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metarate

Meta-analysis of single incidence rates

Description

Calculation of an overall incidence rate from studies reporting a single incidence rate. 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

metarate(
  event,
  time,
  studlab,
  data = NULL,
  subset = NULL,
  exclude = NULL,
  method = "Inverse",
  sm = gs("smrate"),
  incr = gs("incr"),
  allincr = gs("allincr"),
  addincr = gs("addincr"),
  level = gs("level"),
  level.comb = gs("level.comb"),
  comb.fixed = gs("comb.fixed"),
  comb.random = gs("comb.random"),
  overall = comb.fixed | comb.random,
  overall.hetstat = comb.fixed | comb.random,
  hakn = gs("hakn"),
  adhoc.hakn = gs("adhoc.hakn"),
  method.tau,
  method.tau.ci = gs("method.tau.ci"),
  tau.preset = NULL,
  TE.tau = NULL,
  tau.common = gs("tau.common"),
  prediction = gs("prediction"),
  level.predict = gs("level.predict"),
  null.effect = NA,
  method.bias = gs("method.bias"),
  backtransf = gs("backtransf"),
  irscale = 1,
  irunit = "person-years",
  text.fixed = gs("text.fixed"),
  text.random = gs("text.random"),
  text.predict = gs("text.predict"),
  text.w.fixed = gs("text.w.fixed"),
  text.w.random = gs("text.w.random"),
  title = gs("title"),
  complab = gs("complab"),
  outclab = "",
  byvar,
  bylab,
  print.byvar = gs("print.byvar"),
  byseparator = gs("byseparator"),
  keepdata = gs("keepdata"),
  warn = gs("warn"),
  control = NULL,
  ...
)

Arguments

`event`	Number of events.
`time`	Person time at risk.
`studlab`	An optional vector with study labels.
`data`	An optional data frame containing the study information, i.e., event and time.
`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.
`method`	A character string indicating which method is to be used for pooling of studies. One of `"Inverse"` and `"GLMM"`, can be abbreviated.
`sm`	A character string indicating which summary measure (`"IR"`, `"IRLN"`, `"IRS"`, or `"IRFT"`) is to be used for pooling of studies, see Details.
`incr`	A numeric which is added to the event number of studies with zero events, i.e., studies with an incidence rate of 0.
`allincr`	A logical indicating if `incr` is considered for all studies if at least one study has zero events. If FALSE (default), `incr` is considered only in studies with zero events.
`addincr`	A logical indicating if `incr` is used for all studies irrespective of number of events.
`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.
`overall`	A logical indicating whether overall summaries should be reported. This argument is useful in a meta-analysis with subgroups if overall results should not be reported.
`overall.hetstat`	A logical value indicating whether to print heterogeneity measures for overall treatment comparisons. This argument is useful in a meta-analysis with subgroups if heterogeneity statistics should only be printed on subgroup level.
`hakn`	A logical indicating whether the method by Hartung and Knapp should be used to adjust test statistics and confidence intervals.
`adhoc.hakn`	A character string indicating whether an ad hoc variance correction should be applied in the case of an arbitrarily small Hartung-Knapp variance estimate, see Details.
`method.tau`	A character string indicating which method is used to estimate the between-study variance τ^2 and its square root τ. Either `"DL"`, `"PM"`, `"REML"`, `"ML"`, `"HS"`, `"SJ"`, `"HE"`, or `"EB"`, can be abbreviated.
`method.tau.ci`	A character string indicating which method is used to estimate the confidence interval of τ^2 and τ. Either `"QP"`, `"BJ"`, or `"J"`, or `""`, 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 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 `"Begg"`, `"Egger"`, or `"Thompson"`, can be abbreviated. See function `metabias`.
`backtransf`	A logical indicating whether results for transformed rates (argument `sm != "IR"`) should be back transformed in printouts and plots. If TRUE (default), results will be presented as incidence rates; otherwise transformed rates will be shown.
`irscale`	A numeric defining a scaling factor for printing of rates.
`irunit`	A character string specifying the time unit used to calculate rates, e.g. person-years.
`text.fixed`	A character string used in printouts and forest plot to label the pooled fixed effect estimate.
`text.random`	A character string used in printouts and forest plot to label the pooled random effects estimate.
`text.predict`	A character string used in printouts and forest plot to label the prediction interval.
`text.w.fixed`	A character string used to label weights of fixed effect model.
`text.w.random`	A character string used to label weights of random effects model.
`title`	Title of meta-analysis / systematic review.
`complab`	Comparison label.
`outclab`	Outcome label.
`byvar`	An optional vector containing grouping information (must be of same length as `event`).
`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 the addition of `incr` to studies with zero events should result in a warning.
`control`	An optional list to control the iterative process to estimate the between-study variance τ^2. This argument is passed on to `rma.uni` or `rma.glmm`, respectively.
`...`	Additional arguments passed on to `rma.glmm` function.

Details

This function provides methods for fixed effect and random effects meta-analysis of single incidence rates to calculate an overall rate. Note, you should use R function metainc to compare incidence rates of pairwise comparisons instead of using metarate for each treatment arm separately which will break randomisation in randomised controlled trials.

The following transformations of incidence rates are implemented to calculate an overall rate:

Log transformation (sm = "IRLN", default)
Square root transformation (sm = "IRS")
Freeman-Tukey Double arcsine transformation (sm = "IRFT")
No transformation (sm = "IR")

By default (argument method = "Inverse"), the inverse variance method (Borenstein et al., 2010) is used for pooling by calling metagen internally. A random intercept Poisson regression model (Stijnen et al., 2010) can be utilised instead with argument method = "GLMM" which calls the rma.glmm function from R package metafor.

Default settings are utilised for several arguments (assignments using gs function). These defaults can be changed for the current R session using the settings.meta function.

Furthermore, R function update.meta can be used to rerun a meta-analysis with different settings.

Continuity correction

If the summary measure (argument sm) is equal to "IR" or "IRLN", a continuity correction is applied if any study has zero events, i.e., an incidence rate of 0.

By default, 0.5 is used as continuity correction (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 the Freeman-Tukey (Freeman & Tukey, 1950) and square root transformation as well as GLMMs no continuity correction is used.

Estimation of between-study variance

The following methods to estimate the between-study variance τ^2 are available for the inverse variance method:

DerSimonian-Laird estimator (method.tau = "DL")
Paule-Mandel estimator (method.tau = "PM")
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")

See metagen for more information on these estimators. Note, the maximum-likelihood method is utilized for GLMMs.

Confidence interval for the between-study variance

The following methods to calculate a confidence interval for τ^2 and τ are available.

Argument	Method
`method.tau.ci = "J"`	Method by Jackson
`method.tau.ci = "BJ"`	Method by Biggerstaff and Jackson
`method.tau.ci = "QP"`	Q-Profile method

See metagen for more information on these methods. For GLMMs, no confidence intervals for τ^2 and τ are calculated. Likewise, no confidence intervals for τ^2 and τ are calculated if method.tau.ci = "".

Hartung-Knapp method

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; Langan et al., 2019) show improved coverage probabilities compared to the classic random effects method.

In rare settings with very homogeneous treatment estimates, the Hartung-Knapp variance estimate can be arbitrarily small resulting in a very narrow confidence interval (Knapp and Hartung, 2003; Wiksten et al., 2016). In such cases, an ad hoc variance correction has been proposed by utilising the variance estimate from the classic random effects model with the HK method (Knapp and Hartung, 2003; IQWiQ, 2020). An alternative approach is to use the wider confidence interval of classic fixed or random effects meta-analysis and the HK method (Wiksten et al., 2016; Jackson et al., 2017).

Argument adhoc.hakn can be used to choose the ad hoc method:

Argument	Ad hoc method
`adhoc.hakn = ""`	not used
`adhoc.hakn = "se"`	use variance correction if HK standard error is smaller
	than standard error from classic random effects
	meta-analysis (Knapp and Hartung, 2003)
`adhoc.hakn = "iqwig6"`	use variance correction if HK confidence interval
	is narrower than CI from classic random effects model
	with DerSimonian-Laird estimator (IQWiG, 2020)
`adhoc.hakn = "ci"`	use wider confidence interval of classic random effects
	and HK meta-analysis
	(Hybrid method 2 in Jackson et al., 2017)

For GLMMs, a method similar to Knapp and Hartung (2003) is implemented, see description of argument tdist in rma.glmm, and the ad hoc variance correction is not available.

Prediction interval

A prediction interval for the proportion in 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.

Subgroup analysis

Argument byvar can be used to conduct subgroup analysis for a categorical covariate. The metareg function can be used instead for more than one categorical covariate or continuous covariates.

Specify the null hypothesis of test for an overall effect

Argument null.effect can be used to specify the rate used under the null hypothesis in a test for an overall effect.

By default (null.effect = NA), no hypothesis test is conducted as it is unclear which value is a sensible choice for the data at hand. An overall rate of 2, for example, could be tested by setting argument null.effect = 2.

Note, all tests for an overall effect are two-sided with the alternative hypothesis that the effect is unequal to null.effect.

Exclusion of studies from meta-analysis

Arguments subset and exclude can be used to exclude studies from the meta-analysis. Studies are removed completely from the meta-analysis using argument subset, while excluded studies are shown in printouts and forest plots using argument exclude (see Examples in metagen). Meta-analysis results are the same for both arguments.

Presentation of meta-analysis results

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.

Argument irscale can be used to rescale rates, 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.

Value

An object of class c("metarate", "meta") with corresponding print, summary, and forest functions. The object is a list containing the following components:

`event, n, studlab, exclude,`	As defined above.
`sm, incr, allincr, addincr, method.ci,`	As defined above.
`level, level.comb,`	As defined above.
`comb.fixed, comb.random,`	As defined above.
`overall, overall.hetstat,`	As defined above.
`hakn, adhoc.hakn, method.tau, method.tau.ci,`	As defined above.
`tau.preset, TE.tau, null.hypothesis,`	As defined above.
`method.bias, tau.common, title, complab, outclab,`	As defined above.
`byvar, bylab, print.byvar, byseparator, warn`	As defined above.
`TE, seTE`	Estimated (un)transformed incidence rate and its standard error for 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 (un)transformed incidence rate and standard error (fixed effect model).
`lower.fixed, upper.fixed`	Lower and upper confidence interval limits (fixed effect model).
`statistic.fixed, pval.fixed`	z-value and p-value for test of overall effect (fixed effect model).
`TE.random, seTE.random`	Estimated overall (un)transformed incidence rate and standard error (random effects model).
`lower.random, upper.random`	Lower and upper confidence interval limits (random effects model).
`statistic.random, pval.random`	z-value or t-value and corresponding p-value for test of overall 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.
`pval.Q`	P-value of heterogeneity test.
`Q.LRT`	Heterogeneity statistic for likelihood-ratio test (only if `method = "GLMM"`).
`df.Q.LRT`	Degrees of freedom for likelihood-ratio test
`pval.Q.LRT`	P-value of likelihood-ratio test.
`tau2`	Between-study variance τ^2.
`se.tau2`	Standard error of τ^2.
`lower.tau2, upper.tau2`	Lower and upper limit of confidence interval for τ^2.
`tau`	Square-root of between-study variance τ.
`lower.tau, upper.tau`	Lower and upper limit of confidence interval for τ.
`H`	Heterogeneity statistic H.
`lower.H, upper.H`	Lower and upper confidence limit for heterogeneity statistic H.
`I2`	Heterogeneity statistic I^2.
`lower.I2, upper.I2`	Lower and upper confidence limit for heterogeneity statistic I^2.
`Rb`	Heterogeneity statistic R_b.
`lower.Rb, upper.Rb`	Lower and upper confidence limit for heterogeneity statistic R_b.
`method`	A character string indicating method used for pooling: `"Inverse"`
`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.
`statistic.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.
`statistic.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.w`	Number of events in subgroups - if `byvar` is not missing.
`n.w`	Number of observations 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.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.
`pval.Q.w.fixed`	P-value of within subgroups heterogeneity statistic Q (based on fixed effect model) - if `byvar` is not missing.
`pval.Q.w.random`	P-value of within subgroups heterogeneity statistic Q (based on random effects model) - 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.
`pval.Q.b.fixed`	P-value of between subgroups heterogeneity statistic Q (based on fixed effect model) - if `byvar` is not missing.
`pval.Q.b.random`	P-value of between subgroups heterogeneity statistic Q (based on random effects model) - if `byvar` is not missing.
`tau.w`	Square-root of between-study variance within 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 limit for heterogeneity statistic H within subgroups - if `byvar` is not missing.
`I2.w`	Heterogeneity statistic I^2 within subgroups - if `byvar` is not missing.
`lower.I2.w, upper.I2.w`	Lower and upper confidence limit for heterogeneity statistic I^2 within subgroups - if `byvar` is not missing.
`incr.event`	Increment added to number of events.
`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

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

Freeman MF & Tukey JW (1950): Transformations related to the angular and the square root. Annals of Mathematical Statistics, 21, 607–11

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

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

IQWiG (2020): General Methods: Version 6.0. https://www.iqwig.de/en/about-us/methods/methods-paper/

Jackson D, Law M, Rücker G, Schwarzer G (2017): The Hartung-Knapp modification for random-effects meta-analysis: A useful refinement but are there any residual concerns? Statistics in Medicine, 36, 3923–34

Langan D, Higgins JPT, Jackson D, Bowden J, Veroniki AA, Kontopantelis E, et al. (2019): A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses. Research Synthesis Methods, 10, 83–98

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

Wiksten A, Rücker G, Schwarzer G (2016): Hartung-Knapp method is not always conservative compared with fixed-effect meta-analysis. Statistics in Medicine, 35, 2503–15

Examples

# Apply various meta-analysis methods to estimate incidence rates
#
m1 <- metarate(4:1, c(10, 20, 30, 40))
m2 <- update(m1, sm = "IR")
m3 <- update(m1, sm = "IRS")
m4 <- update(m1, sm = "IRFT")
#
m1
m2
m3
m4
#
forest(m1)
forest(m1, irscale = 100)
forest(m1, irscale = 100, irunit = "person-days")
forest(m1, backtransf = FALSE)
## Not run: 
forest(m2)
forest(m3)
forest(m4)

## End(Not run)

m5 <- metarate(40:37, c(100, 200, 300, 400), sm = "IRFT")
m5