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9 Appendix 3: Using the Optimal Package in R

9 Appendix 3: Using the Optimal Package in R

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48



OPTIMAL DESIGNS FOR THE MEASUREMENT OF PREFERENCES



linear model for paired comparisons that is similar to the model considered by

Scheffe´ (1952) is presented. This will be followed by a discussion of design criteria.

Besides the well-known criteria of D-, A- and E-optimality, we consider the criterion

of stochastic distance optimality originally proposed by Sinha (1970) which has

attracted attention recently (e.g. Liski et al., 1999; SahaRay and Bhandari, 2003).

The idea underlying this criterion is to minimize the distance between the vector of

parameters and its corresponding estimate in a stochastic sense. From an applied

point of view, this criterion appears to be particularly appropriate, since it matches

the conception of a good design as intuitively conceived by many practitioners of

conjoint analysis: a good design should yield estimates which are close to the

parameters. Recently developed D-optimal paired comparison designs (Graßhoff et

al., 2004) with a manageable number of evaluations also happen to be optimal with

regard to the distance criterion. Empirical tests of these designs are performed in

two experiments. In the first experiment, optimal paired comparison designs are

compared with a design derived from procedures presented by Green (1974),

whereas in the second the standard of comparison is defined by a design heuristic

that is used to generate paired comparison designs by ACA (1994), the market

leader for conjoint analysis software over recent years. The name ACA is an

acronym for adaptive conjoint analysis and signifies that in the course of a conjoint

analysis interview previous evaluations are taken into account when the objects

for future pairs have to be chosen. A special feature of the second experiment

consists in the utilization of incomplete descriptions of the objects. That is, the

objects presented for evaluation are characterized by only a subset of the attributes.

These so-called partial profiles are often employed in conjoint analysis as a means

to prevent information overload.



3.2 Conjoint Analysis

Historically, the development of conjoint methods was inspired by what was considered a breakthrough in the branch of mathematical psychology called measurement theory. Luce and Tukey (1964) named this invention conjoint measurement.

The core of conjoint measurement is a representation theorem concerning the

preference relation among objects that are characterized by the levels of certain

attributes. The theorem provides a set of sufficient conditions that guarantee the

existence of an additive representation of the preference relation (for a general

formulation of the theorem, see Krantz et al., 1971, p. 302). That is, when all

conditions are met, there is a utility value for every attribute level so that given any

two objects one is preferred over the other if and only if the sum of the utilities

corresponding to the levels of the preferred object exceeds the respective sum for

the other object. Moreover, the utility values are unique up to certain linear transformations, that is, they are measured on an interval scale. More precisely, the

utility values corresponding to the levels of an attribute are unique up to a linear

transformation with an attribute specific intercept and a slope parameter which is



PAIRED COMPARISON MODELS IN CONJOINT ANALYSIS



49



the same for all attributes. The important consequence of this uniqueness property

is that contrasts between utility values within attributes can be compared across

attributes.

The potential to yield interval scaled utilities from ordinal preference judgements

contributed much to the method’s initial appeal when conjoint measurement was

discovered by marketing researchers. In this vein, Green and Rao (1971) introduced

conjoint measurement to a larger audience in marketing, stressing the fact that

‘its procedures require only rank-ordered input, yet yield interval-scaled output’

(p. 355). With regard to the original formulation of conjoint measurement’s representation theorem, this statement is only valid, however, when all attributes possess

an infinite number of levels and the complete preference relation is known. In order

to determine this relation it would be necessary to rank order the infinite number

of objects that can be generated by combining the attribute levels or to compare all

pairs of objects. Clearly, this is not possible in practice where only a finite number

of levels can be used. For attributes with a finite number of levels the uniqueness

problem of the measurement scale was solved by Fishburn and Roberts (1988).

Despite the aforementioned limitation, conjoint measurement and its numerous

modifications attracted much attention. Since many of these modifications were

only loosely connected with the original conjoint measurement approach, Green

and Srinivasan (1978) introduced the name of conjoint analysis as a collective term

‘to cover models and techniques that emphasize the transformation of subjective

responses into estimated parameters’ (p. 103). A detailed account of the theory

and methods of conjoint analysis is given in Louviere (1988b) where, among other

things, the relationship between conjoint measurement and conjoint analysis is

clarified. Green and Srinivasan (1990) summarize the development of conjoint

methods during the 1980s. In addition to reviewing technical developments, they

address the issues of reliability and validity of measurement. Carroll and Green

(1995) provide a taxonomy of the host of conjoint methods. The commercial use of

conjoint analysis has been documented by Cattin and Wittink (1982), Wittink and

Cattin (1989) as well as Wittink et al. (1994). These papers indicate a continuous

increase of applications in marketing and related areas. Moreover, it can be seen

that the general availability of software for conducting conjoint analysis by the

mid-1980s has fostered the propagation of some methods, whereas the usage of

other methods has steadily declined. In particular, graded paired comparisons play

a prominent role since they represent a major portion of the interview conducted

by the popular ACA program. An edited book by Gustaffson et al. (2000) attests to

the continuing interest in conjoint analysis. Green et al. (2001) integrate findings

from over 30 years of research.



3.3 Paired Comparison Models in Conjoint Analysis

Direct quantitative evaluation of a set of objects is often deemed to yield imprecise

results when the objects are judged with respect to a subjective criterion as is the



50



OPTIMAL DESIGNS FOR THE MEASUREMENT OF PREFERENCES



case with preferences. For example, to assess the sweetness of a number of different

sweeteners represents a considerably difficult task when specimens are tasted one

at a time. Here, the difficulty arises from the lack of a reference standard. A technique that circumvents this problem is the method of paired comparisons.

In a paired comparison task, objects are presented in pairs and the respondent

has to trade off one alternative against the other. The method was originally invented in the field of psychophysics in the middle of the nineteenth century, where

it was used, for example, to measure the perceived heaviness of vessels (see e.g.

David, 1988). However, as was already pointed out by Thurstone (1928) who

employed paired comparisons in connection with his famous law of comparative

judgement for the measurement of attitudes, the method’s domain of application is

much wider. Today’s fields of application include, among others, psychology, economics and sensory analysis.

Models for data arising from paired comparisons can be broadly classified with

regard to the kind of response. Most of the more common models like the one by

Bradley and Terry (1952) assume a qualitative response. That is, it is only observed

which of the objects in a pair dominates the other with respect to the criterion. As

has already been explained, in conjoint analysis, a different type of paired comparisons is most common. In addition to requiring the respondent to indicate the preferred object, these graded paired comparisons demand a quantitative response.

That is, the strength of preference has to be indicated. For such paired comparisons

with a quantitative response, a linear model approach is appropriate (Scheffe´ ,

1952).

The objects that have to be evaluated in a conjoint analysis differ in a variety

of attributes. These attributes can be formally identified with influential factors

occurring in an engineering set-up. Thus it seems attractive to apply methods of

multi-factor approaches from standard linear design theory (Schwabe, 1996). For

that purpose a general linear model has to be developed to fit the data structure

of paired comparisons arising from conjoint analysis. In particular, such a model

should deal simultaneously with both quantitative (continuous) and qualitative

(discrete) types of attributes covering the work by van Berkum (1987) and

Großmann et al. (2001) as well as the more traditional approach based on incomplete block designs (for recent treatments see Mukerjee et al., 2002, and Street et al.,

2001).

As a starting point we assume a general linear model



Y~ tị ẳ ftị0 b ỵ Z~ ðtÞ

for the overall utility Y~ ðtÞ of a single object t which is chosen from a set T of

possible realizations for the object. In general, each object t ¼ ðt1 ; . . . ; tK Þ consists

of a variety of K components that represent the attributes. In this setting, f is a

vector of known regression functions which describe the form of the functional

relationship between the possible t and the corresponding mean response EðY~ ðtÞÞ,



PAIRED COMPARISON MODELS IN CONJOINT ANALYSIS



51



and b is a vector of parameters. Throughout, transposition of vectors and matrices is

indicated by a prime.

In contrast to standard design problems, the utilities of the objects are usually not

directly measurable in paired comparisons. Only preferences can be observed

for one or the other of two objects presented. We assume that the preference is

numerically given as the difference between utilities Yðs; tị ẳ Y~ sị Y~ tị when a

pair of objects s and t are compared. In the given case, the observations are properly

described by the linear model



Yxị ẳ Fxị0 b ỵ Zxị ẳ fsị ftịị0 b ỵ Zs; tị



3:1ị



with settings x ẳ s; tị for the paired comparisons chosen from the design region

X ¼ T Â T of possible pairs. Here, F is the derived regression function defined by

Fxị ẳ Fs; tị ẳ fsị ftị. The comparisons are assumed to result in uncorrelated

and homoscedastic errors Z ¼ Zðs; tÞ with a mean of zero. Note that the above

difference model can be interpreted as a linearization of the Bradley–Terry model

in the particular case of equal utilities (Großmann et al., 2002).

The model can easily accommodate variants of conjoint analysis models like the

so-called vector model, the ideal point model and the part-worth function model

(Green and Srinivasan, 1978). For example, the part-worth function model posits

that the utility of a single object t equals the sum of the utilities corresponding to

its K attribute levels t1 ; . . . ; tK . Formally, this situation can be identified with the

K-way layout without interactions in an ANOVA setting. Here, the attributes are

represented by K factors with v k levels ik ¼ 1; . . . ; v k , k ¼ 1; . . . ; K, each. Let

1fig ð jÞ denote the indicator function which is 1 when j equals i and 0 otherwise.

Then, for



ftị ẳ 1f1g t1 ị; . . . ; 1fv1 g ðt1 Þ; . . . ; 1f1g ðtK Þ; . . . ; 1fv K g ðtK ÞÞ0

and



b ¼ ð 1;1 ; . . . ; 1;v1 ; . . . ;

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