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A Multidimensional Approach to ConjointAnalysis
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Applied Stochastic Models and Data Analysis · December 1998
DOI: 10.1002/(SICI)10990747(199812)14:43.0.CO;2W
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Correspondence to: C. N. Lauro, Dipartimento di Matematica e Statistica, Universita` degli Studi di Napoli,
&&
FedericoII
''
Complesso Monte Sant
'
Angelo, Via Cinthia, 80126 Napoli, Italy.

The paper has been supported by a C.N.R. (no. 9301789) grant:
Analisi dei Dati e Statistica Computazionale
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CCC 8755
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10$17.50
Recei
v
ed June 1997
Copyright
1998 John Wiley & Sons, Ltd.
Re
v
ised June 1998
APPLIED STOCHASTIC MODELS AND DATA ANALYSIS
Appl
.
Stochastic Models Data Anal
.
14
, 265
}
274 (1998)
A MULTIDIMENSIONAL APPROACHTO CONJOINT ANALYSIS

C. N. LAURO,
*
G. GIORDANO, R. VERDE
Dipartimento di Matematica e Statistica, Uni
v
ersita
`
degli Studi di Napoli,
**
Federico II
++
Complesso Monte Sant
+
Angelo,Via Cinthia, 80126 Napoli, Italy
SUMMARY
In this paper we propose an alternative approach to conjoint analysis (CA) based on the principalcomponent analysis onto a reference subspace (PCAR). Following this technique, the results of a CA areconsiderably enriched by some graphical interpretation tools and optimal synthesis of the individualestimated partworth coe
$
cients are also obtained with respect to hierarchized judges subpopulations. Copyright
1998 John Wiley & Sons, Ltd.
KEY WORDS:
conjoint analysis; principal component analysis onto a reference subspace
1. INTRODUCTIONConjoint analysis
deals with
preference judgements
, expressed by
individuals
(hereafter called
judges
), about a set of
stimuli
(products or services). Such stimuli are described by several
attributes
.Conjoint analysis, in particular, aims at evaluating the relative importance of the levels of eachattribute in establishing the global preference associated to the di
!
erent stimuli.In this paper we mainly consider the socalled
metric conjoint analysis
(CA) approach in whichthemultiplelinearregressionmodel is used inorder to estimatethe
part

worth coe
.
cient
or
utility
of each level.For each judge a regression model is established. The dependent variable
>
(
g
"
1,
2
,
G
) isa vector of judgements expressed as ranks on a set of
Q
stimuli; the explanatory variables arebinary indicator vectors associated with the levels of each attribute
X
(
k
"
1,
2
,
K
) based ona suitable experimental design. In order to reduce the number of stimuli, the experimental designis usually an orthogonal fractionated design.The basic CA model is expressed as
>
"
b
X
#2#
b
X
#2#
b
X
#
e
(1)
where
e
is the error term. The regression coe
$
cients
b
can be interpreted as individual
&
part

worth coe
.
cients
1
of each level.Hereinafter we will use the following matrix notation: let
X
be the design matrix of dimension
Q
K
, whose rows refer to
Q stimuli
and the columns are the levels of
p
attributes; each attributehas
K
(
h
"
1,
2
,
p
) levels, the total number of levels is equal to
K
"
K
.We assume that the stimuli are characterized by just one level for each attribute, so that theattributes can be represented by indicator matrices.Thus, the design matrix
X
is a partitioned matrix consisting of
p
juxtaposed indicatormatrices
X
(
h
"
1,
2
,
p
) of general element
x
3
0, 1
, (
i
"
1,
2
,
Q
):
X
"
[
X
X
2
X
2
X
] (2)Let
Y
be the
preference
matrix of dimension
Q
G
, whose columns consist of the judgementsgiven by
G
judges with respect to
Q stimuli
. The judgements, expressed by ranking or ratings, areusually de
"
ned on an ordinal scale ranging from 1 to
Q
; usually the highest value corresponds tothe most preferred stimulus.According to the above notation, the CA model (1) can be rewritten as the followingmultivariate regression model:
Y
"
XB
#
E
(3)where respectively,
B
is the matrix of the
G
individual partworth coe
$
cients associated to the
K
attributelevels and
E
the error term.The global preference of each stimulus can be computed adding up the partworth coe
$
cientsrelatedto the proper levels of the stimulus:
XB
<
u
, where
u
is a
G
dimensionvector of unit elements.Usually the partworth coe
$
cients, estimated by individual models, are summarized by thearithmetic mean
b
M
"
1
G
B
<
u
(4)Sometimes such synthesis are obtained for homogeneous subsets of respondents.The problems in interpreting CA results mainly concern with the following aspects:(i) the synthesis of the
individual
partworth coe
$
cients;(ii) the validation of the additivity hypothesis;(iii) the classical regression hypothesis for interpreting results and, even if CA can be seen asa multivariate model, usual geometrical tools for interpreting results are not considered inliterature.In order to improve the interpretation of conjoint analysis results and overcome the mentionedproblems, in this paper we propose an alternative approach to CA in the context of multidimensional data analysis.With the aim of selecting additive hierarchical patterns of preferences,
"
rstly we propose tosummarize the partworth coe
$
cients by performing a factorial decomposition, of the conjointanalysisestimated parameter matrix, through a principal component analysis. So doing theindividual partworth coe
$
cients are optimally aggregate with respect to each factorial axis bymeans of a suitable weighting system re
#
ecting the judges
'
preferences.266
C. N. LAURO, G. GIORDANO AND R. VERDE
Appl
.
Stochastic Models Data Anal
.
14
, 265
}
274 (1998)Copyright
1998 John Wiley & Sons, Ltd.
Moreover, aiming at improving the CA interpretation, the proposed approach considers therepresentation of the relationships existing between attributelevels and individual preference judgements on a factorial subspace.In order to take into account the explanatory power of the adopted experimental design, wepropose a more re
"
ned geometrical approach based on
principal component analysis ontoa reference subspace
(PCAR).
It is worth noting that the criterion optimized in PCAR (i.e. variance explained by theexperimental design) is consistent with the one maximized in the CA model estimation. Thus, itcan be shown that the predicted preference structure is the same as in CA and the estimatedpartworth coe
$
cients are just related to the PCAR eigenstructure.Furthermore, the PCAR geometrical representations allow to enrich even more CA by jointplots of attributelevels, judges and stimuli.An application of the proposed approach to real data (a co
!
eepreferences analysis) allows tohighlight its properties bringing more insight with respect to the results of Conjoint Analysis. Inthe application, some resamplingbased tools are used for evaluating the stability of the results ina nonparametric context.2. OPTIMAL PARTWORTH DECOMPOSITION: FACTORIAL APPROACHESAccording to the metric conjoint analysis approach, the partworth coe
$
cients estimates
B
<
"
(
X
X
)
X
Y
are obtained by minimizing the expressionmin
B
Y
!
XB
(5)Due to the peculiar structure of
X
, it can be easily seen that its rank is equal to (
K
!
p
), andconsequently we cannot compute the inverse of the matrix
X
X
. Di
!
erent alternatives can be usedas solution (5), for example (
X
X
)
the Moore
}
Penrosegeneralized inverse, or otherwise, considering the inverse of
X
X
, when
X
(
Q
(
K
!
p
)) is a
full

rank
matrix obtained by dropping onecolumn
per
block of
X
, and substracting it from the other columns of the same block.If we are interested in estimating only main e
!
ects of each attribute, then we can consider thediagonal matrix
, obtained by setting to zero the o
!
diagonal elements of (
X
X
). Then, thepartworth coe
$
cients are the elements of the matrix
B
<
computed according to the followingexpression:
B
<
"
X
Y
(6)According to (6) the estimated preference matrix is given by the following expression:
Y
<
"
X
X
Y
(7)The partworth coe
$
cients described by the elements of the columns of
B
<
correspond to themultiple regression coe
$
cients of the individual preferences computed by the ordinary leastsquares.Note that the
non

metricconjoint analysis
approach
aims at computing partworthcoe
$
cientsin such a way that the rank order, reconstructed by the estimated model, is the same as theoriginal, or violates the ordering as little as possible. So, in our approach we can consideralternatively the partworth coe
$
cients computed under the monotone constraints on thedependent variables.
CONJOINT ANALYSIS
267
Appl
.
Stochastic Models Data Anal
.
14
, 265
}
274 (1998)Copyright
1998 John Wiley & Sons, Ltd.
In order to obtain suitable synthesis of
part

worth
coe
$
cients in a lowdimensional subspace,in terms of principal components of
B
<
, we look for the axes of maximum variability of thepreference judgements as solutions of the following equation:
B
<
B
<
"
(
"
1,
2
,
m
)
rank(
B
<
)) (8)under the orthonormality constraints:
"
1 and
"
0
∀
O
; where
and
are,respectively, the
th eigenvalue and eigenvector of the
B
<
B
<
.Aggregate partwork coe
$
cients with respect to the factorial axes are thus obtained as
b
"
B
<
(
"
1,
2
,
m
) (9)They represent
optimal aggregate part

worth coe
.
cients
being obtained for individual
part

worth
by means of a weighting system based on eigenvectors
(
"
1,
2
,
m
). They allow todecompose the preference structure expressed by
B
<
with respect to the axes of maximumvariability in a hierarchicaland additive way. Obviously,this solution for the
b
is not unique andit depends on the particular choice of the inverse of (
X
X
).Nevertheless,this approach, based on a PCA synthesis, does not allow to take into account theexplicative power of attributes of the design matrix
X
explained by the factorial axes.In order to decompose the variance of the preference judgements explained by the attributes,expressed as
trace
(
Y
X
(
X
X
)
X
Y
), we consider the eigensolution, under the usual orthonormality constraints, of the following equation:(
Y
X
(
X
X
)
X
Y
)
z
"
z
(
"
1,
2
,
m
;
m
"
rank(
X
)) (10)In the expressions above the (
X
X
)
is the Moore
}
Penrosegeneralizedinverse matrix, while
and
z
are, respectively, the
th eigenvalue and eigenvector of (
Y
X
(
X
X
)
X
Y
).From a geometrical point of view, this approach is equivalent to the
principal componentanalysis onto a reference subspace
(PCAR) de
"
ned as the principal component analysis of the bestimage of
Y
, and is
Y
<
"
P
Y
, onto the subspace
S
spanned by the columns of the matrix
X
. Infact, according to the properties of the Moore
}
Penrosegeneralized inverse,
P
"
X
(
X
X
)
X
isan orthogonal projection operator onto
S
.An alternative solution can be obtained in relation to the approach proposed by D
'
Ambra,Lauro,
when multiple reference subspaces
S
are considered. It consists in a PCA of the imagesof the column vectors of the preferencejudgements matrix
Y
, on the disjoint subspaces
S
,spanned by the column vectors of the design matrix partitions
X
(
h
"
1,
2
,
p
).The maximuminertia axes are obtained as solution of the PCAR characteristic equation,under orthonormality constraints:(
Y
X
(
X
X
)
X
Y
)
w
"
Y
P
Y
w
"
w
(
"
1,
2
,
m
) (11)where:
P
"
X
(
X
X
)
X
(
h
"
1,
2
,
p
) are orthogonal projectors associated to
S
.Similarly than in (9), the aggregate partwork coe
$
cients can be obtained with respect to thefactorial axes
w
(
"
1,
2
,
m
) as
b
I
"
B
<
w
"
X
Y
w
. (12)268
C. N. LAURO, G. GIORDANO AND R. VERDE
Appl
.
Stochastic Models Data Anal
.
14
, 265
}
274 (1998)Copyright
1998 John Wiley & Sons, Ltd.