meta: metamean – R documentation

Become an expert in R — Interactive courses, Cheat Sheets, certificates and more!

metamean

Meta-analysis of single means

Description

Calculation of an overall mean from studies reporting a single mean using the inverse variance method for pooling; inverse variance weighting is used for pooling.

Usage

metamean(
  n,
  mean,
  sd,
  studlab,
  data = NULL,
  subset = NULL,
  exclude = NULL,
  median,
  q1,
  q3,
  min,
  max,
  method.mean = "Luo",
  method.sd = "Shi",
  approx.mean,
  approx.sd,
  sm = gs("smmean"),
  method.ci = gs("method.ci.cont"),
  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 = NA,
  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"),
  warn = gs("warn"),
  control = NULL
)

Arguments

`n`	Number of observations.
`mean`	Estimated mean.
`sd`	Standard deviation.
`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.
`median`	Median (used to estimate the mean and standard deviation).
`q1`	First quartile (used to estimate the mean and standard deviation).
`q3`	Third quartile (used to estimate the mean and standard deviation).
`min`	Minimum (used to estimate the mean and standard deviation).
`max`	Maximum (used to estimate the mean and standard deviation).
`method.mean`	A character string indicating which method to use to approximate the mean from the median and other statistics (see Details).
`method.sd`	A character string indicating which method to use to approximate the standard deviation from sample size, median, interquartile range and range (see Details).
`approx.mean`	Approximation method to estimate means (see Details).
`approx.sd`	Approximation method to estimate standard deviations (see Details).
`sm`	A character string indicating which summary measure (`"MRAW"` or `"MLN"`) is to be used for pooling of studies.
`method.ci`	A character string indicating which method is used to calculate confidence intervals for individual studies, 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.
`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 should be back transformed in printouts and plots for `sm = "MLN"`. If TRUE (default), results will be presented as means; otherwise logarithm of means 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 `n`).
`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 deviations).
`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 single means to calculate an overall mean; inverse variance weighting is used for pooling. The following transformations of means are implemented to calculate an overall mean:

Raw, i.e. untransformed, means (sm = "MRAW", default)
Log transformed means (sm = "MLN")

Note, you should use R function metacont to compare means of pairwise comparisons instead of using metamean for each treatment arm separately which will break randomisation in randomised controlled trials.

Calculations are conducted on the log scale if sm = "ROM". Accordingly, list elements TE, TE.fixed, and TE.random contain the logarithm of means. In printouts and plots these values are back transformed if argument backtransf = TRUE.

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.

Approximate means from sample sizes, medians and other statistics

Missing means can be derived from

sample size, median, interquartile range and range (arguments n, median, q1, q3, min, and max),
sample size, median and interquartile range (arguments n, median, q1, and q3), or
sample size, median and range (arguments n, median, min, and max).

By default, methods described in Luo et al. (2018) are utilized (argument method.mean = "Luo"):

equation (15) if sample size, median, interquartile range and range are available,
equation (11) if sample size, median and interquartile range are available,
equation (7) if sample size, median and range are available.

Instead the methods described in Wan et al. (2014) are used if argument method.mean = "Wan"):

equation (10) if sample size, median, interquartile range and range are available,
equation (14) if sample size, median and interquartile range are available,
equation (2) if sample size, median and range are available.

By default, missing means are replaced successively using interquartile ranges and ranges (if available), interquartile ranges (if available) and finally ranges. Argument approx.mean can be used to overwrite this behaviour for each individual study and treatment arm:

use means directly (entry "" in argument approx.mean);
median, interquartile range and range ("iqr.range");
median and interquartile range ("iqr");
median and range ("range").

Approximate standard deviations from sample sizes, medians and other statistics

Missing standard deviations can be derived from

sample size, median, interquartile range and range (arguments n, median, q1, q3, min, and max),
sample size, median and interquartile range (arguments n, median, q1 and q3), or
sample size, median and range (arguments n, median, min and max).

Wan et al. (2014) describe methods to estimate the standard deviation from the sample size, median and additional statistics. Shi et al. (2020) provide an improved estimate of the standard deviation if the interquartile range and range are available in addition to the sample size and median. Accordingly, equation (11) in Shi et al. (2020) is the default (argument method.sd = "Shi"), if the median, interquartile range and range are provided. The method by Wan et al. (2014) is used if argument method.sd = "Wan" and, depending on the sample size, either equation (12) or (13) is used. If only the interquartile range or range is available, equations (15) / (16) and (7) / (9) in Wan et al. (2014) are used, respectively.

By default, missing standard deviations are replaced successively using these method, i.e., interquartile ranges and ranges are used before interquartile ranges before ranges. Argument approx.sd can be used to overwrite this default for each individual study and treatment arms:

sample size, median, interquartile range and range ("iqr.range");
sample size, median and interquartile range ("iqr");
sample size, median and range ("range").

Confidence intervals for individual studies

For untransformed means (argument sm = "MRAW"), the confidence interval for individual studies can be based on the

standard normal distribution (method.ci = "z", default), or
t-distribution (method.ci = "t").

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) 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("metamean", "meta") with corresponding print, summary, and forest functions. The object is a list containing the following components:

`n, mean, sd,`	As defined above.
`studlab, exclude, sm, method.ci,`	As defined above.
`median, q1, q3, min, max,`	As defined above.
`method.mean, method.sd,`	As defined above.
`approx.mean, approx.sd,`	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, method.bias,`	As defined above.
`tau.common, title, complab, outclab,`	As defined above.
`byvar, bylab, print.byvar, byseparator, warn`	As defined above.
`TE, seTE`	Estimated effect (mean or log mean) and standard error of individual studies.
`lower, upper`	Lower and upper confidence interval limits for individual studies.
`statistic, pval`	Statistic 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 effect (mean or log mean) and standard error (fixed effect model).
`lower.fixed, upper.fixed`	Lower and upper confidence interval limits (fixed effect model).
`statistic.fixed, pval.fixed`	Statistic and p-value for test of overall treatment effect (fixed effect model).
`TE.random, seTE.random`	Estimated overall effect (mean or log mean) and standard error (random effects model).
`lower.random, upper.random`	Lower and upper confidence interval limits (random effects model).
`statistic.random, pval.random`	Statistic and 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.
`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`	Pooling method: `"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 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`	Statistics and p-values for test of treatment 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`	Statistics and p-values 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 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

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

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 <- metamean(rep(100, 3), 1:3, rep(1, 3))
m1

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

# With test for overall mean equal to 2
#
update(m1, null.effect = 2)
update(m2, null.effect = 2)

# Print results without back-transformation
#
update(m1, backtransf = FALSE)
update(m2, backtransf = FALSE)
update(m1, null.effect = 2, backtransf = FALSE)
update(m2, null.effect = 2, backtransf = FALSE)