Corrected item-test correlations

Previous papers on this subject derive the correlation between an item and the remainder of the test. This correlation is unsatisfactory because the reliability of the remainder varies inversely with the reliability of the item omitted. The present paper derives the correlation between an item and the total test, with that item replaced by a rationally equivalent item. The general formula is then modified, for dichotomus items, to give the corrected point-biserial, biserial, and Brogden biserial correlations. The results apply strictly only to factorially homogeneous tests: those in which the same trait or combination of traits is measured (apart from error) by every item.