Calculation of fixed and random effects estimates for metaanalyses with continuous outcome data; inverse variance weighting is used for pooling.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  metacont(n.e, mean.e, sd.e, n.c, mean.c, sd.c, studlab,
data=NULL, subset=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"), 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"))

n.e 
Number of observations in experimental group. 
mean.e 
Estimated mean in experimental group. 
sd.e 
Standard deviation in experimental group. 
n.c 
Number of observations in control group. 
mean.c 
Estimated mean in control group. 
sd.c 
Standard deviation in control group. 
studlab 
An optional vector with study labels. 
data 
An optional data frame containing the study information. 
subset 
An optional vector specifying a subset of studies to be used. 
level 
The level used to calculate confidence intervals for individual studies. 
level.comb 
The level used to calculate confidence intervals for pooled estimates. 
comb.fixed 
A logical indicating whether a fixed effect metaanalysis should be conducted. 
comb.random 
A logical indicating whether a random effects metaanalysis should be conducted. 
prediction 
A logical indicating whether a prediction interval should be printed. 
level.predict 
The level used to calculate prediction interval for a new study. 
hakn 
A logical indicating whether the method by Hartung and Knapp should be used to adjust test statistics and confidence intervals. 
method.tau 
A character string indicating which method is used
to estimate the betweenstudy variance τ^2. Either

tau.preset 
Prespecified value for the squareroot of the betweenstudy variance τ^2. 
TE.tau 
Overall treatment effect used to estimate the betweenstudy variance tausquared. 
tau.common 
A logical indicating whether tausquared should be the same across subgroups. 
method.bias 
A character string indicating which test is to be
used. Either 
backtransf 
A logical indicating whether results for ratio of
means ( 
title 
Title of metaanalysis / systematic review. 
complab 
Comparison label. 
outclab 
Outcome label. 
label.e 
Label for experimental group. 
label.c 
Label for control group. 
label.left 
Graph label on left side of forest plot. 
label.right 
Graph label on right side of forest plot. 
sm 
A character string indicating which summary measure
( 
pooledvar 
A logical indicating if a pooled variance should be
used for the mean difference ( 
method.smd 
A character string indicating which method is used
to estimate the standardised mean difference
( 
sd.glass 
A character string indicating which standard
deviation is used in the denominator for Glass' method to estimate
the standardised mean difference. Either 
exact.smd 
A logical indicating whether exact formulae should be used in estimation of the standardised mean difference and its standard error (see Details). 
byvar 
An optional vector containing grouping information (must
be of same length as 
bylab 
A character string with a label for the grouping variable. 
print.byvar 
A logical indicating whether the name of the grouping variable should be printed in front of the group labels. 
byseparator 
A character string defining the separator between label and levels of grouping variable. 
keepdata 
A logical indicating whether original data (set) should be kept in meta object. 
warn 
A logical indicating whether warnings should be printed (e.g., if studies are excluded from metaanalysis due to zero standard deviations). 
Calculation of fixed and random effects estimates for metaanalyses 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"
)
Metaanalysis of ratio of means – also called response ratios – is described in Hedges et al. (1999) and Friedrich et al. (2008).
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"
.
Calculations are conducted on the log scale for ratio of means
(sm="ROM"
). Accordingly, list elements TE
,
TE.fixed
, and TE.random
contain the logarithm of ratio
of means. In printouts and plots these values are back transformed
if argument backtransf=TRUE
.
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 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
.
The function metagen
is called internally to calculate
individual and overall treatment estimates and standard errors.
A prediction interval for 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 metacont object by only specifying
arguments which should be changed.
For the random effects, the method by Hartung and Knapp (2003) is
used to adjust test statistics and confidence intervals if argument
hakn=TRUE
.
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
τ^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.
An object of class c("metacont", "meta")
with corresponding
print
, summary
, plot
function. The object is a
list containing the following components:
n.e, mean.e, sd.e, 

n.c, mean.c, sd.c, 

studlab, sm, level, level.comb, 

comb.fixed, comb.random, 

pooledvar, method.smd, sd.glass, 

hakn, method.tau, tau.preset, TE.tau, method.bias, 

tau.common, title, complab, outclab, 

label.e, label.c, label.left, label.right, 

byvar, bylab, print.byvar, byseparator, warn 
As defined above. 
TE, seTE 
Estimated treatment effect and standard error of individual studies. 
lower, upper 
Lower and upper confidence interval limits for individual studies. 
zval, pval 
zvalue and pvalue for test of treatment effect for individual studies. 
w.fixed, w.random 
Weight of individual studies (in fixed and random effects model). 
TE.fixed, seTE.fixed 
Estimated overall treatment effect and standard error (fixed effect model). 
lower.fixed, upper.fixed 
Lower and upper confidence interval limits (fixed effect model). 
zval.fixed, pval.fixed 
zvalue and pvalue for test of overall treatment effect (fixed effect model). 
TE.random, seTE.random 
Estimated overall treatment effect and standard error (random effects model). 
lower.random, upper.random 
Lower and upper confidence interval limits (random effects model). 
zval.random, pval.random 
zvalue or tvalue and corresponding pvalue for test of overall treatment effect (random effects model). 
prediction, level.predict 
As defined above. 
seTE.predict 
Standard error utilised for prediction interval. 
lower.predict, upper.predict 
Lower and upper limits of prediction interval. 
k 
Number of studies combined in metaanalysis. 
Q 
Heterogeneity statistic. 
tau 
Squareroot of betweenstudy variance. 
se.tau 
Standard error of squareroot of betweenstudy variance. 
C 
Scaling factor utilised internally to calculate common tausquared across subgroups. 
method 
Pooling method: 
df.hakn 
Degrees of freedom for test of treatment effect for
HartungKnapp method (only if 
bylevs 
Levels of grouping variable  if 
TE.fixed.w, seTE.fixed.w 
Estimated treatment effect and
standard error in subgroups (fixed effect model)  if 
lower.fixed.w, upper.fixed.w 
Lower and upper confidence
interval limits in subgroups (fixed effect model)  if

zval.fixed.w, pval.fixed.w 
zvalue and pvalue for test of
treatment effect in subgroups (fixed effect model)  if

TE.random.w, seTE.random.w 
Estimated treatment effect and
standard error in subgroups (random effects model)  if

lower.random.w, upper.random.w 
Lower and upper confidence
interval limits in subgroups (random effects model)  if

zval.random.w, pval.random.w 
zvalue or tvalue and
corresponding pvalue for test of treatment effect in subgroups
(random effects model)  if 
w.fixed.w, w.random.w 
Weight of subgroups (in fixed and
random effects model)  if 
df.hakn.w 
Degrees of freedom for test of treatment effect for
HartungKnapp method in subgroups  if 
n.harmonic.mean.w 
Harmonic mean of number of observations in
subgroups (for back transformation of FreemanTukey Double arcsine
transformation)  if 
n.e.w 
Number of observations in experimental group in
subgroups  if 
n.c.w 
Number of observations in control group in subgroups 
if 
k.w 
Number of studies combined within subgroups  if

k.all.w 
Number of all studies in subgroups  if 
Q.w 
Heterogeneity statistics within subgroups  if

Q.w.fixed 
Overall within subgroups heterogeneity statistic Q
(based on fixed effect model)  if 
Q.w.random 
Overall within subgroups heterogeneity statistic Q
(based on random effects model)  if 
df.Q.w 
Degrees of freedom for test of overall within
subgroups heterogeneity  if 
Q.b.fixed 
Overall between subgroups heterogeneity statistic Q
(based on fixed effect model)  if 
Q.b.random 
Overall between subgroups heterogeneity statistic
Q (based on random effects model)  if 
df.Q.b 
Degrees of freedom for test of overall between
subgroups heterogeneity  if 
tau.w 
Squareroot of betweenstudy variance within subgroups
 if 
C.w 
Scaling factor utilised internally to calculate common
tausquared across subgroups  if 
H.w 
Heterogeneity statistic H within subgroups  if

lower.H.w, upper.H.w 
Lower and upper confidence limti for
heterogeneity statistic H within subgroups  if 
I2.w 
Heterogeneity statistic I2 within subgroups  if

lower.I2.w, upper.I2.w 
Lower and upper confidence limti for
heterogeneity statistic I2 within subgroups  if 
keepdata 
As defined above. 
data 
Original data (set) used in function call (if

subset 
Information on subset of original data used in
metaanalysis (if 
call 
Function call. 
version 
Version of R package meta used to create object. 
Guido Schwarzer sc@imbi.unifreiburg.de
Borenstein et al. (2009), Introduction to MetaAnalysis, 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), Metaanalysis 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 metaanalysis: A simulation study. BMC Med Res Methodol, 8, 32.
Glass G (1976), Primary, secondary, and metaanalysis of research. Educational Researcher, 5, 3–8.
Hartung J & Knapp G (2001), On tests of the overall treatment effect in metaanalysis with normally distributed responses. Statistics in Medicine, 20, 1771–82. doi: 10.1002/sim.791 .
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 metaanalysis of response ratios in experimental ecology. Ecology, 80, 1150–6.
Higgins JPT, Thompson SG, Spiegelhalter DJ (2009), A reevaluation of randomeffects metaanalysis. Journal of the Royal Statistical Society: Series A, 172, 137–59.
Knapp G & Hartung J (2003), Improved Tests for a Random Effects Metaregression with a Single Covariate. Statistics in Medicine, 22, 2693–710, doi: 10.1002/sim.1482 .
Paule RC & Mandel J (1982), Consensus values and weighting factors. Journal of Research of the National Bureau of Standards, 87, 377–85.
Review Manager (RevMan) [Computer program]. Version 5.3. Copenhagen: The Nordic Cochrane Centre, The Cochrane Collaboration, 2014.
Viechtbauer W (2010), Conducting MetaAnalyses in R with the Metafor Package. Journal of Statistical Software, 36, 1–48.
White IR, Thomas J (2005), Standardized mean differences in individuallyrandomized and clusterrandomized trials, with applications to metaanalysis. Clinical Trials, 2, 141–51.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  data(Fleiss93cont)
meta1 < metacont(n.e, mean.e, sd.e, n.c, mean.c, sd.c, data=Fleiss93cont, sm="SMD")
meta1
forest(meta1)
meta2 < metacont(Fleiss93cont$n.e, Fleiss93cont$mean.e,
Fleiss93cont$sd.e,
Fleiss93cont$n.c, Fleiss93cont$mean.c,
Fleiss93cont$sd.c,
sm="SMD")
meta2
data(amlodipine)
meta3 < metacont(n.amlo, mean.amlo, sqrt(var.amlo),
n.plac, mean.plac, sqrt(var.plac),
data=amlodipine, studlab=study)
summary(meta3)
# Use pooled variance
#
meta4 < metacont(n.amlo, mean.amlo, sqrt(var.amlo),
n.plac, mean.plac, sqrt(var.plac),
data=amlodipine, studlab=study,
pooledvar=TRUE)
summary(meta4)
# Use Cohen's d instead of Hedges' g as effect measure
#
meta5 < metacont(n.e, mean.e, sd.e, n.c, mean.c, sd.c, data=Fleiss93cont,
sm="SMD", method.smd="Cohen")
meta5
# Use Glass' delta instead of Hedges' g as effect measure
#
meta6 < metacont(n.e, mean.e, sd.e, n.c, mean.c, sd.c, data=Fleiss93cont,
sm="SMD", method.smd="Glass")
meta6
# Use Glass' delta based on the standard deviation in the experimental group
#
meta7 < metacont(n.e, mean.e, sd.e, n.c, mean.c, sd.c, data=Fleiss93cont,
sm="SMD", method.smd="Glass", sd.glass="experimental")
meta7
# Calculate Hedges' g based on exact formulae
#
update(meta1, exact.smd=TRUE)
#
# Metaanalysis of response ratios (Hedges et al., 1999)
#
data(woodyplants)
meta8 < metacont(n.elev, mean.elev, sd.elev,
n.amb, mean.amb, sd.amb,
data=woodyplants, sm="ROM")
summary(meta8)
summary(meta8, backtransf=FALSE)

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