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There are two popular statistical models for meta-analysis, the fixed-effect model and the random-effects model. The fact that these two models employ similar sets of formulas to compute statistics, and sometimes yield similar estimates for the various parameters, may lead people to believe that the models are interchangeable. In fact, though, the models represent fundamentally different assumptions about the data. The selection of the appropriate model is important to ensure that the various statistics are estimated correctly. Additionally, and more fundamentally, the model serves to place the analysis in context. It provides a framework for the goals of the analysis as well as for the interpretation of the statistics. In this paper we explain the key assumptions of each model, and then outline the differences between the models. We conclude with a discussion of factors to consider when choosing between the two models. Copyright © 2010 John Wiley & Sons, Ltd.

There are 2 families of statistical procedures in meta-analysis: fixed-and randomeffects procedures. They were developed for somewhat different inference goals: making inferences about the effect parameters in the studies that have been observed versus making inferences about the distribution of effect parameters in a population of studies from a random sample of studies. The authors evaluate the performance of confidence intervals and hypothesis tests when each type of statistical procedure is used for each type of inference and confirm that each procedure is best for making the kind of inference for which it was designed. Conditionally random-effects procedures (a hybrid type) are shown to have properties in between those of fixed-and random-effects procedures.The use of quantitative methods to summarize the results of several empirical research studies, or metaanalysis, is now widely used in psychology, medicine, and the social sciences. Meta-analysis usually involves describing the results of each study by means of a numerical index (an estimate of effect size, such as a correlation coefficient, a standardized mean difference, or an odds ratio) and then combining these estimates across studies to obtain a summary. Two somewhat different statistical models have been developed for inference about average effect size from a collection of studies, called the fixed-effects and random-effects models. (A third alternative, the mixedeffects model, arises in conjunction with analyses involving study-level covariates or moderator variables, which we do not consider in this article; see Hedges, 1992.) Fixed-effects models treat the effect-size parameters as fixed but unknown constants to be estimated and usually (but not necessarily) are used in conjunction with assumptions about the homogeneity of effect parameters (see, e.g., Hedges, 1982;Rosenthal & Rubin, 1982). Random-effects models treat the effectsize parameters as if they were a random sample from

Glass's estimator of effect size, the sample mean difference divided by the sample standard deviation, is studied in the context of an explicit statistical model. The exact distribution of Glass's estimator is obtained and the estimator is shown to have a small sample bias. The minimum variance unbiased estimator is obtained and shown to have uniformly smaller variance than Glass's (biased) estimator. Measurement error is shown to attenuate estimates of effect size and a correction is given. The effects of measurement invalidity are discussed. Expressions for weights that yield the most precise weighted estimate of effect size are also derived.

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Conventional meta-analytic techniques rely on the assumption that effect size estimates from different studies are independent and have sampling distributions with known conditional variances. The independence assumption is violated when studies produce several estimates based on the same individuals or there are clusters of studies that are not independent (such as those carried out by the same investigator or laboratory). This paper provides an estimator of the covariance matrix of meta-regression coefficients that are applicable when there are clusters of internally correlated estimates. It makes no assumptions about the specific form of the sampling distributions of the effect sizes, nor does it require knowledge of the covariance structure of the dependent estimates. Moreover, this paper demonstrates that the meta-regression coefficients are consistent and asymptotically normally distributed and that the robust variance estimator is valid even when the covariates are random. The theory is asymptotic in the number of studies, but simulations suggest that the theory may yield accurate results with as few as 20-40 studies. Copyright © 2010 John Wiley & Sons, Ltd.

We propose to change the default P-value threshold for statistical significance from 0.05 to 0.005 for claims of new discoveries. T he lack of reproducibility of scientific studies has caused growing concern over the credibility of claims of new discoveries based on 'statistically significant' findings. There has been much progress toward documenting and addressing several causes of this lack of reproducibility (for example, multiple testing, P-hacking, publication bias and under-powered studies). However, we believe that a leading cause of non-reproducibility has not yet been adequately addressed: statistical standards of evidence for claiming new discoveries in many fields of science are simply too low. Associating statistically significant findings with P < 0.05 results in a high rate of false positives even in the absence of other experimental, procedural and reporting problems.For fields where the threshold for defining statistical significance for new discoveries is P < 0.05, we propose a change to P < 0.005. This simple step would immediately improve the reproducibility of scientific research in many fields. Results that would currently be called significant but do not meet the new threshold should instead be called suggestive. While statisticians have known the relative weakness of using P ≈ 0.05 as a threshold for discovery and the proposal to lower it to 0.005 is not new 1,2 , a critical mass of researchers now endorse this change.We restrict our recommendation to claims of discovery of new effects. We do not address the appropriate threshold for confirmatory or contradictory replications of existing claims. We also do not advocate changes to discovery thresholds in fields that have already adopted more stringent standards (for example, genomics and high-energy physics research; see the 'Potential objections' section below).We also restrict our recommendation to studies that conduct null hypothesis significance tests. We have diverse views about how best to improve reproducibility, and many of us believe that other ways of summarizing the data, such as Bayes factors or other posterior summaries based on clearly articulated model assumptions, are preferable to P values. However, changing the P value threshold is simple, aligns with the training undertaken by many researchers, and might quickly achieve broad acceptance.