Graded Responses and Joining Categories: A Rejoinder to Andrich' “Models For Measurement, Precision, and Nondichotomization of Graded Responses”

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Abstract
Andrich (1995) claims that the “probability distribution [of graded responses] reflects the precision with which the data are collected” (p. 7), and that an “increase in precision of responses [ . . . ] destroys the joining assumption” (p. 22). He stressed “that Jansen and Roskam simply asserted this equivalence [of the joining assumption and ξ-invariance], and did not derive it.” However, Jansen and Roskam (1986) and Roskam and Jansen (1989)—in the sequel referred to as JR—have neither asserted an equivalence between ξ-invariance and the joining assumption, nor defined ξ-invariance such that it could be considered in terms of estimation.