Multi-trait analysis of gene-by-environment interactions in large-scale genetic studies.

Identifying genotype-by-environment interaction (GEI) is challenging because the GEI analysis generally has low power. Large-scale consortium-based studies are ultimately needed to achieve adequate power for identifying GEI. We introduce Multi-Trait Analysis of Gene-Environment Interactions (MTAGEI), a powerful, robust, and computationally efficient framework to test gene-environment interactions on multiple traits in large data sets, such as the UK Biobank (UKB). To facilitate the meta-analysis of GEI studies in a consortium, MTAGEI efficiently generates summary statistics of genetic associations for multiple traits under different environmental conditions and integrates the summary statistics for GEI analysis. MTAGEI enhances the power of GEI analysis by aggregating GEI signals across multiple traits and variants that would otherwise be difficult to detect individually. MTAGEI achieves robustness by combining complementary tests under a wide spectrum of genetic architectures. We demonstrate the advantages of MTAGEI over existing single-trait-based GEI tests through extensive simulation studies and the analysis of the whole exome sequencing data from the UKB.

A systematic review and meta-analysis of pollutants in environmental media.

Environmental pollutants are ubiquitous in our environmental media, resulting in detrimental impacts on both humans and the environment. An evidence-based review, particularly a systematic review and meta-analysis, performs a crucial function in assessing the pollution status of pollutants in environmental media at national and global scales. We selected and thoroughly investigated 76 papers focusing on systematic reviews and meta-analyses of contaminants in environmental media. The need to broaden the scope of studies was observed with an increase in the total number of publications, and there were greater focuses on food safety, water pollution, biological pollution, and environmental risks. Furthermore, this review outlined the fundamental procedures involved in a systematic review and meta-analysis, including literature searching, screening of articles, study quality analysis, data extraction and synthesis, and meta-analysis. A meta-analysis typically comprises fixed- and/or random-effects meta-analysis, identifying and measuring heterogeneity, sensitivity analysis, publication bias, subgroup analysis, and meta-regression. We specifically explored the application of meta-analysis to assess the presence of contaminants in environmental media based on two different pollutant categories, namely, non-biological and biological pollutants. The mean value is commonly utilized to assess the pooled concentration of non-biological pollutants, while the prevalence serves as the effect size of biological pollutants. Additionally, we summarized the innovative applications, frequent misuses, and problems encountered in systematic reviews and meta-analyses. Finally, we proposed several suggestions for future research endeavors.

From the Individual to the Group: Using Idiographic Analyses and Two-Stage Random Effects Meta-Analysis to Obtain Population Level Inferences for within-Person Processes.

In psychology, the use of portable technology and wearable devices to ease participant burden in data collection is on the rise. This creates increased interest in collecting real-time or near real-time data from individuals within their natural environments. As a result, vast amounts of observational time series data are generated. Often, motivation for collecting this data hinges on understanding within-person processes that underlie psychological phenomena. Motivated by the body of Dr. Peter Molenaar's life work calling for analytical approaches that consider potential heterogeneity and non-ergodicity, the focus of this paper is on using idiographic analyses to generate population inferences for within-person processes. Meta-analysis techniques using one-stage and two-stage random effects meta-analysis as implemented in single-case experimental designs are presented. The case for preferring a two-stage approach for meta-analysis of single-subject observational time series data is made and demonstrated using an empirical example. This provides a novel implementation of the methodology as prior implementations focus on applications to short time series with experimental designs. Inspired by Dr. Molenaar's work, we describe how an approach, two-stage random effects meta-analysis (2SRE-MA), aligns with recent calls to consider idiographic approaches when making population-level inferences regarding within-person processes.

Rare events meta-analysis using the Bayesian beta-binomial model.

In meta-analyses of rare events, it can be challenging to obtain a reliable estimate of the pooled effect, in particular when the meta-analysis is based on a small number of studies. Recent simulation studies have shown that the beta-binomial model is a promising candidate in this situation, but have thus far only investigated its performance in a frequentist framework. In this study, we aim to make the beta-binomial model for meta-analysis of rare events amenable to Bayesian inference by proposing prior distributions for the effect parameter and investigating the models' robustness to different specifications of priors for the scale parameter. To evaluate the performance of Bayesian beta-binomial models with different priors, we conducted a simulation study with two different data generating models in which we varied the size of the pooled effect, the degree of heterogeneity, the baseline probability, and the sample size. Our results show that while some caution must be exercised when using the Bayesian beta-binomial in meta-analyses with extremely sparse data, the use of a weakly informative prior for the effect parameter is beneficial in terms of mean bias, mean squared error, and coverage. For the scale parameter, half-normal and exponential distributions are identified as candidate priors in meta-analysis of rare events using the Bayesian beta-binomial model.

Adjusted closed-form confidence interval formulas for network meta-analysis with a small number of studies.

We derive simple formulas for closed-form confidence intervals for the Wald statistic, likelihood ratio statistic, and score statistic for network meta-analysis (NMA). Additionally, we consider resolutions of concerns that network meta-analyzes with a small number of studies cannot maintain a nominal confidence level. For bias adjustment in analyzes with a small number of studies, the Bartlett-type adjustment is a well-known method. Many Bartlett-type adjustment-type methods are based on maximum likelihood estimators (MLEs). However, NMA often uses restricted MLEs that have not been extensively discussed with respect to the Bartlett-type adjustment. In this article, we propose a Bartlett-type adjustment method for the Wald statistic, likelihood ratio statistic, and score statistic when nuisance parameters are estimated by not only the maximum likelihood method but also the restricted maximum likelihood method. We can compute closed-form confidence intervals adjusted using the Bartlett-type adjustment immediately without any numerical calculations (eg, bootstrap method). Additionally, we propose a higher-order adjustment by applying the bootstrap method to Bartlett-type adjusted statistics. Using a computer simulation, we confirmed that the adjusted confidence intervals maintained a nominal confidence level. Additionally, we confirmed that the confidence intervals of the Wald statistic, likelihood ratio statistic, and score statistic based on the restricted maximum likelihood method performed well without further bootstrap adjustment and the performances of the three adjusted confidence intervals were comparable. Finally, we demonstrated that confidence intervals were adjusted for actual NMA. In the actual NMA, the adjusted confidence intervals of the Wald statistic were wider, the adjusted confidence intervals of the likelihood ratio statistic were also wider, and the adjusted confidence intervals of the score statistic were narrower. We recommend using the likelihood ratio test statistic with the restricted maximum likelihood estimator; however, just in case, we recommend applying the Bartlett-type adjustment to remove the second order bias. From demonstrations in actual studies, we confirmed that the adjusted confidence intervals improved compared with the naive confidence intervals.

Random-effects meta-analysis models for the odds ratio in the case of rare events under different data-generating models: A simulation study.

Meta-analysis of binary data is challenging when the event under investigation is rare, and standard models for random-effects meta-analysis perform poorly in such settings. In this simulation study, we investigate the performance of different random-effects meta-analysis models in terms of point and interval estimation of the pooled log odds ratio in rare events meta-analysis. First and foremost, we evaluate the performance of a hypergeometric-normal model from the family of generalized linear mixed models (GLMMs), which has been recommended, but has not yet been thoroughly investigated for rare events meta-analysis. Performance of this model is compared to performance of the beta-binomial model, which yielded favorable results in previous simulation studies, and to the performance of models that are frequently used in rare events meta-analysis, such as the inverse variance model and the Mantel-Haenszel method. In addition to considering a large number of simulation parameters inspired by real-world data settings, we study the comparative performance of the meta-analytic models under two different data-generating models (DGMs) that have been used in past simulation studies. The results of this study show that the hypergeometric-normal GLMM is useful for meta-analysis of rare events when moderate to large heterogeneity is present. In addition, our study reveals important insights with regard to the performance of the beta-binomial model under different DGMs from the binomial-normal family. In particular, we demonstrate that although misalignment of the beta-binomial model with the DGM affects its performance, it shows more robustness to the DGM than its competitors.

A straightforward meta-analysis approach for oncology phase I dose-finding studies.

Phase I early-phase clinical studies aim at investigating the safety and the underlying dose-toxicity relationship of a drug or combination. While little may still be known about the compound's properties, it is crucial to consider quantitative information available from any studies that may have been conducted previously on the same drug. A meta-analytic approach has the advantages of being able to properly account for between-study heterogeneity, and it may be readily extended to prediction or shrinkage applications. Here we propose a simple and robust two-stage approach for the estimation of maximum tolerated dose(s) utilizing penalized logistic regression and Bayesian random-effects meta-analysis methodology. Implementation is facilitated using standard R packages. The properties of the proposed methods are investigated in Monte Carlo simulations. The investigations are motivated and illustrated by two examples from oncology.

Meta-analysis methods for multiple related markers: Applications to microbiome studies with the results on multiple α-diversity indices.

Meta-analysis is a practical and powerful analytic tool that enables a unified statistical inference across the results from multiple studies. Notably, researchers often report the results on multiple related markers in each study (eg, various α-diversity indices in microbiome studies). However, univariate meta-analyses are limited to combining the results on a single common marker at a time, whereas existing multivariate meta-analyses are limited to the situations where marker-by-marker correlations are given in each study. Thus, here we introduce two meta-analysis methods, multi-marker meta-analysis (mMeta) and adaptive multi-marker meta-analysis (aMeta), to combine multiple studies throughout multiple related markers with no priori results on marker-by-marker correlations. mMeta is a statistical estimator for a pooled estimate and its SE across all the studies and markers, whereas aMeta is a statistical test based on the test statistic of the minimum P-value among marker-specific meta-analyses. mMeta conducts both effect estimation and hypothesis testing based on a weighted average of marker-specific pooled estimates while estimating marker-by-marker correlations non-parametrically via permutations, yet its power is only moderate. In contrast, aMeta closely approaches the highest power among marker-specific meta-analyses, yet it is limited to hypothesis testing. While their applications can be broader, we illustrate the use of mMeta and aMeta to combine microbiome studies throughout multiple α-diversity indices. We evaluate mMeta and aMeta in silico and apply them to real microbiome studies on the disparity in α-diversity by the status of human immunodeficiency virus (HIV) infection. The R package for mMeta and aMeta is freely available at https://github.com/hk1785/mMeta.

Bounds for the weight of external data in shrinkage estimation.

Shrinkage estimation in a meta-analysis framework may be used to facilitate dynamical borrowing of information. This framework might be used to analyze a new study in the light of previous data, which might differ in their design (e.g., a randomized controlled trial and a clinical registry). We show how the common study weights arise in effect and shrinkage estimation, and how these may be generalized to the case of Bayesian meta-analysis. Next we develop simple ways to compute bounds on the weights, so that the contribution of the external evidence may be assessed a priori. These considerations are illustrated and discussed using numerical examples, including applications in the treatment of Creutzfeldt-Jakob disease and in fetal monitoring to prevent the occurrence of metabolic acidosis. The target study's contribution to the resulting estimate is shown to be bounded below. Therefore, concerns of evidence being easily overwhelmed by external data are largely unwarranted.

A CD-based mapping method for combining multiple related parameters from heterogeneous intervention trials.

Effect size can differ as a function of the elapsed time since treatment or as a function of other key covariates, such as sex or age. In evidence synthesis, a better understanding of the precise conditions under which treatment does work or does not work well has been highly valued. With increasingly accessible individual patient or participant data (IPD), more precise and informative inference can be within our reach. However, simultaneously combining multiple related parameters across heterogeneous studies is challenging because each parameter from each study has a specific interpretation within the context of the study and other covariates in the model. This paper proposes a novel mapping method to combine study-specific estimates of multiple related parameters across heterogeneous studies, which ensures valid inference at all inference levels by combining sample-dependent functions known as Confidence Distributions (CD). We describe the "CD-based mapping method" and provide a data application example for a multivariate random-effects meta-analysis model. We estimated up to 13 study-specific regression parameters for each of 14 individual studies using IPD in the first step, and subsequently combined the study-specific vectors of parameters, yielding a full vector of hyperparameters in the second step of meta-analysis. Sensitivity analysis indicated that the CD-based mapping method is robust to model misspecification. This novel approach to multi-parameter synthesis provides a reasonable methodological solution when combining complex evidence using IPD.

Comparison of random-effects meta-analysis models for the relative risk in the case of rare events: A simulation study.

Pooling the relative risk (RR) across studies investigating rare events, for example, adverse events, via meta-analytical methods still presents a challenge to researchers. The main reason for this is the high probability of observing no events in treatment or control group or both, resulting in an undefined log RR (the basis of standard meta-analysis). Other technical challenges ensue, for example, the violation of normality assumptions, or bias due to exclusion of studies and application of continuity corrections, leading to poor performance of standard approaches. In the present simulation study, we compared three recently proposed alternative models (random-effects [RE] Poisson regression, RE zero-inflated Poisson [ZIP] regression, binomial regression) to the standard methods in conjunction with different continuity corrections and to different versions of beta-binomial regression. Based on our investigation of the models' performance in 162 different simulation settings informed by meta-analyses from the Cochrane database and distinguished by different underlying true effects, degrees of between-study heterogeneity, numbers of primary studies, group size ratios, and baseline risks, we recommend the use of the RE Poisson regression model. The beta-binomial model recommended by Kuss (2015) also performed well. Decent performance was also exhibited by the ZIP models, but they also had considerable convergence issues. We stress that these recommendations are only valid for meta-analyses with larger numbers of primary studies. All models are applied to data from two Cochrane reviews to illustrate differences between and issues of the models. Limitations as well as practical implications and recommendations are discussed; a flowchart summarizing recommendations is provided.

Short- and intermediate-term exposure to NO2 and mortality: A multi-county analysis in China.

Nitrogen dioxide (NO2) is a well-established traffic emissions tracer and has been associated with multiple adverse health outcomes. Short- and long-term exposure to NO2 has been studied and is well-documented in existing literature, but information on intermediate-term NO2 effects and mortality is lacking, despite biological plausibility. We obtained daily NO2 and mortality data from 42 counties in China from 2013 to 2015. Distributed-lag non-linear models were employed to investigate the relationship between non-accidental mortality and NO2 up to 30 days before the event, including PM2.5, temperature, relative humidity, and holidays as covariates in a random effects meta-analysis pooling county-specific estimates. We repeated the analysis for cardiovascular- and respiratory-related mortality, and explored sex-stratified associations. Per 10 µg/m3 increase in NO2, we estimated a 0.13% (95%CI: 0.03, 0.23%), 0.57% (95%CI: -0.04, 1.18%), and -0.14% (95%CI: -1.63, 1.37%) change in non-accidental mortality for same-day and previous-day NO2 (lag0-1 cumulated), in the preceding 7 days (lag0-7 cumulated), and in the preceding 30 days (lag0-30 cumulated), respectively. The strongest estimate was observed for respiratory-related mortality in the lag0-30 cumulated effect for women (3.12%; 95%CI: -1.66, 8.13%). We observed a trend of higher effect estimates of intermediate-term NO2 exposure on respiratory mortality compared to that of the short-term, although the differences were not statistically significant. Our results at longer lags for all-cause and cardiovascular mortality were sensitive to modeling choices. Future work should further investigate intermediate-term air pollution exposure given their potential biological relevance, but in larger scale settings.

Random-effects meta-analysis of few studies involving rare events.

Meta-analyses of clinical trials targeting rare events face particular challenges when the data lack adequate numbers of events for all treatment arms. Especially when the number of studies is low, standard random-effects meta-analysis methods can lead to serious distortions because of such data sparsity. To overcome this, we suggest the use of weakly informative priors (WIPs) for the treatment effect parameter of a Bayesian meta-analysis model, which may also be seen as a form of penalization. As a data model, we use a binomial-normal hierarchical model (BNHM) that does not require continuity corrections in case of zero counts in one or both arms. We suggest a normal prior for the log-odds ratio with mean 0 and standard deviation 2.82, which is motivated (a) as a symmetric prior centered around unity and constraining the odds ratio within a range from 1/250 to 250 with 95% probability and (b) as consistent with empirically observed effect estimates from a set of 37 773 meta-analyses from the Cochrane Database of Systematic Reviews. In a simulation study with rare events and few studies, our BNHM with a WIP outperformed a Bayesian method without a WIP and a maximum likelihood estimator in terms of smaller bias and shorter interval estimates with similar coverage. Furthermore, the methods are illustrated by a systematic review in immunosuppression of rare safety events following pediatric transplantation. A publicly available R package, MetaStan, is developed to automate a Bayesian implementation of meta-analysis models using WIPs.

Dynamically borrowing strength from another study through shrinkage estimation.

Meta-analytic methods may be used to combine evidence from different sources of information. Quite commonly, the normal-normal hierarchical model (NNHM) including a random-effect to account for between-study heterogeneity is utilized for such analyses. The same modeling framework may also be used to not only derive a combined estimate, but also to borrow strength for a particular study from another by deriving a shrinkage estimate. For instance, a small-scale randomized controlled trial could be supported by a non-randomized study, e.g. a clinical registry. This would be particularly attractive in the context of rare diseases. We demonstrate that a meta-analysis still makes sense in this extreme case, effectively based on a synthesis of only two studies, as illustrated using a recent trial and a clinical registry in Creutzfeld-Jakob disease. Derivation of a shrinkage estimate within a Bayesian random-effects meta-analysis may substantially improve a given estimate even based on only a single additional estimate while accounting for potential effect heterogeneity between the studies. Alternatively, inference may equivalently be motivated via a model specification that does not require a common overall mean parameter but considers the treatment effect in one study, and the difference in effects between the studies. The proposed approach is quite generally applicable to combine different types of evidence originating, e.g. from meta-analyses or individual studies. An application of this more general setup is provided in immunosuppression following liver transplantation in children.

Cocaine dependence: "Side effects" and syndrome formation within 1-12 months after first cocaine use.

BACKGROUND: This project offers new epidemiological estimates for DSM-IV cocaine dependence among sub-groups of newly incident cocaine users in the United States (US), including estimated attack rates for 21 dependence-related cocaine side effect problems and experiences occurring <12 months after onset. METHOD: In 2002-2016, US National Surveys on Drug Use and Health (NSDUH) sampled, recruited, and assessed cocaine experiences of non-institutionalized civilians. Unweighted estimates for year-pairs (2002-3,…,2015-16) are from 3488 cocaine powder-only initiates and 275 powder-then-crack initiates (all evaluated <12 months after onset). Analysis-weighted attack rate estimates are incidence proportions with 95% confidence intervals (CI), summarized via meta-analysis. RESULTS: Evaluated <12 months after onset, meta-analysis summaries show 5% of powder-only initiates developed cocaine dependence (95% CI = 4%, 6%) versus 22% of powder-then-crack initiates (95% CI = 17%, 29%). For several cocaine side effect problems and experiences (e.g., 'loss of control' indicators) there is a statistically robust crack-associated excess risk. CONCLUSIONS: Three interpretations of observed crack-associated excess risk are especially cogent and deserving of continued inquiry: (1) Powder-then-crack initiates start with heightened dependence risk susceptibilities (i.e., pre-dating onset); (2) Powder-using initiates become cocaine dependent and then start using crack; (3) The cocaine delivery variant of 'crack-smoking' is more toxic than powder insufflation. For powder-then-crack initiates, the cocaine dependence risk (22%) is modestly lower but statistically undifferentiable from a recently estimated risk of heroin dependence <12 months after heroin onset (30%). Clinicians can use these side effect estimates in an evidence-based diagnostic workup when patients disclose new onsets of cocaine use.

A meta-analysis of the P3 amplitude in tasks requiring deception in legal and social contexts.

In deception tasks the parietal P3 amplitude of the event-related potential indicates either recognition of salient stimuli (larger P3 following salient information) or mental effort (smaller P3 following demanding information). This meta-analysis (k = 77) investigated population effect sizes (δ) for conceptual and methodological a-priori moderators (study design, pre-task scenario, context of deception tasks, and P3 quantification). Within-subject designs show evidence of the underlying cognitive processes, between-subject designs allow for comparisons of cognitive processes in culprits vs. innocents. Committed vs. imagined mock crime scenarios yield larger δ. Deception tasks with a legal context result in almost twice as large δ than deception tasks with social-evaluative and social-biographical contexts. Peak-to-peak P3 quantification resulted in larger δ than other quantifications. Counter-measure techniques in 3-stimulus protocols reduce the discriminability of concealed vs. truthful P3 amplitudes. Depending on stimulus knowledge, deception tasks provide evidence for the salience hypothesis and the mental effort hypothesis, respectively.

Effects of hemoglobin levels during pregnancy on adverse maternal and infant outcomes: a systematic review and meta-analysis.

Maternal anemia affects approximately 56 million women worldwide and increases the risk of adverse pregnancy outcomes. Our study aimed to summarize the evidence for the association between maternal hemoglobin (Hb) concentrations and maternal or infant outcomes, evaluating it in a continuous manner. In this systematic review and meta-analysis, we conducted an electronic search on PubMed, Embase, CINAHL, and Web of Science from inception to April 19, 2017, and further updated to November 21, 2018, applying subject heading terms related to pregnant women with anemia. We included 117 studies with 4,127,430 pregnancies. Maternal anemia increased the risk of low birth weight (odds ratio (OR), 1.65; 95% confidence interval (CI): 1.45-1.87), preterm birth (PTB) (OR, 2.11; 95% CI: 1.76-2.53), perinatal mortality (PNM) (OR, 3.01; 95% CI: 1.92-4.73), stillbirth (OR, 1.95; 95% CI: 1.15-3.31), and maternal mortality (OR, 3.20; 95% CI: 1.16-8.85). A nonlinear relationship was found between maternal Hb and adverse maternal and infant outcomes. The OR of outcomes such as PTB, small-for-gestational age, PNM, preeclampsia, gestational hypertension, and postpartum hemorrhage was increased by two to three times. Assessing Hb as a continuous variable is important to determine the associated risk of adverse outcomes with decreasing or increasing levels.

Bayesian log-Gaussian Cox process regression: with applications to meta-analysis of neuroimaging working memory studies.

Working memory (WM) was one of the first cognitive processes studied with functional magnetic resonance imaging (fMRI). With now over 20 years of studies on WM, each study with tiny sample sizes, there is a need for meta-analysis to identify the brain regions consistently activated by WM tasks, and to understand the inter-study variation in those activations. However, current methods in the field cannot fully account for the spatial nature of neuroimaging meta-analysis data or the heterogeneity observed among WM studies. In this work, we propose a fully Bayesian random-effects meta-regression model based on log-Gaussian Cox processes, which can be used for meta-analysis of neuroimaging studies. An efficient MCMC scheme for posterior simulations is presented which makes use of some recent advances in parallel computing using graphics processing units (GPUs). Application of the proposed model to a real dataset provides valuable insights regarding the function of the WM.

Likelihood-based random-effects meta-analysis with few studies: empirical and simulation studies.

BACKGROUND: Standard random-effects meta-analysis methods perform poorly when applied to few studies only. Such settings however are commonly encountered in practice. It is unclear, whether or to what extent small-sample-size behaviour can be improved by more sophisticated modeling. METHODS: We consider likelihood-based methods, the DerSimonian-Laird approach, Empirical Bayes, several adjustment methods and a fully Bayesian approach. Confidence intervals are based on a normal approximation, or on adjustments based on the Student-t-distribution. In addition, a linear mixed model and two generalized linear mixed models (GLMMs) assuming binomial or Poisson distributed numbers of events per study arm are considered for pairwise binary meta-analyses. We extract an empirical data set of 40 meta-analyses from recent reviews published by the German Institute for Quality and Efficiency in Health Care (IQWiG). Methods are then compared empirically as well as in a simulation study, based on few studies, imbalanced study sizes, and considering odds-ratio (OR) and risk ratio (RR) effect sizes. Coverage probabilities and interval widths for the combined effect estimate are evaluated to compare the different approaches. RESULTS: Empirically, a majority of the identified meta-analyses include only 2 studies. Variation of methods or effect measures affects the estimation results. In the simulation study, coverage probability is, in the presence of heterogeneity and few studies, mostly below the nominal level for all frequentist methods based on normal approximation, in particular when sizes in meta-analyses are not balanced, but improve when confidence intervals are adjusted. Bayesian methods result in better coverage than the frequentist methods with normal approximation in all scenarios, except for some cases of very large heterogeneity where the coverage is slightly lower. Credible intervals are empirically and in the simulation study wider than unadjusted confidence intervals, but considerably narrower than adjusted ones, with some exceptions when considering RRs and small numbers of patients per trial-arm. Confidence intervals based on the GLMMs are, in general, slightly narrower than those from other frequentist methods. Some methods turned out impractical due to frequent numerical problems. CONCLUSIONS: In the presence of between-study heterogeneity, especially with unbalanced study sizes, caution is needed in applying meta-analytical methods to few studies, as either coverage probabilities might be compromised, or intervals are inconclusively wide. Bayesian estimation with a sensibly chosen prior for between-trial heterogeneity may offer a promising compromise.

Multistep estimators of the between-study variance: The relationship with the Paule-Mandel estimator.

A wide variety of estimators of the between-study variance are available in random-effects meta-analysis. Many, but not all, of these estimators are based on the method of moments. The DerSimonian-Laird estimator is widely used in applications, but the Paule-Mandel estimator is an alternative that is now recommended. Recently, DerSimonian and Kacker have developed two-step moment-based estimators of the between-study variance. We extend these two-step estimators so that multiple (more than two) steps are used. We establish the surprising result that the multistep estimator tends towards the Paule-Mandel estimator as the number of steps becomes large. Hence, the iterative scheme underlying our new multistep estimator provides a hitherto unknown relationship between two-step estimators and Paule-Mandel estimator. Our analysis suggests that two-step estimators are not necessarily distinct estimators in their own right; instead, they are quantities that are closely related to the usual iterative scheme that is used to calculate the Paule-Mandel estimate. The relationship that we establish between the multistep and Paule-Mandel estimator is another justification for the use of the latter estimator. Two-step and multistep estimators are perhaps best conceptualized as approximate Paule-Mandel estimators.