limitmeta  R Documentation 
Implementation of the limit metaanalysis method by Rücker et al. (2011) to adjust for bias in metaanalysis.
limitmeta( x, method.adjust = "beta0", level = x$level, level.ma = x$level.ma, backtransf = x$backtransf, title = x$title, complab = x$complab, outclab = x$outclab )
x 
An object of class 
method.adjust 
A character string indicating which adjustment
method is to be used. One of 
level 
The level used to calculate confidence intervals for individual studies. 
level.ma 
The level used to calculate confidence intervals for pooled estimates. 
backtransf 
A logical indicating whether results should be
back transformed in printouts and plots. If

title 
Title of metaanalysis / systematic review. 
complab 
Comparison label. 
outclab 
Outcome label. 
This function provides the method by Rücker et al. (2011) to estimate an effect estimate adjusted for bias in metaanalysis. The underlying model is an extended random effects model that takes account of possible small study effects by allowing the treatment effect to depend on the standard error:
theta(i) = beta + sqrt(SE(i)^2 + tau^2)(epsilon(i) + alpha),
where epsilon(i) follows a standard normal distribution. Here theta(i) is the observed effect in study i, beta the global mean, SE(i) the withinstudy standard error, and tau^2 the betweenstudy variance. The parameter alpha represents the bias introduced by smallstudy effects. On the one hand, alpha can be interpreted as the expected shift in the standardized treatment effect if precision is very small. On the other hand, theta(adj) = beta + tau*alpha is interpreted as the limit treatment effect for a study with infinite precision (corresponding to SE(i) = 0).
Note that as alpha is included in the model equation, beta has a different interpretation as in the usual random effects model. The two models agree only if alpha=0. If there are genuine smallstudy effects, the model includes a component making the treatment effect depend on the standard error. The expected treatment effect of a study of infinite precision, beta + tau*alpha, is used as an adjusted treatment effect estimate.
The maximum likelihood estimates for alpha and beta can be interpreted as intercept and slope in linear regression on a socalled generalised radial plot, where the xaxis represents the inverse of sqrt(SE(i)^2 + tau^2) and the yaxis represents the treatment effect estimates, divided by sqrt(SE(i)^2 + tau^2).
Two further adjustments are available that use a shrinkage procedure. Based on the extended random effects model, a limit metaanalysis is defined by inflating the precision of each study with a common factor. The limit metaanalysis yields shrunken estimates of the studyspecific effects, comparable to empirical Bayes estimates. Based on the extended random effects model, we obtain three different treatment effect estimates that are adjusted for smallstudy effects:
an estimate based on the
expectation of the extended random effects model, beta0 = beta +
tau*alpha (method.adjust="beta0"
)
the extended random
effects model estimate of the limit metaanalysis, including bias
parameter (method.adjust="betalim"
)
the usual random
effects model estimate of the limit metaanalysis, excluding bias
parameter (method.adjust="mulim"
)
See Rücker, Schwarzer et al. (2011), Section 7, for the definition
of G^2 and the three heterogeneity statisticics Q
,
Q.small
, and Q.resid
.
For comparison, the original random effects metaanalysis is always printed in the sensitivity analysis.
An object of class "limitmeta"
with corresponding
print
, summary
and funnel
function. The
object is a list containing the following components:
x, level, level.ma,method.adjust,title, complab,
outclab 
As defined above. 
TE, seTE 
Estimated treatment effect and standard error of individual studies. 
TE.limit, seTE.limit 
Shrunken estimates and standard error of individual studies. 
studlab 
Study labels. 
TE.random, seTE.random 
Unadjusted overall treatment effect 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 corresponding pvalue for test of overall treatment effect (random effects model). 
w.random 
Weight of individual studies (in random effects model). 
tau 
Squareroot of betweenstudy variance. 
TE.adjust, seTE.adjust 
Adjusted overall effect and standard error (random effects model). 
lower.adjust, upper.adjust 
Lower and upper confidence interval limits for adjusted effect estimate (random effects model). 
statistic.adjust, pval.adjust 
Statistic and corresponding pvalue for test of overall treatment effect for adjusted estimate (random effects model). 
alpha.r 
Intercept of the linear regression line on the generalised radial plot, here interpreted as bias parameter in an extended random effects model. Represents the expected shift in the standardized treatment effect if precision is very small. 
beta.r 
Slope of the linear regression line on the generalised radial plot. 
Q 
Heterogeneity statistic. 
Q.small 
Heterogeneity statistic for small study effects. 
Q.resid 
Heterogeneity statistic for residual heterogeneity beyond small study effects. 
G.squared 
Heterogeneity statistic G^2 (ranges from 0 to 100%). 
k 
Number of studies combined in metaanalysis. 
call 
Function call. 
version 
Version of R package metasens used to create object. 
Gerta Rücker gerta.ruecker@uniklinikfreiburg.de, Guido Schwarzer guido.schwarzer@uniklinikfreiburg.de
Rücker G, Carpenter JR, Schwarzer G (2011): Detecting and adjusting for smallstudy effects in metaanalysis. Biometrical Journal, 53, 351–68
Rücker G, Schwarzer G, Carpenter JR, Binder H, Schumacher M (2011): Treatmenteffect estimates adjusted for smallstudy effects via a limit metaanalysis. Biostatistics, 12, 122–42
funnel.limitmeta
, print.limitmeta
data(Moore1998) m1 < metabin(succ.e, nobs.e, succ.c, nobs.c, data = Moore1998, sm = "OR", method = "Inverse") print(limitmeta(m1), digits = 2)
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