meta (version 4.9-6)

metagen: Generic inverse variance meta-analysis

Description

Fixed effect and random effects meta-analysis based on estimates (e.g. log hazard ratios) and their standard errors. The inverse variance method is used for pooling.

Usage

metagen(TE, seTE, studlab, data = NULL, subset = NULL,
exclude = NULL, sm = "", level = gs("level"),
level.comb = gs("level.comb"), comb.fixed = gs("comb.fixed"),
comb.random = gs("comb.random"), hakn = gs("hakn"),
method.tau = gs("method.tau"), 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"), n.e = NULL, n.c = NULL, pval, df,
lower, upper, level.ci = 0.95, median, q1, q3, min, max, approx.TE,
approx.seTE, backtransf = gs("backtransf"), pscale = 1,
irscale = 1, irunit = "person-years", title = gs("title"),
complab = gs("complab"), outclab = "", label.e = gs("label.e"),
label.c = gs("label.c"), label.left = gs("label.left"),
label.right = gs("label.right"), byvar, bylab,
print.byvar = gs("print.byvar"), byseparator = gs("byseparator"),
keepdata = gs("keepdata"), warn = gs("warn"), control = NULL)

Arguments

TE

Estimate of treatment effect, e.g., log hazard ratio or risk difference.

seTE

Standard error of treatment estimate.

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.

sm

A character string indicating underlying summary measure, e.g., "RD", "RR", "OR", "ASD", "HR", "MD", "SMD", or "ROM".

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.

hakn

A logical indicating whether 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 $$\tau^2$$. 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 $$\tau^2$$.

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 "rank", "linreg", or "mm", can be abbreviated. See function metabias

n.e

Number of observations in experimental group.

n.c

Number of observations in control group.

pval

P-value (used to estimate the standard error).

df

Degrees of freedom (used in test or to construct confidence interval).

lower

Lower limit of confidence interval (used to estimate the standard error).

upper

Upper limit of confidence interval (used to estimate the standard error).

level.ci

Level of confidence interval.

median

Median (used to estimate the treatment effect and standard error).

q1

First quartile (used to estimate the treatment effect and standard error).

q3

Third quartile (used to estimate the treatment effect and standard error).

min

Minimum (used to estimate the treatment effect and standard error).

max

Maximum (used to estimate the treatment effect and standard error).

approx.TE

Approximation method to estimate treatment estimate (see Details).

approx.seTE

Approximation method to estimate standard error (see Details).

backtransf

A logical indicating whether results should be back transformed in printouts and plots. If backtransf = TRUE (default), results for sm = "OR" are printed as odds ratios rather than log odds ratios and results for sm = "ZCOR" are printed as correlations rather than Fisher's z transformed correlations, for example.

pscale

A numeric giving scaling factor for printing of single event probabilities or risk differences, i.e. if argument sm is equal to "PLOGIT", "PLN", "PRAW", "PAS", "PFT", or "RD".

irscale

A numeric defining a scaling factor for printing of single incidence rates or incidence rate differences, i.e. if argument sm is equal to "IR", "IRLN", "IRS", "IRFT", or "IRD".

irunit

A character specifying the time unit used to calculate rates, e.g. person-years.

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 TE).

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 errors).

control

An optional list to control the iterative process to estimate the between-study variance tau^2. This argument is passed on to rma.uni.

Value

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

TE, seTE, studlab, exclude, n.e, n.c

As defined above.

sm, level, level.comb,

As defined above.

comb.fixed, comb.random,

As defined above.

hakn, method.tau, tau.preset, TE.tau, method.bias,

As defined above.

tau.common, title, complab, outclab,

As defined above.

label.e, label.c, label.left, label.right,

As defined above.

byvar, bylab, print.byvar, byseparator, warn

As defined above.

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.

null.effect

As defined above.

k

Number of studies combined in meta-analysis.

Q

Heterogeneity statistic.

df.Q

Degrees of freedom for heterogeneity statistic.

pval.Q

P-value of heterogeneity test.

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.

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 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.

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.

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).

call

Function call.

version

Version of R package meta used to create object.

Details

This function provides the generic inverse variance method for meta-analysis which requires treatment estimates and their standard errors (Borenstein et al., 2010). The method is useful, e.g., for pooling of survival data (using log hazard ratio and standard errors as input). Arguments TE and seTE can be used to provide treatment estimates and standard errors directly. However, it is possible to derive these quantities from other information.

For several arguments defaults settings are utilised (see assignments with gs under Usage). These defaults can be changed using settings.meta.

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

Approximate treatment estimates

Missing treatment estimates can be derived from

1. confidence limits provided by arguments lower and upper;

2. median, interquartile range and range (arguments median, q1, q3, min, and max);

3. median and interquartile range (arguments median, q1 and q3);

4. median and range (arguments median, min and max).

For confidence limits, the treatment estimate is defined as the center of the confidence interval (on the log scale for relative effect measures like the odds ratio or hazard ratio). For median, interquartile range and range, equation (10) in Wan et al. (2014) is used to approximate the treatment effect (i.e., mean). Similarly, equations (14) and (2) in Wan et al. (2014) are used if median and interquartile range or range, respectively, are provided.

By default, missing treatment estimates are replaced successively using these method, e.g., confidence limits are utilised before interquartile ranges. Argument approx.TE can be used to overwrite this default for each individual study:

• Use treatment estimate directly (entry "" in argument approx.TE);

• confidence limits ("ci" in argument approx.TE);

• median, interquartile range and range ("iqr.range");

• median and interquartile range ("iqr");

• median and range ("range").

Approximate standard errors

Missing standard errors can be derived from

1. p-value provided by arguments pval and (optional) df;

2. confidence limits (arguments lower, upper, and (optional) df);

3. sample size, median, interquartile range and range (arguments n.e and / or n.c, median, q1, q3, min, and max);

4. sample size, median and interquartile range (arguments n.e and / or n.c, median, q1 and q3);

5. sample size, median and range (arguments n.e and / or n.c, median, min and max).

For p-values and confidence limits, calculations are either based on the standard normal or t distribution if argument df is provided. Furthermore, argument level.ci can be used to provide the level of the confidence interval. For median, interquartile range and range, depending on the sample size, equation (12) or (13) in Wan et al. (2014) is used to approximate the standard error. Similarly, equations (15) / (16) and (7) / (9) in Wan et al. (2014) are used if median and interquartile range or range, respectively, are provided. The sample size of individual studies must be provided with arguments n.e and / or n.c. The total sample size is calculated as n.e + n.c if both arguments are provided.

By default, missing standard errors are replaced successively using these method, e.g., p-value before confidence limits before interquartile range and range. Argument approx.seTE can be used to overwrite this default for each individual study:

• Use standard error directly (entry "" in argument approx.seTE);

• p-value ("pval" in argument approx.seTE);

• confidence limits ("ci");

• median, interquartile range and range ("iqr.range");

• median and interquartile range ("iqr");

• median and range ("range").

Estimation of between-study variance

The following methods are available to estimate the between-study variance $$\tau^2$$.

 Argument Method method.tau = "DL" DerSimonian-Laird estimator (DerSimonian and Laird, 1986) method.tau = "PM" Paule-Mandel estimator (Paule and Mandel, 1982) method.tau = "REML" Restricted maximum-likelihood estimator (Viechtbauer, 2005) method.tau = "ML" Maximum-likelihood estimator (Viechtbauer, 2005) method.tau = "HS" Hunter-Schmidt estimator (Hunter and Schmidt, 2015) method.tau = "SJ" Sidik-Jonkman estimator (Sidik and Jonkman, 2005) method.tau = "HE" Hedges estimator (Hedges and Olkin, 1985)

Historically, the DerSimonian-Laird method was the de facto standard to estimate the between-study variance $$\tau^2$$ and is still the default in many software packages including Review Manager 5 (RevMan 5) and R package meta. However, its role has been challenged and especially the Paule-Mandel and REML estimators have been recommended (Veroniki et al., 2016). Accordingly, the following R command can be used to use the Paule-Mandel estimator in all meta-analyses of the R session: settings.meta(method.tau = "PM")

The DerSimonian-Laird and Paule-Mandel estimators are implemented in R package meta. The other estimators are available if R package metafor (Viechtbauer 2010) is installed by internally calling R function rma.uni.

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; Langen et al., 2018) show improved coverage probabilities compared to the classic random effects method. However, in rare settings with very homogeneous treatment estimates, the Hartung-Knapp method can be anti-conservative (Wiksten et al., 2016). The Hartung-Knapp method is used if argument hakn = TRUE.

Prediction interval

A prediction interval for the treatment effect of 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.

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

Argument pscale can be used to rescale single proportions or risk differences, e.g. pscale = 1000 means that proportions are expressed as events per 1000 observations. This is useful in situations with (very) low event probabilities.

Argument irscale can be used to rescale single rates or rate differences, 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.

Default settings for comb.fixed, comb.random, pscale, irscale, irunit and several other arguments can be set for the whole R session using settings.meta.

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

Cooper H & Hedges LV (1994): The Handbook of Research Synthesis. Newbury Park, CA: Russell Sage Foundation

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

Hedges LV & Olkin I (1985): Statistical methods for meta-analysis. San Diego, CA: Academic Press

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

Hunter JE & Schmidt FL (2015): Methods of Meta-Analysis: Correcting Error and Bias in Research Findings (Third edition). Thousand Oaks, CA: Sage

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

Langan D, Higgins JPT, Jackson D, Bowden J, Veroniki AA, Kontopantelis E, et al. (2018): A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses. Research Synthesis Methods

Morris CN (1983): Parametric empirical Bayes inference: Theory and applications (with discussion). Journal of the American Statistical Association 78, 47--65

Paule RC & Mandel J (1982): Consensus values and weighting factors. Journal of Research of the National Bureau of Standards, 87, 377--85

Review Manager (RevMan) [Computer program]. Version 5.3. Copenhagen: The Nordic Cochrane Centre, The Cochrane Collaboration, 2014

Sidik K & Jonkman JN (2005): Simple heterogeneity variance estimation for meta-analysis. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54, 367--84

Veroniki AA, Jackson D, Viechtbauer W, Bender R, Bowden J, Knapp G, et al. (2016): Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research Synthesis Methods, 7, 55--79

Viechtbauer W (2005): Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30, 261--93

Viechtbauer W (2010): Conducting Meta-Analyses in R with the metafor Package. Journal of Statistical Software, 36, 1--48

Wan X, Wang W, Liu J, Tong T (2014): Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Medical Research Methodology, 14, 135

Wiksten A, R<U+00FC>cker G, Schwarzer G (2016): Hartung-Knapp method is not always conservative compared with fixed-effect meta-analysis. Statistics in Medicine, 35, 2503--15

update.meta, metabin, metacont, print.meta, settings.meta

Examples

Run this code
# NOT RUN {
data(Fleiss93)
m1 <- metabin(event.e, n.e, event.c, n.c,
data = Fleiss93, sm = "RR", method = "I")
m1
# Identical results by using the generic inverse variance method
metagen(m1$TE, m1$seTE, sm = "RR")
#
forest(metagen(m1$TE, m1$seTE, sm = "RR"))

# Meta-analysis with prespecified between-study variance
#
summary(metagen(m1$TE, m1$seTE, sm = "RR", tau.preset = sqrt(0.1)))

# Meta-analysis of survival data:
#
logHR <- log(c(0.95, 1.5))
selogHR <- c(0.25, 0.35)
metagen(logHR, selogHR, sm = "HR")

# Paule-Mandel method to estimate between-study variance for data
# from Paule & Mandel (1982)
#
average <- c(27.044, 26.022, 26.340, 26.787, 26.796)
variance <- c(0.003, 0.076, 0.464, 0.003, 0.014)
#
summary(metagen(average, sqrt(variance), sm = "MD", method.tau = "PM"))

# Conduct meta-analysis using hazard ratios and 95% confidence intervals
#
# Data from Steurer et al. (2006), Analysis 1.1 Overall survival
# https://www.cochranelibrary.com/cdsr/doi/10.1002/14651858.CD004270.pub2/abstract
#
study <- c("FCG on CLL 1996", "Leporrier 2001", "Rai 2000", "Robak 2000")
HR <- c(0.55, 0.92, 0.79, 1.18)
lower.HR <- c(0.28, 0.79, 0.59, 0.64)
upper.HR <- c(1.09, 1.08, 1.05, 2.17)
#
# Input must be log hazard ratios, not hazard ratios
#
metagen(log(HR), lower = log(lower.HR), upper = log(upper.HR),
studlab = study, sm = "HR")

# }


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