Relation To The Scientific Definition of Measurement
Satisfaction of the conditions of conjoint measurement means that measurements of the levels of A and X can be expressed as either ratios between magnitudes or ratios between magnitude differences. It is most commonly interpreted as the latter, given that most behavioural scientists consider that their tests and surveys "measure" attributes on so-called "interval scales" (Kline 1998). That is, they believe tests do not identify absolute zero levels of psychological attributes.
Formally, if P, A and X form an additive conjoint structure, then there exist functions from A and X into the real numbers such that for a and b in A and x and y in X:
.
If and are two other real valued functions satisfying the above expression, there exist and real valued constants satisfying:
and .
That is, and are measurements of A and X unique up to affine transformation (i.e. each is an interval scale in Stevens’ (1946) parlance). The mathematical proof of this result is given in (Krantz et al. 1971, pp. 261–6).
This means that the levels of A and X are magnitude differences measured relative to some kind of unit difference. Each level of P is a difference between the levels of A and X. However, it is not clear from the literature as to how a unit could be defined within an additive conjoint context. van der Ven 1980 proposed a scaling method for conjoint structures but he also did not discuss the unit.
The theory of conjoint measurement, however, is not restricted to the quantification of differences. If each level of P is a product of a level of A and a level of X, then P is another different quantity whose measurement is expressed as a magnitude of A per unit magnitude of X. For example, A consists of masses and X consists of volumes, then P consists of densities measured as mass per unit of volume. In such cases, it would appear that one level of A and one level of X must be identified as a tentative unit prior to the application of conjoint measurement.
If each level of P is the sum of a level of A and a level of X, then P is the same quantity as A andX. For example, A and X are lengths so hence must be P. All three must therefore be expressed in the same unit. In such cases, it would appear that a level of either A or X must be tentatively identified as the unit. Hence it would seem that application of conjoint measurement requires some prior descriptive theory of the relevant natural system.
Read more about this topic: Theory Of Conjoint Measurement
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