Systematic reviews aim to summarize all the available evidence relevant to a particular research question. If appropriate, the data from identified studies are quantitatively combined in a meta-analysis. Often only few studies regarding a particular research question exist. In these settings the estimation of the between-study heterogeneity is challenging. Furthermore, the assessment of publication bias is difficult as standard methods such as visual inspection or formal hypothesis tests in funnel plots do not provide adequate guidance. Previously, Henmi and Copas (Statistics in Medicine 2010, 29: 2969-2983) proposed a confidence interval for the overall effect in random-effects meta-analysis that is robust to publication bias to some extent. As is evident from their simulations, the confidence intervals have improved coverage compared with standard methods. To our knowledge, the properties of their method has never been assessed for meta-analyses including fewer than five studies. In this manuscript, we propose a variation of the method by Henmi and Copas employing an improved estimator of the between-study heterogeneity, in particular when dealing with few studies only. In a simulation study, the proposed method is compared to several competitors. Overall, we found that our method outperforms the others in terms. In particular, an improvement in coverage probability of the new method compared with the proposal by Henmi and Copas is demonstrated. The work is motivated and illustrated by a systematic review and meta-analysis in paediatric immunosuppression following liver transplantations.


INTRODUCTION
Systematic reviews aim to summarize all the available evidence relevant to a particular research question. If appropriate, the data from identified studies are quantitatively combined in a meta-analysis. If the true effect is the same in all studies to be combined in a meta-analysis, then the so-called common-effect or fixed-effect model is appropriate. In practical applications this assumption often appears to be too strict as some level of between-trial heterogeneity in the effects is suspected. Then the random-effects model is applied which treats the latent study-specific effects as random from an assumed distribution, often a normal distribution. Inference regarding the overall effect in the standard model relies on normality assumptions which apply if the number of trials and the trials themselves are sufficiently large. A number of estimators for the between-trial heterogeneity have been proposed [1]; among them the standard approach by DerSimonian and Laird [2].
Recently the topic of meta-analysis of few studies, say 2 -5, got more attention, since this case is very common in practice. With few studies, however, confidence intervals of the overall effect based on normal quantiles tend to be too short as they ignore the uncertainty in estimating the between-trial heterogeneity. As remedies, methods based on t-quantiles have been proposed [5,6,7,8]. With few studies only, however, they are often conservative and so long that they are uninformative [9]. Also various likelihood-based methods have recently been assessed in the specific situation of few studies and not found to be a solution to the problem [10]. Between-trial heterogeneity estimates mentioned above often result in zero [4], with the notable exception of the method proposed by Chung et al [3]. Chung et al suggested the so-called Bayes modal (BM) estimator, which uses in a Bayesian framework a weakly informative prior for the between-trial heterogeneity to avoid zero estimates of the heterogeneity. Furthermore, a fully Bayesian approach to random-effects meta-analysis with weakly informative priors for the between-trial heterogeneity parameter has some advantages in this situation, since zero estimates are avoided as with the BM estimator and in addtition the uncertainty in estimating the heterogeneity is accounted for [4,11]. Of course the Bayesian credible intervals would not necessarily have frequentist properties. Evaluating the operating characteristics in extensive simulation studies, it was found that the frequentist coverage probabilities are often above the nominal level with conservative choices of the prior for the between-trial heterogeneity [4,11]. Bender et al [12] recently provided an overview on the topic of meta-analyses with few studies.
In systematic reviews, relevant evidence is identified through systematic searches of literature databases. If all relevant studies would be published, this would be sufficient. However, this is not always the case. The problem was first described as the 'file drawer problem' [13]. Today various types of reporting biases are carefully distinguished including publication bias, time lag bias, citation bias and outcome reporting bias to name but a few. Studies might not be published at all for various reasons or only with a certain delay or in journals or languages that are more difficult to access (see e.g. Table 7.2.a in [14]). In the following we focus here on the aspect of publication bias. Prospective registration of clinical trials is one way to tackle this problem. It has become standard practice to search not only at least two electronic databases of the literature but also to search at least one registry for clinical studies such as clinicaltrials.gov. The idea would be to include unpublished studies in systematic reviews. However, access to the unpublished results is often challenging as it requires the cooperation of investigators, sponsors etc.
A number of methods have been proposed over the years to deal with publication bias [15]. A popular way to interrogate data for publication bias is the visualization in form of a so-called funnel plot. In this scatter plot, each study contributes an estimate of an effect measure and its estimated standard error. The former is plotted on the x-axis, the latter on the y-axis. If no publication bias is present, we would expect the plot to be symmetric to the vertical line running through the average effect. Any absence of this symmetry might be interpreted as a signal that some form of reporting bias might be present. As this can be difficult to judge, formal hypothesis tests have been proposed (see e.g. [16]). The problem with the visual inspection as well as with the formal tests is that they become more powerful with larger number of studies, but are less sensitive with few studies only. In the context of funnel plots, trim-and-fill methods have been proposed to correct the overall effect for potential publication bias [17]. Following an alternative approach, several sensitivity analysis methods have been suggested based on selection functions describing the selective publication process [18,19,20]. For instance, Copas and Jackson [19] investigated the maximum bias over all possible selection functions which satisfy the (fairly weak) condition that studies with smaller standard errors are at least as likely to be selected than studies with larger standard errors. Building on their work, Henmi et al [20] developed sensitivity analyses that, in contrast to the proposal by Copas and Jackson [19], account for uncertainty in estimation. Again, the methods are not designed for the setting of few studies only.
In contrast to the approaches described above, Henmi and Copas [21] proposed a method for random-effects meta-analysis that is robust to selection. They modified the DerSimonian-Laird (DL) confidence interval [2] by replacing the random-effect estimator by the fixed-effect estimator of the overall effect and by replacing the normal quantiles by more accurate ones. The latter depends on the between-trial heterogeneity. The DL estimator is used in the computation of the quantiles. Therefore, with few studies this approach may not work well. Here we propose a modification of the Henmi-Copas method by replacing the estimator of the between-study heterogeneity in the computation of the quantiles by the one developed by Chung et al [3]. The properties of the new approach are assessed and compared to alternative methods including the Henmi-Copas approach in Monte Carlo simulation studies considering in particular the case of few studies with and without publication bias.
Our work is motivated by a systematic review and meta-analysis of controlled clinical trials assessing efficacy and safety of Interleukin-2 receptor antagonists (IL-2RA) in children having undergone liver transplantations [22], a rare surgical procedure in children. In total only six relevant studies were identified with little standardization with regard to the design of the studies implying some level of heterogeneity. Although the authors carefully checked for publication bias using standard techniques, it cannot be excluded that in particular some smaller studies were not published if they resulted in inconclusive treatment effects.
The manuscript is organized as follows. In the next section the new confidence interval for the overall effect is developed starting by introducing notation and reviewing existing methodology. The simulation study assessing the properties of the new confidence interval in comparison to existing methods is presented in Section 3. In Section 4 the proposed method is applied to the motivating example. We close with a brief discussion of our findings and their limitations.

METHODS
Adopting the notation by Henmi and Copas [21], the true effect of an individual study i out of n independent studies is denoted by θ i. Estimates y i of the effects θ i are observed with stand errors σ i. Here we consider the normal-normal hierarchical model (NNHM), which is the standard model for random-effects meta-analysis. In the NNHM, it is assumed that the θ i are from a normal distribution with expectation θ and variance τ 2, i.e.
Furthermore, the effect estimators Y i follow (at least approximately) a normal distribution with expectation θ i and variance σ 2 i, i.e.
From Equations (1) and (2) they follow the marginal model If the between-trial heterogeneity τ 2 is 0, then the random-effects model reduces to the so-called fixed-effect or common-effect model. The focus of our study is inference regarding θ, the overall effect. A standard method to construct an estimator and a (1−α) confidence interval for θ was proposed by DerSimonian and Laird [2] (DL). In short, the DL estimator of θ is given bŷ Here, the DL estimatorτ 2 DL of the between-study heterogeneity τ 2 is given byτ The weights w i are