limitmeta: Limit meta-analysis

In guido-s/metasens: Statistical Methods for Sensitivity Analysis in Meta-Analysis

Limit meta-analysis

Description

Implementation of the limit meta-analysis method by Rücker et al. (2011) to adjust for bias in meta-analysis.

Usage

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
)

Arguments

x	An object of class meta.
method.adjust	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.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 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 Rücker 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:

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 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 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 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.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 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).
statistic.adjust, pval.adjust	Statistic 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.

Author(s)

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

References

Rücker G, Carpenter JR, Schwarzer G (2011): Detecting and adjusting for small-study effects in meta-analysis. Biometrical Journal, 53, 351–68

Rücker 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, 122–42

See Also

funnel.limitmeta, print.limitmeta

Examples

data(Moore1998)
m1 <- metabin(succ.e, nobs.e, succ.c, nobs.c,
  data = Moore1998, sm = "OR", method = "Inverse")

print(limitmeta(m1), digits = 2)

guido-s/metasens documentation built on April 5, 2023, 2:30 p.m.