Under my umbrella: Rating scales obscure statistical power and effect size heterogeneity

Data from rating scales underlie very specific restrictions: They have a lower limit, an upper limit, and they only consist of a few integers. These characteristics produce particular dependencies between means and standard deviations. A mean that is a non-integer, for example, can never be associated with zero variability, while a mean equal to one of the scale's limits can only be associated with zero variability. The relationship can be described by umbrella plots for which we present a formalization. We use that formalization to explore implications for statistical power and for the relationship between heterogeneity in unstandardized and standardized effect sizes. The analysis illustrates that power is not only affected by the mean difference and sample size, but also by the position of a mean within the respective scale. Further, the umbrella restrictions of rating scales can impede interpretability of meta-analytic heterogeneity. Estimations of relative heterogeneity can diverge between unstandardized and standardized effects, raising questions about which of the two patterns of heterogeneity we would want to explain (for example, through moderators). We reanalyze data from the Many Labs projects to illustrate the issue and finally discuss the implications of our observations as well as ways to utilize these properties of rating scales. To facilitate in-depth exploration and practical application of our formalization, we developed the Shiny Umbrellas app, which is publicly available at https://www.apps.meta-rep.lmu.de/shiny_umbrellas/ .