Metaanalysis of incidence rates
Description
Calculation of fixed effect and random effects estimates (incidence
rate ratio or incidence rate difference) for metaanalyses with
event counts. MantelHaenszel, Cochran, inverse variance method, and
generalised linear mixed model (GLMM) are available for pooling. For
GLMMs, the rma.glmm
function from R package
metafor (Viechtbauer 2010) is called internally.
Usage
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  metainc(event.e, time.e, event.c, time.c, studlab,
data=NULL, subset=NULL, method="MH",
sm=.settings$sminc,
incr=.settings$incr, allincr=.settings$allincr,
addincr=.settings$addincr,
model.glmm = "UM.FS",
level=.settings$level, level.comb=.settings$level.comb,
comb.fixed=.settings$comb.fixed, comb.random=.settings$comb.random,
hakn=.settings$hakn,
method.tau=
ifelse(!is.na(charmatch(tolower(method), "glmm", nomatch = NA)),
"ML", .settings$method.tau),
tau.preset=NULL, TE.tau=NULL,
tau.common=.settings$tau.common,
prediction=.settings$prediction, level.predict=.settings$level.predict,
method.bias=.settings$method.bias,
n.e=NULL, n.c=NULL,
backtransf=.settings$backtransf,
title=.settings$title, complab=.settings$complab, outclab="",
label.e=.settings$label.e, label.c=.settings$label.c,
label.left=.settings$label.left, label.right=.settings$label.right,
byvar, bylab, print.byvar=.settings$print.byvar,
byseparator = .settings$byseparator,
keepdata=.settings$keepdata,
warn=.settings$warn,
...)

Arguments
event.e 
Number of events in experimental group. 
time.e 
Person time at risk in experimental group. 
event.c 
Number of events in control group. 
time.c 
Person time at risk in control group. 
studlab 
An optional vector with study labels. 
data 
An optional data frame containing the study information, i.e., event.e, time.e, event.c, and time.c. 
subset 
An optional vector specifying a subset of studies to be used. 
method 
A character string indicating which method is to be
used for pooling of studies. One of 
sm 
A character string indicating which summary measure
( 
incr 
A numerical value which is added to each cell frequency for studies with a zero cell count, see Details. 
allincr 
A logical indicating if 
addincr 
A logical indicating if 
model.glmm 
A character string indicating which GLMM should be
used. One of 
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 τ^2. 
tau.common 
A logical indicating whether tausquared should be the same across subgroups. 
method.bias 
A character string indicating which test for
funnel plot asymmetry is to be used. Either 
n.e 
Number of observations in experimental group (optional). 
n.c 
Number of observations in control group (optional). 
backtransf 
A logical indicating whether results for incidence
rate ratio ( 
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. 
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 
... 
Additional arguments passed on to

Details
Treatment estimates and standard errors are calculated for each study. The following measures of treatment effect are available:
Incidence Rate Ratio (
sm="IRR"
)Incidence Rate Difference (
sm="IRD"
)
For several arguments defaults settings are utilised (assignments
with .settings$
). 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
.
By default, both fixed effect and random effects models are
considered (see arguments comb.fixed
and
comb.random
). If method
is "MH"
(default), the
MantelHaenszel method is used to calculate the fixed effect
estimate (Greenland & Robbins, 1985); if method
is
"Inverse"
, inverse variance weighting is used for pooling; if
method
is "Cochran"
, the Cochran method is used for
pooling (BayneJones, 1964, Chapter 8).
A distinctive and frequently overlooked advantage of incidence rates
is that individual patient data (IPD) can be extracted from count
data. Accordingly, statistical methods for IPD, i.e., generalised
linear mixed models, can be utilised in a metaanalysis of incidence
rate ratios (Stijnen et al., 2010). These methods are available
(argument method = "GLMM"
) by calling the
rma.glmm
function from R package
metafor internally. Three different GLMMs are available for
metaanalysis of incidence rate ratios using argument
model.glmm
(which corresponds to argument model
in the
rma.glmm
function):
Poisson regression model with fixed study effects (default)
[] (
model.glmm = "UM.FS"
, i.e., Unconditional Model  Fixed Study effects)Mixedeffects Poisson regression model with random study effects
[] (
model.glmm = "UM.RS"
, i.e., Unconditional Model  Random Study effects)Generalised linear mixed model (conditional PoissonNormal)
[] (
model.glmm = "CM.EL"
, i.e., Conditional Model  Exact Likelihood)
Details on these three GLMMs as well as additional arguments which
can be provided using argument '...
' in metainc
are
described in rma.glmm
where you can also find
information on the iterative algorithms used for estimation. Note,
regardless of which value is used for argument model.glmm
,
results for two different GLMMs are calculated: fixed effect model
(with fixed treatment effect) and random effects model (with random
treatment effects).
For studies with a zero cell count, by default, 0.5 is added to all
cell frequencies of these studies (argument incr
). This
continuity correction is used both to calculate individual study
results with confidence limits and to conduct metaanalysis based on
the inverse variance method. For MantelHaenszel method, Cochran
method, and GLMMs, nothing is added to zero cell counts.
Accordingly, estimates for these methods are not defined if the
number of events is zero in all studies either in the experimental
or control group.
Argument byvar
can be used to conduct subgroup analysis for
all methods but GLMMs. Instead use the metareg
function for GLMMs which can also be used for continuous covariates.
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 metainc 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.
Value
An object of class c("metainc", "meta")
with corresponding
print
, summary
, plot
function. The object is a
list containing the following components:
event.e, time.e, event.c, time.c, studlab, 

sm, method, incr, allincr, addincr, model.glmm, warn, 

level, level.comb, comb.fixed, comb.random, 

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 
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 Q. 
df.Q 
Degrees of freedom for 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. 
sparse 
Logical flag indicating if any study included in metaanalysis has any zero cell frequencies. 
incr.event 
Increment added to number of events. 
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 
event.e.w 
Number of events in experimental group in subgroups
 if 
time.e.w 
Total person time in subgroups (experimental group)
 if 
n.e.w 
Number of observations in experimental group in
subgroups  if 
event.c.w 
Number of events in control group in subgroups  if

time.c.w 
Total person time in subgroups (control group)  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 
.glmm.fixed 
GLMM object generated by call of

.glmm.random 
GLMM object generated by call of

call 
Function call. 
version 
Version of R package meta used to create object. 
version.metafor 
Version of R package metafor used for GLMMs. 
Author(s)
Guido Schwarzer sc@imbi.unifreiburg.de
References
BayneJones S et al. (1964), Smoking and Health: Report of the Advisory Committee to the Surgeon General of the United States. U23 Department of Health, Education, and Welfare. Public Health Service Publication No. 1103. http://profiles.nlm.nih.gov/ps/retrieve/ResourceMetadata/NNBBMQ
DerSimonian R & Laird N (1986), Metaanalysis in clinical trials. Controlled Clinical Trials, 7, 177–188.
Greenland S & Robins JM (1985), Estimation of a common effect parameter from sparse followup data. Biometrics, 41, 55–68.
Hartung J & Knapp G (2001), A Refined Method for the Metaanalysis of Controlled Clinical Trials with Binary Outcome. Statistics in Medicine, 20, 3875–89.
Higgins JPT, Thompson SG, Spiegelhalter DJ (2009), A reevaluation of randomeffects metaanalysis. Journal of the Royal Statistical Society: Series A, 172, 137–159.
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, 377–385.
Stijnen T, Hamza TH, Ozdemir P (2010), Random effects metaanalysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29, 3046–67.
Viechtbauer W (2010), Conducting MetaAnalyses in R with the Metafor Package. Journal of Statistical Software, 36, 1–48.
See Also
metabin
, update.meta
, print.meta
Examples
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 58 59 60 61 62 63 64  data(smoking)
m1 < metainc(d.smokers, py.smokers,
d.nonsmokers, py.nonsmokers,
data=smoking, studlab=study)
print(m1, digits=2)
m2 < metainc(d.smokers, py.smokers,
d.nonsmokers, py.nonsmokers,
data=smoking, studlab=study,
method="Cochran")
print(m2, digits=2)
data(lungcancer)
m3 < metainc(d.smokers, py.smokers,
d.nonsmokers, py.nonsmokers,
data=lungcancer, studlab=study)
print(m3, digits=2)
# Redo Cochran metaanalysis with inflated standard errors
#
# All cause mortality
#
TEa < log( (smoking$d.smokers/smoking$py.smokers) /
(smoking$d.nonsmokers/smoking$py.nonsmokers)
)
seTEa < sqrt(1/smoking$d.smokers +
1/smoking$d.nonsmokers + 2.5/smoking$d.nonsmokers)
#
metagen(TEa, seTEa, sm="IRR", studlab=smoking$study)
# Lung cancer mortality
#
TEl < log( (lungcancer$d.smokers/lungcancer$py.smokers) /
(lungcancer$d.nonsmokers/lungcancer$py.nonsmokers)
)
seTEl < sqrt(1/lungcancer$d.smokers +
1/lungcancer$d.nonsmokers + 2.25/lungcancer$d.nonsmokers)
#
metagen(TEl, seTEl, sm="IRR", studlab=lungcancer$study)
#
# Metaanalysis using generalised linear mixed models
# (only if R packages 'metafor' and 'lme4' are available)
#
if (suppressMessages(require(metafor, quietly = TRUE, warn = FALSE)) &
require(lme4, quietly = TRUE))
#
# Poisson regression model (fixed study effects)
#
m4 < metainc(d.smokers, py.smokers, d.nonsmokers, py.nonsmokers,
data = smoking, studlab = study, method = "GLMM")
m4
#
# Mixedeffects Poisson regression model (random study effects)
#
update(m4, model.glmm = "UM.RS", nAGQ = 1)
#
# Generalised linear mixed model (conditional PoissonNormal)
#
update(m4, model.glmm = "CM.EL")

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