Fixed effect and random effects meta-analysis based on estimates (e.g. log hazard ratios) and their standard errors. The inverse variance method is used for pooling.

```
metagen(TE, seTE, studlab, data = NULL, subset = NULL,
exclude = NULL, sm = "", 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"), null.effect = 0,
method.bias = gs("method.bias"), n.e = NULL, n.c = NULL, pval, df,
lower, upper, level.ci = 0.95, median, q1, q3, min, max, approx.TE,
approx.seTE, backtransf = gs("backtransf"), pscale = 1,
irscale = 1, irunit = "person-years", 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)
```

TE

Estimate of treatment effect, e.g., log hazard ratio or risk difference.

seTE

Standard error of treatment estimate.

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 underlying summary measure,
e.g., `"RD"`

, `"RR"`

, `"OR"`

, `"ASD"`

,
`"HR"`

, `"MD"`

, `"SMD"`

, or `"ROM"`

.

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 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.

null.effect

A numeric value specifying the effect under the null hypothesis.

method.bias

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

, `"linreg"`

, or `"mm"`

,
can be abbreviated. See function `metabias`

n.e

Number of observations in experimental group.

n.c

Number of observations in control group.

pval

P-value (used to estimate the standard error).

df

Degrees of freedom (used in test or to construct confidence interval).

lower

Lower limit of confidence interval (used to estimate the standard error).

upper

Upper limit of confidence interval (used to estimate the standard error).

level.ci

Level of confidence interval.

median

Median (used to estimate the treatment effect and standard error).

q1

First quartile (used to estimate the treatment effect and standard error).

q3

Third quartile (used to estimate the treatment effect and standard error).

min

Minimum (used to estimate the treatment effect and standard error).

max

Maximum (used to estimate the treatment effect and standard error).

approx.TE

Approximation method to estimate treatment estimate (see Details).

approx.seTE

Approximation method to estimate standard error (see Details).

backtransf

A logical indicating whether results should be
back transformed in printouts and plots. If ```
backtransf =
TRUE
```

(default), results for `sm = "OR"`

are printed as odds
ratios rather than log odds ratios and results for ```
sm =
"ZCOR"
```

are printed as correlations rather than Fisher's z
transformed correlations, for example.

pscale

A numeric giving scaling factor for printing of
single event probabilities or risk differences, i.e. if argument
`sm`

is equal to `"PLOGIT"`

, `"PLN"`

,
`"PRAW"`

, `"PAS"`

, `"PFT"`

, or `"RD"`

.

irscale

A numeric defining a scaling factor for printing of
single incidence rates or incidence rate differences, i.e. if
argument `sm`

is equal to `"IR"`

, `"IRLN"`

,
`"IRS"`

, `"IRFT"`

, or `"IRD"`

.

irunit

A character specifying the time unit used to calculate rates, e.g. person-years.

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 `TE`

).

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 errors).

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("metagen", "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.

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.

As defined above.

Number of studies combined in meta-analysis.

Heterogeneity statistic.

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.

Pooling method: `"Inverse"`

.

Degrees of freedom for test of treatment effect for
Hartung-Knapp method (only if `hakn = TRUE`

).

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`

.

Harmonic mean of number of observations in
subgroups (for back transformation of Freeman-Tukey Double
arcsine transformation) - if `byvar`

is not missing.

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 limti 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.

This function provides the *generic inverse variance method*
for meta-analysis which requires treatment estimates and their
standard errors (Borenstein et al., 2010). The method is useful,
e.g., for pooling of survival data (using log hazard ratio and
standard errors as input). Arguments `TE`

and `seTE`

can
be used to provide treatment estimates and standard errors
directly. However, it is possible to derive these quantities from
other information.

For several arguments defaults settings are utilised (see
assignments with `gs`

under **Usage**). These
defaults can be changed using `settings.meta`

.

Furthermore, R function `update.meta`

can be used to
rerun a meta-analysis with different settings.

Missing treatment estimates can be derived from

confidence limits provided by arguments

`lower`

and`upper`

;median, interquartile range and range (arguments

`median`

,`q1`

,`q3`

,`min`

, and`max`

);median and interquartile range (arguments

`median`

,`q1`

and`q3`

);median and range (arguments

`median`

,`min`

and`max`

).

By default, missing treatment estimates are replaced successively
using these method, e.g., confidence limits are utilised before
interquartile ranges. Argument `approx.TE`

can be used to
overwrite this default for each individual study:

Use treatment estimate directly (entry

`""`

in argument`approx.TE`

);confidence limits (

`"ci"`

in argument`approx.TE`

);median, interquartile range and range (

`"iqr.range"`

);median and interquartile range (

`"iqr"`

);median and range (

`"range"`

).

Missing standard errors can be derived from

p-value provided by arguments

`pval`

and (optional)`df`

;confidence limits (arguments

`lower`

,`upper`

, and (optional)`df`

);sample size, median, interquartile range and range (arguments

`n.e`

and / or`n.c`

,`median`

,`q1`

,`q3`

,`min`

, and`max`

);sample size, median and interquartile range (arguments

`n.e`

and / or`n.c`

,`median`

,`q1`

and`q3`

);sample size, median and range (arguments

`n.e`

and / or`n.c`

,`median`

,`min`

and`max`

).

`df`

is provided. Furthermore, argument `level.ci`

can be
used to provide the level of the confidence interval. For median,
interquartile range and range, depending on the sample size,
equation (12) or (13) in Wan et al. (2014) is used to approximate
the standard error. Similarly, equations (15) / (16) and (7) / (9)
in Wan et al. (2014) are used if median and interquartile range or
range, respectively, are provided. The sample size of individual
studies must be provided with arguments `n.e`

and / or
`n.c`

. The total sample size is calculated as `n.e`

+
`n.c`

if both arguments are provided.By default, missing standard errors are replaced successively using
these method, e.g., p-value before confidence limits before
interquartile range and range. Argument `approx.seTE`

can be
used to overwrite this default for each individual study:

Use standard error directly (entry

`""`

in argument`approx.seTE`

);p-value (

`"pval"`

in argument`approx.seTE`

);confidence limits (

`"ci"`

);median, interquartile range and range (

`"iqr.range"`

);median and interquartile range (

`"iqr"`

);median and range (

`"range"`

).

The following methods are available to estimate the between-study variance \(\tau^2\).

Argument |
Method |

`method.tau = "DL"` |
DerSimonian-Laird estimator (DerSimonian and Laird, 1986) |

`method.tau = "PM"` |
Paule-Mandel estimator (Paule and Mandel, 1982) |

`method.tau = "REML"` |
Restricted maximum-likelihood estimator (Viechtbauer, 2005) |

`method.tau = "ML"` |
Maximum-likelihood estimator (Viechtbauer, 2005) |

`method.tau = "HS"` |
Hunter-Schmidt estimator (Hunter and Schmidt, 2015) |

`method.tau = "SJ"` |
Sidik-Jonkman estimator (Sidik and Jonkman, 2005) |

`method.tau = "HE"` |
Hedges estimator (Hedges and Olkin, 1985) |

Historically, the DerSimonian-Laird method was the de facto
standard to estimate the between-study variance \(\tau^2\) and is
still the default in many software packages including Review
Manager 5 (RevMan 5) and R package meta. However, its role
has been challenged and especially the Paule-Mandel and REML
estimators have been recommended (Veroniki et al.,
2016). Accordingly, the following R command can be used to use the
Paule-Mandel estimator in all meta-analyses of the R session:
`settings.meta(method.tau = "PM")`

The DerSimonian-Laird and Paule-Mandel estimators are implemented
in R package meta. The other estimators are available if R
package metafor (Viechtbauer 2010) is installed by internally
calling R function `rma.uni`

.

Hartung and Knapp (2001a,b) proposed an alternative method for
random effects meta-analysis based on a refined variance estimator
for the treatment estimate. Simulation studies (Hartung and Knapp,
2001a,b; IntHout et al., 2014; Langen et al., 2018) 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 treatment effect of 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.

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`

. For example, functions
`print.meta`

and `forest.meta`

will not
show results for the random effects model if ```
comb.random =
FALSE
```

.

Argument `pscale`

can be used to rescale single proportions or
risk differences, e.g. `pscale = 1000`

means that proportions
are expressed as events per 1000 observations. This is useful in
situations with (very) low event probabilities.

Argument `irscale`

can be used to rescale single rates or rate
differences, e.g. `irscale = 1000`

means that rates are
expressed as events per 1000 time units, e.g. person-years. This is
useful in situations with (very) low rates. Argument `irunit`

can be used to specify the time unit used in individual studies
(default: "person-years"). This information is printed in summaries
and forest plots if argument `irscale`

is not equal to 1.

Default settings for `comb.fixed`

, `comb.random`

,
`pscale`

, `irscale`

, `irunit`

and several other
arguments can be set for the whole R session using
`settings.meta`

.

Borenstein M, Hedges LV, Higgins JP, Rothstein HR (2010):
A basic introduction to fixed-effect and random-effects models for
meta-analysis.
*Research Synthesis Methods*,
**1**, 97--111

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

Hedges LV & Olkin I (1985):
*Statistical methods for meta-analysis*.
San Diego, CA: Academic Press

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

Hunter JE & Schmidt FL (2015):
*Methods of Meta-Analysis: Correcting Error and Bias in
Research Findings* (Third edition).
Thousand Oaks, CA: Sage

Hartung J, Knapp G (2001a):
On tests of the overall treatment effect in meta-analysis with
normally distributed responses.
*Statistics in Medicine*,
**20**, 1771--82

Hartung J, Knapp G (2001b):
A refined method for the meta-analysis of controlled clinical
trials with binary outcome.
*Statistics in Medicine*,
**20**, 3875--89

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

Langan D, Higgins JPT, Jackson D, Bowden J, Veroniki AA,
Kontopantelis E, et al. (2018):
A comparison of heterogeneity variance estimators in simulated
random-effects meta-analyses.
*Research Synthesis Methods*

Morris CN (1983):
Parametric empirical Bayes inference: Theory and applications (with
discussion).
*Journal of the American Statistical Association*
**78**, 47--65

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

Sidik K & Jonkman JN (2005):
Simple heterogeneity variance estimation for meta-analysis.
*Journal of the Royal Statistical Society: Series C (Applied Statistics)*,
**54**, 367--84

Veroniki AA, Jackson D, Viechtbauer W, Bender R, Bowden J, Knapp G,
et al. (2016):
Methods to estimate the between-study variance and its uncertainty
in meta-analysis.
*Research Synthesis Methods*,
**7**, 55--79

Viechtbauer W (2005):
Bias and efficiency of meta-analytic variance estimators in the
random-effects model.
*Journal of Educational and Behavioral Statistics*,
**30**, 261--93

Viechtbauer W (2010):
Conducting Meta-Analyses in R with the metafor Package.
*Journal of Statistical Software*,
**36**, 1--48

Wan X, Wang W, Liu J, Tong T (2014):
Estimating the sample mean and standard deviation from the sample
size, median, range and/or interquartile range.
*BMC Medical Research Methodology*,
**14**, 135

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(Fleiss93)
m1 <- metabin(event.e, n.e, event.c, n.c,
data = Fleiss93, sm = "RR", method = "I")
m1
# Identical results by using the generic inverse variance method
metagen(m1$TE, m1$seTE, sm = "RR")
#
forest(metagen(m1$TE, m1$seTE, sm = "RR"))
# Meta-analysis with prespecified between-study variance
#
summary(metagen(m1$TE, m1$seTE, sm = "RR", tau.preset = sqrt(0.1)))
# Meta-analysis of survival data:
#
logHR <- log(c(0.95, 1.5))
selogHR <- c(0.25, 0.35)
metagen(logHR, selogHR, sm = "HR")
# Paule-Mandel method to estimate between-study variance for data
# from Paule & Mandel (1982)
#
average <- c(27.044, 26.022, 26.340, 26.787, 26.796)
variance <- c(0.003, 0.076, 0.464, 0.003, 0.014)
#
summary(metagen(average, sqrt(variance), sm = "MD", method.tau = "PM"))
# Conduct meta-analysis using hazard ratios and 95% confidence intervals
#
# Data from Steurer et al. (2006), Analysis 1.1 Overall survival
# https://www.cochranelibrary.com/cdsr/doi/10.1002/14651858.CD004270.pub2/abstract
#
study <- c("FCG on CLL 1996", "Leporrier 2001", "Rai 2000", "Robak 2000")
HR <- c(0.55, 0.92, 0.79, 1.18)
lower.HR <- c(0.28, 0.79, 0.59, 0.64)
upper.HR <- c(1.09, 1.08, 1.05, 2.17)
#
# Input must be log hazard ratios, not hazard ratios
#
metagen(log(HR), lower = log(lower.HR), upper = log(upper.HR),
studlab = study, sm = "HR")
# }
```

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