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metacor

Meta-analysis of correlations

Description

Calculation of fixed and random effects estimates for meta-analyses with correlations; inverse variance weighting is used for pooling.

Usage

metacor(
  cor,
  n,
  studlab,
  data = NULL,
  subset = NULL,
  exclude = NULL,
  sm = gs("smcor"),
  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 = gs("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 = 0,
  method.bias = gs("method.bias"),
  backtransf = gs("backtransf"),
  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"),
  control = NULL
)

Arguments

`cor`	Correlation.
`n`	Number of observations.
`studlab`	An optional vector with study labels.
`data`	An optional data frame containing the study information, i.e., cor and n.
`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 (`"ZCOR"` or `"COR"`) is to be used for pooling of studies.
`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 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 Fisher's z transformed correlations (`sm = "ZCOR"`) should be back transformed in printouts and plots. If TRUE (default), results will be presented as correlations; otherwise Fisher's z transformed correlations will be shown.
`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.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.
`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`.

Details

Fixed effect and random effects meta-analysis of correlations based either on Fisher's z transformation of correlations (sm = "ZCOR") or direct combination of (untransformed) correlations (sm = "COR") (see Cooper et al., p264-5 and p273-4). Only few statisticians would advocate the use of untransformed correlations unless sample sizes are very large (see Cooper et al., p265). The artificial example given below shows that the smallest study gets the largest weight if correlations are combined directly because the correlation is closest to 1.

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.

Estimation of between-study variance

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

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.

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. No confidence intervals for τ^2 and τ are calculated if method.tau.ci = "".

Hartung-Knapp method

Hartung and Knapp (2001) and Knapp and Hartung (2003) proposed an alternative method for random effects meta-analysis based on a refined variance estimator for the treatment estimate. Simulation studies (Hartung and Knapp, 2001; 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)

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.

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. functions print.meta and forest.meta will not print results for the random effects model if comb.random = FALSE.

Value

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

`cor, n, studlab, exclude,`	As defined above.
`sm, level, level.comb,`	As defined above.
`comb.fixed, comb.random,`	As defined above.
`hakn, adhoc.hakn, method.tau, method.tau.ci,`	As defined above.
`tau.preset, TE.tau, method.bias,`	As defined above.
`method.bias, tau.common, title, complab, outclab,`	As defined above.
`byvar, bylab, print.byvar, byseparator`	As defined above.
`TE, seTE`	Either Fisher's z transformation of correlations (`sm = "ZCOR"`) or correlations (`sm="COR"`) for individual studies.
`lower, upper`	Lower and upper confidence interval limits for individual studies.
`zval, pval`	z-value and p-value for test of effect in individual studies.
`w.fixed, w.random`	Weight of individual studies (in fixed and random effects model).
`TE.fixed, seTE.fixed`	Estimated overall effect (Fisher's z transformation of correlation or correlation) 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 effect (Fisher's z transformation of correlation or correlation) 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.
`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.
`df.hakn`	Degrees of freedom for test of effect for Hartung-Knapp method (only if `hakn = TRUE`).
`method`	Pooling method: `"Inverse"`.
`bylevs`	Levels of grouping variable - if `byvar` is not missing.
`TE.fixed.w, seTE.fixed.w`	Estimated 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 effect in subgroups (fixed effect model) - if `byvar` is not missing.
`TE.random.w, seTE.random.w`	Estimated 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 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 effect for Hartung-Knapp method in subgroups - if `byvar` is not missing and `hakn = TRUE`.
`n.e.w`	Number of observations in experimental group in subgroups - 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.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.
`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`).
`call`	Function call.
`version`	Version of R package meta used to create object.

Note

The function metagen is called internally to calculate individual and overall treatment estimates and standard errors.

Author(s)

Guido Schwarzer sc@imbi.uni-freiburg.de

References

Cooper H, Hedges LV, Valentine JC (2009): The Handbook of Research Synthesis and Meta-Analysis, 2nd Edition. New York: Russell Sage Foundation

DerSimonian R & Laird N (1986): Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177–88

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

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

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

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

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

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

m1 <- metacor(c(0.85, 0.7, 0.95), c(20, 40, 10))

# Print correlations (back transformed from Fisher's z
# transformation)
#
m1

# Print Fisher's z transformed correlations 
#
print(m1, backtransf = FALSE)

# Forest plot with back transformed correlations
#
forest(m1)

# Forest plot with Fisher's z transformed correlations
#
forest(m1, backtransf = FALSE)

m2 <- update(m1, sm = "cor")
m2

# Identical forest plots (as back transformation is the identity
# transformation)
# forest(m2)
# forest(m2, backtransf = FALSE)