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explains that this is a foundational concept in measurement theory: “any single score from a measurement is to represent a single quality” (p. 107). This is technically referred to as making a CA unidimensional (technically stated, a unidimensional test “measures only one dimension or only one latent trait” [AERA et al., 2014, p. 224]). The notion that unidimensionality is foundational to test theory can be traced back to the middle of the 1900s. For example, in a foundational article on measurement theory in 1959, Frederic M. Lord notes that a test is a “collection of tasks; the examinee’s performance on these tasks is taken as an index of [a student’s] standing along some psychological dimension” (p. 473). Over forty years later, David Thissen and Howard Wainer (2001) explain:

      Before the responses to any set of items are combined into a single score that is taken to be, in some sense, representative of the responses to all of the items, we must ascertain the extent to which the items “measure the same thing.” (p. 10)

      Without unidimensionality, a score on a test is difficult to interpret. For example, assume that two students receive a score of 70 on the same test, but that test measures two dimensions. This is depicted in figure 1.1.

      Note: Black = patterns; gray = data analysis. Total possible points for black (patterns) = sixty; total possible points for gray (data analysis) = forty.

      Figure 1.1: Two students’ scores on a two-dimensional test.

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