# limitmeta: Limit meta-analysis In metasens: Statistical Methods for Sensitivity Analysis in Meta-Analysis

 limitmeta R Documentation

## 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,
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 gerta.ruecker@uniklinik-freiburg.de, Guido Schwarzer guido.schwarzer@uniklinik-freiburg.de

### 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

`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)
```

metasens documentation built on March 7, 2023, 7:51 p.m.