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"),
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)
```

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.

exclude

An optional vector specifying studies to exclude from meta-analysis, however, to include in printouts and forest plots.

sm

A character string indicating which summary measure
(`"MD"`

, `"SMD"`

, or `"ROM"`

) is to be used for
pooling of studies.

pooledvar

A logical indicating if a pooled variance should
be used for the mean difference (`sm="MD"`

).

method.smd

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.

sd.glass

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.

exact.smd

A logical indicating whether exact formulae should be used in estimation of the standardised mean difference and its standard error (see Details).

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 meta-analysis should be conducted.

comb.random

A logical indicating whether a random effects meta-analysis should be conducted.

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 between-study variance \(\tau^2\). Either
`"DL"`

, `"PM"`

, `"REML"`

, `"ML"`

,
`"HS"`

, `"SJ"`

, `"HE"`

, or `"EB"`

, can be
abbreviated.

tau.preset

Prespecified value for the square-root of the between-study variance \(\tau^2\).

TE.tau

Overall treatment effect used to estimate the between-study variance tau-squared.

tau.common

A logical indicating whether tau-squared should be the same across subgroups.

prediction

A logical indicating whether a prediction interval should be printed.

level.predict

The level used to calculate prediction interval for a new study.

method.bias

A character string indicating which test is to
be used. Either `"rank"`

, `"linreg"`

, or `"mm"`

,
can be abbreviated. See function `metabias`

backtransf

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

Title of meta-analysis / 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.

byvar

An optional vector containing grouping information
(must be of same length as `n.e`

).

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 meta-analysis due to zero standard deviations).

control

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.

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.

Square-root of between-study variance.

Standard error of square-root of between-study variance.

Scaling factor utilised internally to calculate common tau-squared across subgroups.

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.

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`

).

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"`

)

Meta-analysis 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 the 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
meta-analysis 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 DerSimonian-Laird estimate (1986) is used in the random effects
model if `method.tau = "DL"`

. The iterative Paule-Mandel
method (1982) to estimate the between-study 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 between-study variance
\(\tau^2\) (argument `method.tau`

) are also available:

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"`

)

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
between-study variance.

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

Knapp G & Hartung J (2003):
Improved tests for a random effects meta-regression with a single
covariate.
*Statistics in Medicine*,
**22**, 2693--710

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 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

```
# 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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