[Rasch] Conjoint measurement VS. Conjoint analysis

Andrew Kyngdon AKyngdon at lexile.com
Wed Nov 3 12:03:39 EST 2010

```Gudeta,

Conjoint analysis and the theory of conjoint measurement are two very different things indeed.

The theory of conjoint measurement (Debreu, 1960; Luce & Tukey, 1964) is a general theory of quantity. It describes the kind of relations that exist between the levels of quantitative attributes. Theoretically, it can enable the quantification of attributes for which there exists no natural concatenation operation (i.e., unrestricted side-by-side combination). We are able to concatenate rigid steel rods and so hence evidence of length being a quantity can be readily observed. This is typically not the case with psychological attributes. Combining people's heads does not demonstrate that intelligence is a quantity. But it does not logically follow, from the absence of concatenation operations alone, that intelligence is not a quantity. This remains a coherent hypothesis. The theory of conjoint measurement may be useful in testing this hypothesis.

It is important to note that the scientific definition of a measurement is the product of a real number and a unit. This is the definition of measurement found without exception in physics and metrology (c.f. Emerson, 2008). For example, 273m is a measurement of length where "m" represents the International System of Units (S.I.) (Bureau International des Poids et Mesures, 2006) unit length quantity of the metre. It must be noted that application of the theory of conjoint measurement does not yield the product of a real number and a unit. It's name is unfortunately misleading in this respect. Nevertheless, as I have shown on the "Talking Measurement" forum, the cancellation axioms of the theory of conjoint measurement can be algebraically deduced from the scientific definition of measurement using the work of German mathematician Hoelder (1901).

In order to scientifically measure an attribute, one must have a descriptive theory of the natural system within which that attribute behaves (however sketchy and incomplete that theory is) and a well defined unit of measurement. These are things that are external to the theory of conjoint measurement.

The most basic theoretical form of conjoint measurement is A = f(X, Y), where A, X and Y are quantities and f is an additive, non-interactive function of some kind (Michell, 1990). It can be that X and Y are the same quantity, which also means that A is the same quantity. But it can also be that X and Y are different quantities, in which case A is a different quantity again. For example, X might be mass and Y volume, in which case A is density. A psychological example might be that A is performance upon an intelligence test whilst X is working memory capacity and Y is motivation (Stankov & Cregan, 1993). If certain relations between the degrees (levels) of A are empirically observed, it follows that A, X and Y are quantities. These relations are specified by what are known as the cancellation axioms of conjoint measurement.

A form of conjoint measurement - tradeoff or concomitant variation - has been practiced for hundreds of years before the theory of conjoint measurement was formally articulated.  This involves experimentally manipulating the levels of X, for example, such that levels of A remain constant per unit of Y. Formally, this is a special case of conjoint measurement where the cells of the diagonal of the conjoint array (the levels of A) are equal.

The best introduction to the theory of conjoint measurement was written by Michell (1990). But the theory also has more complex forms, such as A = X1 + X2 +... + Xn and A = Z(X + Y). I have had a paper recently accepted by the British Journal of Mathematical and Statistical Psychology which applied distributive conjoint measurement (the A = Z(X + Y) version) to data simulated using Steve Humphry's Extended Frame of Reference Rasch Model.

The most notable use of the theory of conjoint measurement in psychology occurred in the field of the utility of gains and losses under conditions of risk and uncertainty. David Krantz used it in the formal proof to Kahneman & Tversky's (1979) "prospect theory". In 2002 Kahneman received the Nobel Economics Prize for prospect theory (Tversky died in 1996).

In contrast, conjoint analysis has nothing to do with quantity and scientific measurement. It is essentially the name given to an empirical process employed in the field of marketing. Essentially, one constructs hypothetical goods or services from combinations of attributes. These are known as "profiles". These profiles are either ranked or rated by participants. Sometimes, when there are a small number of attributes, pairwise preference data can be elicited. Once the data has been collected, the attributes are then either dummy coded or effects coded and the data analysed using a multinomial probit or logit model. The parameters of the resulting equation are then considered as "part-worth utilities" of the attributes. When one attribute is price, the model parameters for price can be interpreted as indicative of the "willingness to pay" for the product or service.

Louviere, Hensher & Swait (2000) presented a detailed overview of conjoint analysis. But it is quite technical in places.

I hope this helps.

Andrew

From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf Of Gudeta Fufaa
Sent: Tuesday, 2 November 2010 10:13 AM
To: Parisa Daftari Fard; rasch list
Subject: [Rasch] Conjoint measurement VS. Conjoint analysis

Dear Colleagues,

I appreciate it very much of some body can briefly explain to me the difference between cojoint measurement and conjoint analysis preferably using an example. Thanks.

Gudeta

-------------- next part --------------
An HTML attachment was scrubbed...
URL: https://mailinglist.acer.edu.au/pipermail/rasch/attachments/20101102/417f80ea/attachment.html
```