A character string indicating which adjustment
method is to be used. One of "beta0", "betalim", or
"mulim", can be abbreviated.
level
The level used to calculate confidence intervals for
individual studies.
level.comb
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
backtransf=FALSE, results for the odds ratio are printed as
log odds ratios rather than odds ratio, for example.
title
Title of meta-analysis / systematic review.
complab
Comparison label.
outclab
Outcome label.
Details
This function provides the method by R<c3><83><c2><bc>cker et al. (2011) to
estimate an effect estimate adjusted for bias in meta-analysis. 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:
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 within-study standard error, and tau^2 the between-study
variance. The parameter alpha represents the bias introduced by
small-study 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 small-study
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
so-called generalised radial plot, where the x-axis represents the
inverse of sqrt(SE(i)^2 + tau^2) and the y-axis 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
meta-analysis is defined by inflating the precision of each study
with a common factor. The limit meta-analysis yields shrunken
estimates of the study-specific effects, comparable to empirical
Bayes estimates. Based on the extended random effects model, we
obtain three different treatment effect estimates that are adjusted
for small-study 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 meta-analysis, including bias parameter (method.adjust="betalim")
the usual random effects model estimate of the limit meta-analysis, excluding bias parameter (method.adjust="mulim")
See R<c3><83><c2><bc>cker, 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 meta-analysis is always
printed in the sensitivity analysis.
Value
An object of class "limitmeta" with corresponding
print, summary and funnel function. The object
is a list containing the following components:
x, level, level.com
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).
zval.random, pval.random
z-value and corresponding p-value
for test of overall treatment effect (random effects model).
w.random
Weight of individual studies (in random effects model).
tau
Square-root of between-study 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).
zval.adjust, pval.adjust
z-value and corresponding p-value
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 meta-analysis.
call
Function call.
version
Version of R package metasens used to create object.
R<c3><83><c2><bc>cker G, Carpenter JR, Schwarzer G (2011),
Detecting and adjusting for small-study effects in meta-analysis.
Biometrical Journal, 53, 351–368.
R<c3><83><c2><bc>cker G, Schwarzer G, Carpenter JR, Binder H, Schumacher M (2011),
Treatment-effect estimates adjusted for small-study effects via a limit meta-analysis.
Biostatistics, 12, 122–142.