The quadratic relationship between difficulty of intelligence test items and their correlations with working memory

Fluid intelligence (Gf) is a crucial cognitive ability that involves abstract reasoning in order to solve novel problems. Recent research demonstrated that Gf strongly depends on the individual effectiveness of working memory (WM). We investigated a popular claim that if the storage capacity underlay the WM–Gf correlation, then such a correlation should increase with an increasing number of items or rules (load) in a Gf-test. As often no such link is observed, on that basis the storage-capacity account is rejected, and alternative accounts of Gf (e.g., related to executive control or processing speed) are proposed. Using both analytical inference and numerical simulations, we demonstrated that the load-dependent change in correlation is primarily a function of the amount of floor/ceiling effect for particular items. Thus, the item-wise WM correlation of a Gf-test depends on its overall difficulty, and the difficulty distribution across its items. When the early test items yield huge ceiling, but the late items do not approach floor, that correlation will increase throughout the test. If the early items locate themselves between ceiling and floor, but the late items approach floor, the respective correlation will decrease. For a hallmark Gf-test, the Raven-test, whose items span from ceiling to floor, the quadratic relationship is expected, and it was shown empirically using a large sample and two types of WMC tasks. In consequence, no changes in correlation due to varying WM/Gf load, or lack of them, can yield an argument for or against any theory of WM/Gf. Moreover, as the mathematical properties of the correlation formula make it relatively immune to ceiling/floor effects for overall moderate correlations, only minor changes (if any) in the WM–Gf correlation should be expected for many psychological tests.