Meta-analysis of incidence rates

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Description

Calculation of fixed effect and random effects estimates (incidence rate ratio or incidence rate difference) for meta-analyses with event counts. Mantel-Haenszel, 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

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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 "MH", "Inverse", "Cochran", or "GLMM" can be abbreviated.

sm

A character string indicating which summary measure ("IRR" or "IRD") is to be used for pooling of studies, see Details.

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 incr is added to each cell frequency of all studies if at least one study has a zero cell count. If FALSE (default), incr is added only to each cell frequency of studies with a zero cell count.

addincr

A logical indicating if incr is added to each cell frequency of all studies irrespective of zero cell counts.

model.glmm

A character string indicating which GLMM should be used. One of "UM.FS", "UM.RS", and "CM.EL", 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.

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 between-study variance τ^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 τ^2.

TE.tau

Overall treatment effect used to estimate the between-study variance τ^2.

tau.common

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

method.bias

A character string indicating which test for funnel plot asymmetry is to be used. Either "linreg" or "rank", can be abbreviated. See function metabias

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 (sm="IRR") should be back transformed in printouts and plots. If TRUE (default), results will be presented as incidence rate ratios; otherwise log incidence rate ratios 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 event.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 incr is added to studies with zero cell frequencies).

...

Additional arguments passed on to rma.glmm function.

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 Mantel-Haenszel 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 (Bayne-Jones, 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 meta-analysis 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 meta-analysis 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)

  • Mixed-effects Poisson regression model with random study effects

  • [] (model.glmm = "UM.RS", i.e., Unconditional Model - Random Study effects)

  • Generalised linear mixed model (conditional Poisson-Normal)

  • [] (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 meta-analysis based on the inverse variance method. For Mantel-Haenszel 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 meta-analysis 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 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 τ^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.

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

z-value and p-value 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

z-value and p-value 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

z-value or t-value and corresponding p-value 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 meta-analysis.

Q

Heterogeneity statistic Q.

df.Q

Degrees of freedom for heterogeneity statistic.

tau

Square-root of between-study variance.

se.tau

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

C

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

sparse

Logical flag indicating if any study included in meta-analysis has any zero cell frequencies.

incr.event

Increment added to number of events.

df.hakn

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

bylevs

Levels of grouping variable - if byvar is not missing.

TE.fixed.w, seTE.fixed.w

Estimated treatment effect and standard error in subgroups (fixed effect model) - if byvar is not missing.

lower.fixed.w, upper.fixed.w

Lower and upper confidence interval limits in subgroups (fixed effect model) - if byvar is not missing.

zval.fixed.w, pval.fixed.w

z-value and p-value for test of treatment effect in subgroups (fixed effect model) - if byvar is not missing.

TE.random.w, seTE.random.w

Estimated treatment effect and standard error in subgroups (random effects model) - if byvar is not missing.

lower.random.w, upper.random.w

Lower and upper confidence interval limits in subgroups (random effects model) - if byvar is not missing.

zval.random.w, pval.random.w

z-value or t-value and corresponding p-value for test of treatment effect in subgroups (random effects model) - if byvar is not missing.

w.fixed.w, w.random.w

Weight of subgroups (in fixed and random effects model) - if byvar is not missing.

df.hakn.w

Degrees of freedom for test of treatment effect for Hartung-Knapp method in subgroups - if byvar is not missing and hakn=TRUE.

n.harmonic.mean.w

Harmonic mean of number of observations in subgroups (for back transformation of Freeman-Tukey Double arcsine transformation) - if byvar is not missing.

event.e.w

Number of events in experimental group in subgroups - if byvar is not missing.

time.e.w

Total person time in subgroups (experimental group) - if byvar is not missing.

n.e.w

Number of observations in experimental group in subgroups - if byvar is not missing.

event.c.w

Number of events in control group in subgroups - if byvar is not missing.

time.c.w

Total person time in subgroups (control group) - if byvar is not missing.

n.c.w

Number of observations in control group in subgroups - if byvar is not missing.

k.w

Number of studies combined within subgroups - if byvar is not missing.

k.all.w

Number of all studies in subgroups - if byvar is not missing.

Q.w

Heterogeneity statistics within subgroups - if byvar is not missing.

Q.w.fixed

Overall within subgroups heterogeneity statistic Q (based on fixed effect model) - if byvar is not missing.

Q.w.random

Overall within subgroups heterogeneity statistic Q (based on random effects model) - if byvar is not missing (only calculated if argument tau.common is TRUE).

df.Q.w

Degrees of freedom for test of overall within subgroups heterogeneity - if byvar is not missing.

Q.b.fixed

Overall between subgroups heterogeneity statistic Q (based on fixed effect model) - if byvar is not missing.

Q.b.random

Overall between subgroups heterogeneity statistic Q (based on random effects model) - if byvar is not missing.

df.Q.b

Degrees of freedom for test of overall between subgroups heterogeneity - if byvar is not missing.

tau.w

Square-root of between-study variance within subgroups - if byvar is not missing.

C.w

Scaling factor utilised internally to calculate common tau-squared across subgroups - if byvar is not missing.

H.w

Heterogeneity statistic H within subgroups - if byvar is not missing.

lower.H.w, upper.H.w

Lower and upper confidence limti for heterogeneity statistic H within subgroups - if byvar is not missing.

I2.w

Heterogeneity statistic I2 within subgroups - if byvar is not missing.

lower.I2.w, upper.I2.w

Lower and upper confidence limti for heterogeneity statistic I2 within subgroups - if byvar is not missing.

keepdata

As defined above.

data

Original data (set) used in function call (if keepdata=TRUE).

subset

Information on subset of original data used in meta-analysis (if keepdata=TRUE).

.glmm.fixed

GLMM object generated by call of rma.glmm function (fixed effect model).

.glmm.random

GLMM object generated by call of rma.glmm function (random effects model).

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.uni-freiburg.de

References

Bayne-Jones S et al. (1964), Smoking and Health: Report of the Advisory Committee to the Surgeon General of the United States. U-23 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), Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177–188.

Greenland S & Robins JM (1985), Estimation of a common effect parameter from sparse follow-up data. Biometrics, 41, 55–68.

Hartung J & Knapp G (2001), A Refined Method for the Meta-analysis of Controlled Clinical Trials with Binary Outcome. Statistics in Medicine, 20, 3875–89.

Higgins JPT, Thompson SG, Spiegelhalter DJ (2009), A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A, 172, 137–159.

Knapp G & Hartung J (2003), Improved Tests for a Random Effects Meta-regression 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 meta-analysis 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 Meta-Analyses in R with the Metafor Package. Journal of Statistical Software, 36, 1–48.

See Also

metabin, update.meta, print.meta

Examples

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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 meta-analysis 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)


#
# Meta-analysis 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
 #
 # Mixed-effects Poisson regression model (random study effects)
 #
 update(m4, model.glmm = "UM.RS", nAGQ = 1)
 #
 # Generalised linear mixed model (conditional Poisson-Normal)
 #
 update(m4, model.glmm = "CM.EL")

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