By contrast, if second-order sampling error explains all the observed true variance, then one is entitled to conclude that the same mechanism is likely to occur in the populations studied in the different meta-analyses. In this case, corrected first-order meta-analytic means are closer, but not necessarily identical, to the grand mean than the uncorrected means. If second-order sampling error accounts only for a portion of the variance (i.e., true variance is not null σ 2 > 0), then one must assume different mechanisms for at least some of the results obtained in the individual meta-analyses. Then, the proportion of the between-meta-analysis variance explained by second-order sampling error is calculated and used to produce more accurate estimates in first-order meta-analyses. First, first-order meta-analytic means (ḡ i) are used to calculate a weighted grand mean (g̿). Second-order meta-analysis aims to estimate to what extent second-order sampling error accounts for the difference across overall meta-analytic means in a set of first-order meta-analyses regarding a particular topic. In brief, meta-analysis offers the best way to build reliable cumulative knowledge, because it makes it possible, among other things, to summarize a large number of studies, identify and correct publication biases, and measure the impact of study methodological features on the effect sizes that are observed. The role of covariates in accounting for true heterogeneity is evaluated via meta-regression. Finally, meta-analysis provides measures of true (i.e., not due to random error) between-study heterogeneity necessary to assess the degree of consistency between studies in the same field. Furthermore, the asymmetry in the distribution of the effect sizes due to the systematic suppression of non-significant effects and p-hacking can be detected and corrected by publication bias analysis. That allows researchers to produce more precise measures of an effect than the single primary study. In fact, one of the major advantages of meta-analysis is reducing sampling error by merging effect sizes from different sample sizes. As argued by Schmidt and Oh ( 2013, 2016), integrating the findings in a particular area via meta-analysis is the most effective way for evaluating whether the existing replication studies corroborate or refute the original findings. In order to overcome the problem of low statistical power and reliably estimate the size of effects, researchers have extensively employed meta-analysis. The lack of generalization of skills acquired by training is thus an invariant of human cognition. That is, no impact on far-transfer measures was observed regardless of the type of population and cognitive-training program. Crucially, when placebo effects and publication bias were controlled for, the overall effect size and true variance equaled zero. By contrast, Models 2 and 3 highlighted that far-transfer effects are small or null. Model 1 showed that working-memory training does induce near transfer, and that the size of this effect is moderated by the type of population. Model 3 ( k = 233) extended Model 2 by adding six meta-analyses assessing the far-transfer effects of other cognitive-training programs (video-games, music, chess, and exergames). In Models 1 ( k = 99) and 2 ( k = 119), we investigated the impact of working-memory training on near-transfer (i.e., memory) and far-transfer (e.g., reasoning, speed, and language) measures, respectively, and whether it is mediated by the type of population. We addressed this issue by using second-order meta-analysis. However, it is yet to be established whether the effects differ across cognitive-training programs and populations (children, adults, and older adults). Recent replication attempts and large meta-analytic investigations have shown that the benefits of cognitive-training programs hardly go beyond the trained task and similar tasks. Second-order meta-analysis (i.e., a meta-analysis of meta-analyses) offers a powerful tool for achieving this aim, and we use this technique to illuminate the controversial field of cognitive training. Theory building in science requires replication and integration of findings regarding a particular research question.
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