Theory of Conjoint Measurement - Applications of Conjoint Measurement

Applications of Conjoint Measurement

Empirical applications of the theory of conjoint measurement have been sparse (Cliff 1992; Michell 2009).

Levelt, Riemersma & Bunt 1972 applied the theory to the psychophysics of binaural loudness. They found the double cancellation axiom was rejected. Gigerenzer & Strube 1983 conducted a similar investigation and replicated Levelt, et al.' (1972) findings.

Michell 1990 applied the theory to L.L. Thurstone's (1927) theory of paired comparisons, multidimensional scaling and Coombs' (1964) theory of unidimensional unfolding. He found support of the cancellation axioms only with Coombs' (1964) theory. However, the statistical techniques employed by Michell (1990) in testing Thurstone's theory and multidimensional scaling did not take into consideration the ordinal constraints imposed by the cancellation axioms (van der Linden 1994).

(Johnson 2001), Kyngdon (2006), Michell (1994) and (Sherman 1993) tested the cancellation axioms of upon the interstimulus midpoint orders obtained by the use of Coombs' (1964) theory of unidimensional unfolding. Coombs' theory in all three studies was applied to a set of six statements. These authors found that the axioms were satisfied, however, these were applications biased towards a positive result. With six stimuli, the probability of an interstimulus midpoint order satisfying the double cancellation axioms at random is .5874 (Michell, 1994). This is not an unlikely event. Kyngdon & Richards (2007) employed eight statements and found the interstimulus midpoint orders rejected the double cancellation condition.

Perline, Wright & Wainer 1979 applied conjoint measurement to item response data to a convict parole questionnaire and to intelligence test data gathered from Danish troops. They found considerable violation of the cancellation axioms in the parole questionnaire data, but not in the intelligence test data. Moreover, they recorded the supposed "no - test" instances of double cancellation. Interpreting these correctly as instances in support of double cancellation (Michell, 1988), the results of Perline, Wright & Wainer 1979 are better than what they believed.

Stankov & Cregan 1993 applied conjoint measurement to performance on sequence completion tasks. The columns of their conjoint arrays (X) were defined by the demand placed upon working memory capacity through increasing numbers of working memory place keepers in letter series completion tasks. The rows were defined by levels of motivation (A), which consisted in different amount of times available for compelting the test. Their data (P) consisted of completion times and average number of series correct. They found support for the cancellation axioms, however, their study was biased by the small size of the conjoint arrays (3 × 3 is size) and by statistical techniques that did not take into consideration the ordinal restrictions imposed by the cancellation axioms.

Kyngdon (2011) used Karabatsos' (2001) order restricted inference framework to test a conjoint matrix of reading item response proportions (P) where the examinee reading ability comprised the rows of the conjoint array (A) and the difficulty of the reading items formed the columns of the array (X). The levels of reading ability were identified via raw total test score and the levels of reading item difficulty were identified by the Lexile Framework for Reading (Stenner et al. 2006). Kyngdon found that satisfaction of the cancellation axioms was obtained only through permutation of the matrix in a manner inconsistent with the putative Lexile measures of item difficulty. Kyngdon also tested simulated ability test response data using polynomial conjoint measurement. The data were generated using Humphry's extended frame of reference Rasch model (Humphry & Andrich 2008). He found support of distributive, single and double cancellation consistent with a distributive polynomial conjoint structure in three variables (Krantz & Tversky 1971).