netmeta: Network meta-analysis using graph-theoretical method

In netmeta: Network Meta-Analysis using Frequentist Methods

Network meta-analysis using graph-theoretical method

Description

Network meta-analysis is a generalisation of pairwise meta-analysis that compares all pairs of treatments within a number of treatments for the same condition. The graph-theoretical approach for network meta-analysis uses methods that were originally developed in electrical network theory. It has been found to be equivalent to the frequentist approach to network meta-analysis which is based on weighted least squares regression (Rücker, 2012).

Print method for objects of class netmeta.

Usage

netmeta(
  TE,
  seTE,
  treat1,
  treat2,
  studlab,
  data = NULL,
  subset = NULL,
  sm,
  level = gs("level"),
  level.ma = gs("level.ma"),
  common = gs("common"),
  random = gs("random") | !is.null(tau.preset),
  prediction = FALSE,
  level.predict = gs("level.predict"),
  reference.group,
  baseline.reference = TRUE,
  small.values = "desirable",
  all.treatments = NULL,
  seq = NULL,
  method.tau = "DL",
  tau.preset = NULL,
  tol.multiarm = 0.001,
  tol.multiarm.se = NULL,
  details.chkmultiarm = FALSE,
  sep.trts = ":",
  nchar.trts = 666,
  nchar.studlab = 666,
  func.inverse = invmat,
  n1 = NULL,
  n2 = NULL,
  event1 = NULL,
  event2 = NULL,
  incr = NULL,
  sd1 = NULL,
  sd2 = NULL,
  time1 = NULL,
  time2 = NULL,
  backtransf = gs("backtransf"),
  title = "",
  keepdata = gs("keepdata"),
  control = NULL,
  warn = TRUE,
  warn.deprecated = gs("warn.deprecated"),
  nchar = nchar.trts,
  ...
)

## S3 method for class 'netmeta'
print(
  x,
  common = x$common,
  random = x$random,
  prediction = x$prediction,
  reference.group = x$reference.group,
  baseline.reference = x$baseline.reference,
  all.treatments = x$all.treatments,
  backtransf = x$backtransf,
  nchar.trts = x$nchar.trts,
  header = TRUE,
  digits = gs("digits"),
  digits.stat = gs("digits.stat"),
  digits.pval = max(gs("digits.pval"), 2),
  digits.pval.Q = max(gs("digits.pval.Q"), 2),
  digits.Q = gs("digits.Q"),
  digits.tau2 = gs("digits.tau2"),
  digits.tau = gs("digits.tau"),
  digits.I2 = gs("digits.I2"),
  scientific.pval = gs("scientific.pval"),
  big.mark = gs("big.mark"),
  text.tau2 = gs("text.tau2"),
  text.tau = gs("text.tau"),
  text.I2 = gs("text.I2"),
  legend = TRUE,
  warn.deprecated = gs("warn.deprecated"),
  ...
)

Arguments

TE	Estimate of treatment effect, i.e. difference between first and second treatment (e.g. log odds ratio, mean difference, or log hazard ratio). Or an R object created with pairwise.
seTE	Standard error of treatment estimate.
treat1	Label/Number for first treatment.
treat2	Label/Number for second treatment.
studlab	An optional - but important! - vector with study labels (see Details).
data	An optional data frame containing the study information.
subset	An optional vector specifying a subset of studies to be used.
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 comparisons.
level.ma	The level used to calculate confidence intervals for network estimates.
common	A logical indicating whether results for the common effects model should be printed.
random	A logical indicating whether results for the random effects model should be printed.
prediction	A logical indicating whether prediction intervals should be printed.
level.predict	The level used to calculate prediction intervals for a new study.
reference.group	Reference treatment.
baseline.reference	A logical indicating whether results should be expressed as comparisons of other treatments versus the reference treatment (default) or vice versa. This argument is only considered if reference.group has been specified.
small.values	A character string specifying whether small treatment effects indicate a beneficial ("desirable") or harmful ("undesirable") effect (passed on to netrank, can be abbreviated.
all.treatments	A logical or "NULL". If TRUE, matrices with all treatment effects, and confidence limits will be printed.
seq	A character or numerical vector specifying the sequence of treatments in printouts.
method.tau	A character string indicating which method is used to estimate the between-study variance \tau^2 and its square root \tau. Either "DL", "REML", or "ML", can be abbreviated.
tau.preset	An optional value for manually setting the square-root of the between-study variance \tau^2.
tol.multiarm	A numeric for the tolerance for consistency of treatment estimates in multi-arm studies which are consistent by design.
tol.multiarm.se	A numeric for the tolerance for consistency of standard errors in multi-arm studies which are consistent by design. This check is not conducted if the argument is NULL.
details.chkmultiarm	A logical indicating whether treatment estimates and / or variances of multi-arm studies with inconsistent results or negative multi-arm variances should be printed.
sep.trts	A character used in comparison names as separator between treatment labels.
nchar.trts	A numeric defining the minimum number of characters used to create unique treatment names.
nchar.studlab	A numeric defining the minimum number of characters used to create unique study labels.
func.inverse	R function used to calculate the pseudoinverse of the Laplacian matrix L (see Details).
n1	Number of observations in first treatment group.
n2	Number of observations in second treatment group.
event1	Number of events in first treatment group.
event2	Number of events in second treatment group.
incr	Numerical value added to cell frequencies (for details, see pairwise).
sd1	Standard deviation in first treatment group.
sd2	Standard deviation in second treatment group.
time1	Person time at risk in first treatment group.
time2	Person time at risk in second treatment group.
backtransf	A logical indicating whether results should be back transformed in printouts and forest plots. If backtransf = TRUE, results for sm = "OR" are presented as odds ratios rather than log odds ratios, for example.
title	Title of meta-analysis / systematic review.
keepdata	A logical indicating whether original data (set) should be kept in netmeta object.
control	An optional list to control the iterative process to estimate the between-study variance \tau^2. This argument is passed on to rma.mv.
warn	A logical indicating whether warnings should be printed (e.g., if studies are excluded from meta-analysis due to zero standard errors).
warn.deprecated	A logical indicating whether warnings should be printed if deprecated arguments are used.
nchar	Deprecated argument (replaced by nchar.trts).
...	Additional arguments.
x	An object of class netmeta.
header	A logical indicating whether information on title of meta-analysis, comparison and outcome should be printed at the beginning of the printout.
digits	Minimal number of significant digits, see print.default.
digits.stat	Minimal number of significant digits for tests of overall effect, see print.default.
digits.pval	Minimal number of significant digits for p-value of overall effects, see print.default.
digits.pval.Q	Minimal number of significant digits for p-value of heterogeneity tests, see print.default.
digits.Q	Minimal number of significant digits for heterogeneity statistics, see print.default.
digits.tau2	Minimal number of significant digits for between-study variance, see print.default.
digits.tau	Minimal number of significant digits for square root of between-study variance, see print.default.
digits.I2	Minimal number of significant digits for I-squared statistic, see print.default.
scientific.pval	A logical specifying whether p-values should be printed in scientific notation, e.g., 1.2345e-01 instead of 0.12345.
big.mark	A character used as thousands separator.
text.tau2	Text printed to identify between-study variance \tau^2.
text.tau	Text printed to identify \tau, the square root of the between-study variance \tau^2.
text.I2	Text printed to identify heterogeneity statistic I^2.
legend	A logical indicating whether a legend should be printed.

Details

Network meta-analysis using R package netmeta is described in detail in Schwarzer et al. (2015), Chapter 8.

Let n be the number of different treatments (nodes, vertices) in a network and let m be the number of existing comparisons (edges) between the treatments. If there are only two-arm studies, m is the number of studies. Let TE and seTE be the vectors of observed effects and their standard errors. Let W be the mxm diagonal matrix that contains the inverse variance 1 / seTE^2.

The given comparisons define the network structure. Therefrom an mxn design matrix X (edge-vertex incidence matrix) is formed; for more precise information, see Rücker (2012). Moreover, the nxn Laplacian matrix L and its Moore-Penrose pseudoinverse L+ are calculated (both matrices play an important role in graph theory and electrical network theory). Using these matrices, the variances based on both direct and indirect comparisons can be estimated. Moreover, the hat matrix H can be estimated by H = XL+X^tW = X(X^t W X)^+X^tW and finally consistent treatment effects can be estimated by applying the hat matrix to the observed (potentially inconsistent) effects. H is a projection matrix which maps the observed effects onto the consistent (n-1)-dimensional subspace. This is the Aitken estimator (Senn et al., 2013). As in pairwise meta-analysis, the Q statistic measures the deviation from consistency. Q can be separated into parts for each pairwise meta-analysis and a part for remaining inconsistency between comparisons.

Often multi-arm studies are included in a network meta-analysis. In multi-arm studies, the treatment effects on different comparisons are not independent, but correlated. This is accounted for by reweighting all comparisons of each multi-arm study. The method is described in Rücker (2012) and Rücker and Schwarzer (2014).

Comparisons belonging to multi-arm studies are identified by identical study labels (argument studlab). It is therefore important to use identical study labels for all comparisons belonging to the same multi-arm study, e.g., study label "Willms1999" for the three-arm study in the data example (Senn et al., 2013). The function netmeta then automatically accounts for within-study correlation by reweighting all comparisons of each multi-arm study.

Data entry for this function is in contrast-based format, that is, data are given as contrasts (differences) between two treatments (argument TE) with standard error (argument seTE). In principle, meta-analysis functions from R package meta, e.g. metabin for binary outcomes or metacont for continuous outcomes, can be used to calculate treatment effects separately for each treatment comparison which is a rather tedious enterprise. If data are provided in arm-based format, that is, data are given for each treatment arm separately (e.g. number of events and participants for binary outcomes), a much more convenient way to transform data into contrast-based form is available. Function pairwise can automatically transform data with binary outcomes (using the metabin function from R package meta), continuous outcomes (metacont function), incidence rates (metainc function), and generic outcomes (metagen function). Additional arguments of these functions can be provided (see help page of function pairwise).

Note, all pairwise comparisons must be provided for a multi-arm study. Consider a multi-arm study of p treatments with known variances. For this study, treatment effects and standard errors must be provided for each of p(p - 1) / 2 possible comparisons. For instance, a three-arm study contributes three pairwise comparisons, a four-arm study even six pairwise comparisons. Function pairwise automatically calculates all pairwise comparisons for multi-arm studies.

A simple random effects model assuming that a constant heterogeneity variance is added to each comparison of the network can be defined via a generalised methods of moments estimate of the between-studies variance \tau^2 (Jackson et al., 2012). This is added to the observed sampling variance seTE^2 of each comparison in the network (before appropriate adjustment for multi-arm studies). Then, as in standard pairwise meta-analysis, the procedure is repeated with the resulting enlarged standard errors.

For the random-effects model, the direct treatment estimates are based on the common between-study variance \tau^2 from the network meta-analysis.

Internally, both common and random effects models are calculated regardless of values choosen for arguments common and random. Accordingly, the network estimates for the random effects model can be extracted from component TE.random of an object of class "netmeta" even if argument random = FALSE. However, all functions in R package netmeta will adequately consider the values for common and random. E.g. function print.summary.netmeta will not print results for the random effects model if random = FALSE.

By default, treatment names are not abbreviated in printouts. However, in order to get more concise printouts, argument nchar.trts can be used to define the minimum number of characters for abbreviated treatment names (see abbreviate, argument minlength). R function treats is utilised internally to create abbreviated treatment names.

Names of treatment comparisons are created by concatenating treatment labels of pairwise comparisons using sep.trts as separator (see paste). These comparison names are used in the covariance matrices Cov.common and Cov.random and in some R functions, e.g, decomp.design. By default, a colon is used as the separator. If any treatment label contains a colon the following characters are used as separator (in consecutive order): "-", "_", "/", "+", ".", "|", and "*". If all of these characters are used in treatment labels, a corresponding error message is printed asking the user to specify a different separator.

Value

An object of class netmeta with corresponding print, summary, forest, and netrank functions. The object is a list containing the following components:

studlab, treat1, treat2, TE, seTE	As defined above.
seTE.adj.common, seTE.adj.random	Standard error of treatment estimate, adjusted for multi-arm studies.
design	Design of study providing pairwise comparison.
n1, n2, event1, event2, incr	As defined above.
mean1, mean2, sd1, sd2, time1, time2	As defined above.
sd1, sd2, time1, time2	As defined above.
k	Total number of studies.
m	Total number of pairwise comparisons.
n	Total number of treatments.
d	Total number of designs (corresponding to the unique set of treatments compared within studies).
trts	Treatments included in network meta-analysis.
k.trts	Number of studies evaluating a treatment.
n.trts	Number of observations receiving a treatment (if arguments n1 and n2 are provided).
events.trts	Number of events observed for a treatment (if arguments event1 and event2 are provided).
multiarm	Logical vector to identify pairwise comparisons from multi-arm studies.
n.arms	Number of treatment arms in study providing pairwise comparison.
studies	Vector with unique study labels.
narms	Number of arms for each study.
designs	Vector with unique designs present in the network. A design corresponds to the set of treatments compared within a study.
designs	Vector with unique direct comparisons present in the network.
TE.nma.common, TE.nma.random	A vector of length m of consistent treatment effects estimated by network meta-analysis (nma) (common / random effects model).
seTE.nma.common, seTE.nma.random	A vector of length m of effective standard errors estimated by network meta-analysis (common / random effects model).
lower.nma.common, lower.nma.random	A vector of length m of lower confidence interval limits for consistent treatment effects estimated by network meta-analysis (common effects / random effects model).
upper.nma.common, upper.nma.random	A vector of length m of upper confidence interval limits for the consistent treatment effects estimated by network meta-analysis (common effects / random effects model).
statistic.nma.common, statistic.nma.random	A vector of length m of z-values for test of treatment effect for individual comparisons (common / random effects model).
pval.nma.common, pval.nma.random	A vector of length m of p-values for test of treatment effect for individual comparisons (common / random effects model).
leverage.common	A vector of length m of leverages, interpretable as factors by which variances are reduced using information from the whole network.
w.common, w.random	A vector of length m of weights of individual studies (common / random effects model).
Q.common	A vector of length m of contributions to total heterogeneity / inconsistency statistic.
TE.common, TE.random	nxn matrix with estimated overall treatment effects (common / random effects model).
seTE.common, seTE.random	nxn matrix with standard errors (common / random effects model).
lower.common, upper.common, lower.random, upper.random	nxn matrices with lower and upper confidence interval limits (common / random effects model).
statistic.common, pval.common, statistic.random, pval.random	nxn matrices with z-value and p-value for test of overall treatment effect (common / random effects model).
seTE.predict	nxn matrix with standard errors for prediction intervals.
lower.predict, upper.predict	nxn matrices with lower and upper prediction interval limits.
prop.direct.common, prop.direct.random	A named vector of the direct evidence proportion of each network estimate. (common effects / random effects model).
TE.direct.common, TE.direct.random	nxn matrix with estimated treatment effects from direct evidence (common effects / random effects model).
seTE.direct.common, seTE.direct.random	nxn matrix with estimated standard errors from direct evidence (common effects / random effects model).
lower.direct.common, upper.direct.common, lower.direct.random,	nxn matrices with lower and upper confidence interval limits from direct evidence (common / random effects model).
upper.direct.random	nxn matrices with lower and upper confidence interval limits from direct evidence (common effects / random effects model).
statistic.direct.common, pval.direct.common, statistic.direct.random,	nxn matrices with z-value and p-value for test of overall treatment effect from direct evidence (common / random effects model).
pval.direct.random	nxn matrices with z-value and p-value for test of overall treatment effect from direct evidence (common / random effects model).
TE.indirect.common, TE.indirect.random	nxn matrix with estimated treatment effects from indirect evidence (common / random effects model).
seTE.indirect.common, seTE.indirect.random	nxn matrix with estimated standard errors from indirect evidence (common / random effects model).
lower.indirect.common, upper.indirect.common, lower.indirect.random,	nxn matrices with lower and upper confidence interval limits from indirect evidence (common / random effects model).
upper.indirect.random	nxn matrices with lower and upper confidence interval limits from indirect evidence (common / random effects model).
statistic.indirect.common, pval.indirect.common, statistic.indirect.random,	nxn matrices with z-value and p-value for test of overall treatment effect from indirect evidence (common / random effects model).
pval.indirect.random	nxn matrices with z-value and p-value for test of overall treatment effect from indirect evidence (common / random effects model).
Q	Overall heterogeneity / inconsistency statistic.
df.Q	Degrees of freedom for test of heterogeneity / inconsistency.
pval.Q	P-value for test of heterogeneity / inconsistency.
I2, lower.I2, upper.I2	I-squared, lower and upper confidence limits.
tau	Square-root of between-study variance.
Q.heterogeneity	Overall heterogeneity statistic.
df.Q.heterogeneity	Degrees of freedom for test of overall heterogeneity.
pval.Q.heterogeneity	P-value for test of overall heterogeneity.
Q.inconsistency	Overall inconsistency statistic.
df.Q.inconsistency	Degrees of freedom for test of overall inconsistency.
pval.Q.inconsistency	P-value for test of overall inconsistency.
Q.decomp	Data frame with columns 'treat1', 'treat2', 'Q', 'df' and 'pval.Q', providing heterogeneity statistics for each pairwise meta-analysis of direct comparisons.
A.matrix	Adjacency matrix (nxn).
X.matrix	Design matrix (mxn).
B.matrix	Edge-vertex incidence matrix (mxn).
L.matrix.common, L.matrix.random	Laplacian matrix (nxn).
Lplus.matrix.common, Lplus.matrix.random	Moore-Penrose pseudoinverse of the Laplacian matrix (nxn).
Q.matrix	Matrix of heterogeneity statistics for pairwise meta-analyses, where direct comparisons exist (nxn).
G.matrix	Matrix with variances and covariances of comparisons (mxm). G is defined as BL+B^t.
H.matrix.common, H.matrix.random	Hat matrix (mxm), defined as H = GW = BL+B^tW.
n.matrix	nxn matrix with number of observations in direct comparisons (if arguments n1 and n2 are provided).
events.matrix	nxn matrix with number of events in direct comparisons (if arguments event1 and event2 are provided).
P.common, P.random	nxn matrix with direct evidence proportions (common / random effects model).
Cov.common	Variance-covariance matrix (common effects model)
Cov.random	Variance-covariance matrix (random effects model)
sm, level, level.ma	As defined above.
common, random	As defined above.
prediction, level.predict	As defined above.
reference.group, baseline.reference, small.values, all.treatments	As defined above.
seq, tau.preset, tol.multiarm, tol.multiarm.se	As defined above.
details.chkmultiarm, sep.trts, nchar.trts	As defined above.
backtransf, title, warn, warn.deprecated	As defined above.
call	Function call.
version	Version of R package netmeta used to create object.

Note

R function rma.mv from R package metafor (Viechtbauer 2010) is called internally to estimate the between-study variance \tau^2 for the (restricted) maximum likelihood method. For binary outcomes, incidence rates, and the mean difference, the variance-covariance matrix is calculated if arguments event1, event2, n1, and n2 (binary outcomes); event1, event2, time1, and time2 (incidence rates); n1, n2, sd1, and sd2 (mean difference) are provided. For data sets preprocessed with pairwise the respective variables are selected automatically.

Author(s)

Gerta Rücker [e-mail], Guido Schwarzer [e-mail]

References

Jackson D, White IR, Riley RD (2012): Quantifying the impact of between-study heterogeneity in multivariate meta-analyses. Statistics in Medicine, 31, 3805–20

Rücker G (2012): Network meta-analysis, electrical networks and graph theory. Research Synthesis Methods, 3, 312–24

Rücker G, Schwarzer G (2014): Reduce dimension or reduce weights? Comparing two approaches to multi-arm studies in network meta-analysis. Statistics in Medicine, 33, 4353–69

Schwarzer G, Carpenter JR, Rücker G (2015): Meta-Analysis with R (Use R!). Springer International Publishing, Switzerland

Senn S, Gavini F, Magrez D, Scheen A (2013): Issues in performing a network meta-analysis. Statistical Methods in Medical Research, 22, 169–89

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

See Also

pairwise, forest.netmeta, netrank, metagen

Examples

data(smokingcessation)

# Transform data from arm-based format to contrast-based format
#
p1 <- pairwise(list(treat1, treat2, treat3),
  event = list(event1, event2, event3), n = list(n1, n2, n3),
  data = smokingcessation, sm = "OR")

# Conduct random effects network meta-analysis
#
net1 <- netmeta(p1, common = FALSE)
net1

## Not run: 
data(Senn2013)

# Conduct common effects network meta-analysis
#
net2 <- netmeta(TE, seTE, treat1, treat2, studlab,
  data = Senn2013, sm = "MD", random = FALSE)
net2
net2$Q.decomp

# Comparison with reference group
#
print(net2, reference = "plac")

# Conduct random effects network meta-analysis
#
net3 <- netmeta(TE, seTE, treat1, treat2, studlab,
  data = Senn2013, sm = "MD", common = FALSE)
net3

# Change printing order of treatments with placebo last and use
# long treatment names
#
trts <- c("acar", "benf", "metf", "migl", "piog",
  "rosi", "sita", "sulf", "vild", "plac")
net4 <- netmeta(TE, seTE, treat1.long, treat2.long, studlab,
  data = Senn2013, sm = "MD", common = FALSE,
  seq = trts, reference = "Placebo")
print(net4, digits = 2)

## End(Not run)

netmeta documentation built on May 31, 2023, 5:45 p.m.