Cognitive training and brain stimulation studies have suggested that human cognition, primarily working memory and attention control processes, can be enhanced. Some authors claim that gains (i.e., post-test minus pretest scores) from such interventions are unevenly distributed among people. The magnification account (expressed by the evangelical "who has will more be given") predicts that the largest gains will be shown by the most cognitively efficient people, who will also be most effective in exploiting interventions. In contrast, the compensation account ("who has will less be given") predicts that such people already perform at ceiling, so interventions will yield the largest gains in the least cognitively efficient people. Evidence for this latter account comes from reported negative correlations between the pretest and the training/stimulation gain. In this paper, with the use of mathematical derivations and simulation methods, we show that such correlations are pure statistical artifacts caused by the widely known methodological error called "regression to the mean". Unfortunately, more advanced methods, such as alternative measures, linear models, and control groups do not guarantee correct assessment of the compensation effect either. The only correct method is to use direct modeling of correlations between latent true measures and gain. As to date no training/stimulation study has correctly used this method to provide evidence in favor of the compensation account, we must conclude that most (if not all) of the evidence should be considered inconclusive.
Keywords: Compensation effect; Regression to the mean; Stimulation; Training.