
keepdata = gs("keepdata"),
warn = gs("warn"),
control = NULL
)Number of observations in experimental group.
Estimated mean in experimental group.
Standard deviation in experimental group.
Number of observations in control group.
Estimated mean in control group.
Standard deviation in control group.
An optional vector with study labels.
An optional data frame containing the study information.
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
("MD", "SMD", or "ROM") is to be used for
pooling of studies.
A logical indicating if a pooled variance should
be used for the mean difference (sm="MD").
A character string indicating which method is
used to estimate the standardised mean difference
(sm="SMD"). Either "Hedges" for Hedges' g
(default), "Cohen" for Cohen's d, or "Glass" for
Glass' delta, can be abbreviated.
A character string indicating which standard
deviation is used in the denominator for Glass' method to
estimate the standardised mean difference. Either
"control" using the standard deviation in the control
group (sd.c) or "experimental" using the standard
deviation in the experimental group (sd.e), can be
abbreviated.
A logical indicating whether exact formulae should be used in estimation of the standardised mean difference and its standard error (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 \(\tau^2\) 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 treatment 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 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 ratio of
means (sm="ROM") should be back transformed in printouts
and plots. If TRUE (default), results will be presented as ratio
of means; otherwise log ratio of means will be shown.
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 n.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.
A logical indicating whether warnings should be printed (e.g., if studies are excluded from meta-analysis due to zero standard deviations).
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("metacont", "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.
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 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 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.
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 treatment effect for
Hartung-Knapp method (only if hakn = TRUE).
Pooling method: "Inverse".
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 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.
Calculation of fixed and random effects estimates for meta-analyses with continuous outcome data; inverse variance weighting is used for pooling.
Three different types of summary measures are available for continuous outcomes:
mean difference (argument sm = "MD")
standardised mean difference (sm = "SMD")
ratio of means (sm = "ROM")
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.
For the standardised mean difference three methods are implemented:
Hedges' g (default, method.smd = "Hedges") - see
Hedges (1981)
Cohen's d (method.smd = "Cohen") - see Cohen (1988)
Glass' delta (method.smd = "Glass") - see Glass (1976)
Hedges (1981) calculated the exact bias in Cohen's d which is a
ratio of gamma distributions with the degrees of freedom,
i.e. total sample size minus two, as argument. By default (argument
exact.smd = FALSE), an accurate approximation of this bias
provided in Hedges (1981) is utilised for Hedges' g as well as its
standard error; these approximations are also used in RevMan
5. Following Borenstein et al. (2009) these approximations are not
used in the estimation of Cohen's d. White and Thomas (2005) argued
that approximations are unnecessary with modern software and
accordingly promote to use the exact formulae; this is possible
using argument exact.smd = TRUE. For Hedges' g the exact
formulae are used to calculate the standardised mean difference as
well as the standard error; for Cohen's d the exact formula is only
used to calculate the standard error. In typical applications (with
sample sizes above 10), the differences between using the exact
formulae and the approximation will be minimal.
For Glass' delta, by default (argument sd.glass =
"control"), the standard deviation in the control group
(sd.c) is used in the denominator of the standard mean
difference. The standard deviation in the experimental group
(sd.e) can be used by specifying sd.glass =
"experimental".
Meta-analysis of ratio of means -- also called response ratios --
is described in Hedges et al. (1999) and Friedrich et al. (2008).
Calculations are conducted on the log scale and list elements
TE, TE.fixed, and TE.random contain the
logarithm of the ratio of means. In printouts and plots these
values are back transformed if argument backtransf = TRUE.
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) 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. function
print.meta will not print results for the random
effects model if comb.random = FALSE.
Borenstein M, Hedges LV, Higgins JPT, Rothstein HR (2009): Introduction to Meta-Analysis. Chichester: Wiley
Cohen J (1988): Statistical Power Analysis for the Behavioral Sciences (second ed.). Lawrence Erlbaum Associates
Cooper H & Hedges LV (1994): The Handbook of Research Synthesis. Newbury Park, CA: Russell Sage Foundation
DerSimonian R & Laird N (1986): Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177--88
Friedrich JO, Adhikari NK, Beyene J (2008): The ratio of means method as an alternative to mean differences for analyzing continuous outcome variables in meta-analysis: A simulation study. BMC Medical Research Methodology, 8, 32
Glass G (1976): Primary, secondary, and meta-analysis of research. Educational Researcher, 5, 3--8
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
Hedges LV (1981): Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational and Behavioral Statistics, 6, 107--28
Hedges LV, Gurevitch J, Curtis PS (1999): The meta-analysis of response ratios in experimental ecology. Ecology, 80, 1150--6
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
Review Manager (RevMan) [Computer program]. Version 5.3. Copenhagen: The Nordic Cochrane Centre, The Cochrane Collaboration, 2014
Viechtbauer W (2010): Conducting Meta-Analyses in R with the Metafor Package. Journal of Statistical Software, 36, 1--48
White IR, Thomas J (2005): Standardized mean differences in individually-randomized and cluster-randomized trials, with applications to meta-analysis. Clinical Trials, 2, 141--51
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 {
data(Fleiss93cont)
# Meta-analysis with Hedges' g as effect measure
#
m1 <- metacont(n.e, mean.e, sd.e, n.c, mean.c, sd.c,
data = Fleiss93cont, sm = "SMD")
m1
forest(m1)
# Use Cohen's d instead of Hedges' g as effect measure
#
update(m1, method.smd = "Cohen")
# Use Glass' delta instead of Hedges' g as effect measure
#
update(m1, method.smd = "Glass")
# Use Glass' delta based on the standard deviation in the experimental group
#
update(m1, method.smd = "Glass", sd.glass = "experimental")
# Calculate Hedges' g based on exact formulae
#
update(m1, exact.smd = TRUE)
data(amlodipine)
m2 <- metacont(n.amlo, mean.amlo, sqrt(var.amlo),
n.plac, mean.plac, sqrt(var.plac),
data = amlodipine, studlab = study)
summary(m2)
# Use pooled variance
#
summary(update(m2, pooledvar = TRUE))
# Meta-analysis of response ratios (Hedges et al., 1999)
#
data(woodyplants)
m3 <- metacont(n.elev, mean.elev, sd.elev,
n.amb, mean.amb, sd.amb,
data = woodyplants, sm = "ROM")
summary(m3)
summary(m3, backtransf = FALSE)
# }
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