Impossible Hypotheses and Effect Size Limits

Psychological science is moving toward further specification of effect sizes when formulating hypotheses, performing power-analyses, and considering the relevance of findings. This development has sparked an appreciation for the wider context in which such effect sizes are found, as the importance assigned to specific sizes may vary from situation to situation. We add to this development a crucial, but in psychology hitherto underappreciated, contingency: there are mathematical limits to the magnitudes that population effect sizes can take within the common multivariate context in which psychology is situated, and these limits can be far more restrictive than typically assumed. The implication is that some hypothesized or pre-registered effect sizes may be impossible. At the same time, these restrictions offer a way of statistically triangulating the plausible range of unknown effect sizes. We explain the reason for the existence of these limits, illustrate how to identify them, and offer recommendations and tools for improving hypothesized effect sizes by exploiting the broader multivariate context in which they occur.