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Methods for determining the statistical part worth value of factors in conjoint analysis
MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical
and Computer Modelling 31 (2000) 261-271 www.elsevier.nl/locate/mcm
Methods for Determining the Statistical Part Worth Value of Factors in Conjoint Analysis H. NOGUCHI AND H. ISHII Faculty of Engineering, Osaka University Suita, Osaka 565, Japan Abstract-MONANOVA is one type of conjoint analysis used for measuring the part worth value of factors to the total evaluation, exclusively using preference ranking data of a group of commercial products designed by presorted factors. Its criterion, called Stress, is the same as that of monotone regression in MDS. Consequently, part worth values obtained from MONANOVA do not necessarily lead to definite solutions but give an approximate comparison of each factor’s contribution to the total evaluation of the products. In this paper, we would like to discuss two problems with MONANOVA: namely, its reliability and stability. We would then lie to propose two possible solutions to these problems: the first combines regression and monotone methods; the second employs quadratic fractional programming. With these, we hope to obtain each factor’s contribution to the total evaluation as a partial correlation coefficient and to demonstrate that one can compare the factor’s contribution to the total evaluation with constant stability.@ 2000 Elsevier Science Ltd. All rights reserved.
Keywords-Conjoint analysis, MONotone ANalysis Of VAriance, Partial correlation coefficient, Monotone regression, Quadratic fractional programming.
1. INTRODUCTION Conjoint analysis is a scaling method originally developed in mathematical psychology. It is also used for measuring each factor’s contribution to the whole evaluation of products made from some presorting of the factors. Consequently, in the field of marketing, conjoint analysis has been attracting the world’s attention since the 1980s as the method to preestimate the values of presorted factors from the total evaluation data. The first application of conjoint analysis in industry was made by Green and Rao [l] in 1971. They analyzed data from consumer’s preference ranking, investigating the question, “Which combination of ads would be the most effective?” through printing eight types of advertisements in five different magazines. In Japan, Asano [2] and Noguchi and Isogai [3] have been researching the applications of conjoint analysis in the field of industry. However, even with MONANOVA (MONotone ANalysis Of VAriance) as the representative method of conjoint analysis, one cannot always measure theoretically the definite part worth values from the consumer’s preference ranking data, since the method is based upon the criterion The authors wish to express sincere acknowledgments anonymous referees.
for the helpful comments and suggestions provided by
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262
H. NOGUCHI AND H. ISHII
called Stress (defined result,
the obtained
This particular been widely
undesirability
tried to measure Therefore,
definite
coefficients”
to the total
in MDS, MultiDimensional
cannot
data.
values directly
be applied
is well known;
users to specify
ranking
by default
As a
methods.
the method
to a certain
degree,
there are very few researchers
by elaborating
it is possible
nevertheless,
values of factors
However,
Scaling).
in statistical
has
simply
who have
on MONANOVA.
to apply the scores in statistics
and that it is also possible to consistently
estimate
through
contributions
“partial of factors
evaluation.
In Section
2, we briefly
3, through
part worth values. illustrate
preference
we hope to show that
correlation
Regression”
of MONANOVA
used, since it enables
from the consumers’
In Section
as in “Monotone
scores from MONANOVA
survey
MONANOVA
MONANOVA,
In Section
we explain
4, we propose
them with examples.
as a representative
Finally,
the problems
more elaborate
in Section
method
of conjoint
with reliability
methods
analysis.
and stability
of the
based on MONANOVA
5, we summarize
and
our results.
2. MONANOVA We can classify the various between
a scaling
methods
of preference
of conjoint
analysis,
data and a criterion
Table 1. Classification
Scaling of Data
as in Table 1, according
of goodness
of methods
Criterion Goodness
in conjoint
of
of fit. analysis.
Method
of Fit
paired rank comparison Method
ordinal scale
of
Amount stress
metric
method of least squares
ordinal scale
to paired comparison
maximum
Analysis
analysis
= TRADEOFF Monotone
ordinal scale
transform
of Conjoint
Trade-off
and pairwise sign consistent
Total
to the relation
analysis
[4,5] of variance
= MONANOVA
likelihood
Multiple
regression
Logarithm
estimation
[6,7] analysis
likelihood
= LOGIT
method
[8]
Linear programming Method
of
Individual
minimize violation ordinal scale
Difference
of
techniques
multidimensional
and model space
analysis of preference = LINMAP Weighted
nominal scale ordinal scale
least squares type
the following
notation,
and then introduce
[9]
additive model
based on the alternating least squares = WADDALS
We require
for
both data space
[lo]
MONANOVA.
Y = [?h&,...,Ynl’:
ordinal scale of a consumer’s preference for products where n is the number of products, and ’ means transpose,
z = [%I, 22,. . . ) z,]’ :
order preserving
O-l design matrix products,
transformation
to indicate
of y,
each level of factors
of (1)
Factors
m:
in Conjoint
the number
Analysis
263
of levels of each factor,
b = [bI, b2,. . . , b,]‘:
the part worth values to be estimated,
&=Db:
an additive
We use the goodness
conjoint,
model.
called Stress, as defined
of fit criterion,
by Kruskal
[6,7],
(z-i)‘(z-2) stress =
(2)
(“-q’p-2). \;
In MONANOVA,
the method
forms i relation that it
minimizes
obtains
to y while increasing
monotonically.
Stress with the restriction
may be rather
weak and incomplete).
that
with the restriction
Here, zi will be the expected
zi has a monotonic
relationship
Z
that
value of yi
with yi (though
Now, if
= (Z - Db)‘(Z
R(b)
Stress is minimized
b so that
U(b) = (Db - q’
- Db),
numerator
Db) ,
(Db -
(3)
of S2,
denominator of S2,
(4
then we wish to minimize SE
J
R(b)
(5)
0) ’
with respect, to b. The partial
as
1
‘U’/2
g=db=ij
- [
= -iD’
derivative
$(Z
of S with respect
- R1i2 (U,,‘]
- Db) + (Db - a)
= -;
1,
to b is
[;D’(Z
- Db) + SD’(Db
- a)]
(6)
where a = Db and since = -2D’Z + 2D’Db = -2D’(Z - Db), db XJ d x = x [b’D’Db - 2b’D’a + a’a] = 2D’(Db We derive the gradient
vector gk from (6) using the following bk+l
dS b&k
and IX step width,
where the initial
- a).
(8)
algorithm:
where
= bk - agk,
gk= 2%
iterative
(7)
’
k=O,1,2
)...,
value bo is chosen freely. This operation
(9) stops at the point
when either (a) a predetermined value for Stress is reached, or (b) the maximum iteration determines the minimum.
3. ON THE RELIABILITY STABILITY OF THE PART
AND WORTH
We show an example for which we can easily discuss the reliability and stability of the value of the part worth b.
H. NOGUCHI
264 Table 2. Consumer’s
AND H.ISHII
preference example of six real estate options.
Factors 2. Forms
1. Styles
TYPW of Houses
Eclectic Japanese
Totally t- Japanese
I
I
z
Totally
and Western
Western
A
0
1
0
1
B
0
1
0
0
1
50
C
1
0
0
1
0
40
6a
0
D
1
0
0
0
1
3@l
E
0
0
1
1
0
20
F
0
0
1
0
1
l@l
(blz)
(b13)
@21)
(b22)
(‘w)
qq. . . m
consumer’s
preference order
In Table 2, six different kinds of real estate from A to F are given. These houses consist of two factors (styles and forms) for the house. When a consumer selects them, let us suppose that his preference is in alphabetical order (A 2 B 2 C 2 D 2 E 2 F) according to the order of their scores. By adopting MONANOVA,
let us try to obtain the commercial values of each piece of real
estate by analyzing the two factors (styles and forms) from the consumers’ preference ranking. These commercial values correspond to the part worth value b. We shall consider two numerical examples with each part worth of the two factors having the scores as given in the following tables. Numerical
Example
1
Part Worth
Total Worth’s
Numerical
Score
Example
bll
bi2
bal
bzz
1
5
-3
4
0
A
B
C
D
E
9
2
2
5
2
1
1
1
Stress
= 0
F 2
-3.
2
Part Worth
bll
blz
b13
-4
-2
-6
8
7
0
-4
6
5
0
2
-2
2
4
-2
A Total Worth’s
5
b13
Score
6
0
B 2
5
b-21
C 2
4
2
b22
4
3
2
1
D
E
3
2
2
Stress = 0
F > -
1
In the case that the value b is obtained as in Numerical Example 1, the value Z will be equivalent to the total value. The value Z is not always coincident to the value Z; however, Z will increase as the consumers rank their preference higher; then the orders of Z and Z become parallel. As a result, Kruskal’s Stress becomes 0. Each factor’s contribution to the total evaluation
Factors
can be obtained
by subtracting
the minimum
of styles is bi2 - his = 5 - (-3) contribution
Analysis
265
value from the maximum.
= 8, and that
Example
case will be completely
2, a number
to each other,
of forms to the total
the contribution
of possible
values
so the result
evaluation
will be largely
(1) Which
determined
of the two results
(2) How stable Even though
Stress
will be four, and that
by forms.
of styles.
The value
Example
becomes:
of forms is four times as high as that
the consumers
for b are given.
to the Z score, as in Numerical
equivalent
of Z and Z are equivalent
Z in each
1. The point
gains
= 0. In this case, the
of styles will be one.
Consequently,
Thus,
the preference
In the cases above, two questions
of
remain.
to the initial
vector given?
we have the restrictions
(a) the reappearance
of the order relation
model of the part worth
(c) the minimization
Z and Z,
between
b for Z, and
of S,
the solution for the part worth values are not necessarily unique. utilize the contributions of b for statistical use. In the literature, has been formally
discussed
by Asano
Moreover, it is impossible to the stability of such solutions
[l l], Katahira
[12], Ogawa
der Lans et al. [14]. They considered using the preference order y as a condition unique solution and proposed a method to derive a unique solution in some special neither
the
is more reliable?
are they with respect
(b) an additive
as the above
Hence, the contribution
of forms is bzi - b22 = 4 - 0 = 4. As a result,
of styles is twice as high as that of forms.
In Numerical
contribution
in Conjoint
obtain
a general
statistically. In the next section,
method
based on MONANOVA
we propose
two more elaborate
4. The first method order preserving
PROPOSED
has the objective
function
[13], and Van to obtain a cases. They
nor discuss how to deal with the value b methods
based on MONANOVA.
METHODS (10) as a criterion
of goodness
of fit and uses an
transformation Minimize
c
(Z - 2)’
-+ 0.
The criterion (10) is of least squares type. After we transform using a full rank one-dimensional quantification method similar b = (D*‘D*)-1
(10) the matrix D to the matrix D* to that of Hayashi [15], we obtain
D*‘Z
(where D* is the matrix obtained by removing the first category’s column of each factor after the second factor from D in order to prevent [(D’D)] = 0). We calculate Z by using this b. We then investigate whether the order relation of Z is consistent with that of Z. If it is consistent, this b is the expected value. If the order relations for Z and Z are not consistent,
we transform
Z to Z1 using
an order
preserving transformation. We obtain the value of bi as (D*‘D*)-lD*‘Zr as before. This bl then hopefully gives an expected value with consistent order relations for Zi and Z. We tested out several cases when the order relations for Z and Z were not consistent. We found that the order relation becomes most stable only after an order preserving transformation of Z as Zr . In the case for which the order relation for Z is still not consistent even after transforming to Z1, it is possible to apply the order preserving transformation repeatedly to find the optimum estimate. After such transformations are repeated, it is obvious that the value b will converge to the vector 0. Finally, the part worth values for each factor were centered around zero (so that cj bij = 0), and the partial correlation coefficient between each factor and the consumer’s preference Z was
266
H.
NOGUCHI AND H. ISHII
obtained. The partial correlation coefficient is simply the correlation between Djbj and Z where Dj is the matrix of columns of D for the J‘th factor, and bj is the vector of part-worth values for the jth factor. The statistical null hypothesis that a particular factor is uncorrelated with the preference scores can then be tested using the respective coefficient. We summarize these results for Example 2 in Tables 2 and 3. Table 3. The results of calculation from the example of Table 2. Factor 2
Factor 1
The Value of b
b13
bzl
bzz
TYP_ of Houses
i
Z
-2
0.5
-0.5
A
6
6
==+ B
5
5
c
4
4
constant = 3.5
D
3
3
Spearman’s rank
E
2
2
correlation coefficients
F
1
1
bll
blz
0
2
Range of a Same Factor
12 - (-2)1 = 4
10.5 - (-0.5)1 = 1
Partial Correlation Coefficient
0.956
0.293
Consistent Statistic between 2 and Z
Stress = 0
1.000
In Table 4, we show another example for the evaluation of a business by its president. Table 4 shows the evaluation ranking of eleven departments of an enterprise made by its president. The following evaluation factors have been preset: (1) Achievement (1. fully achieved, 2. unachieved), (2) Economic Environment of the Year (1. favorable, 2. unfavorable), (3) Effort Made Toward Assignments (1. excellent, 2. average, 3. unsatisfactory). The results for the contributions of the factors to the total evaluation of these departments as prepared by the president are given in (a) and (b). The outcomes when conventional MONANOVA is applied are given in (a), while those when our first proposed method is applied are given in (b). Table 4 shows that the result of our first method is better, in the order-relation with Z, than that of MONANOVA. Also, it is possible to consistently compare the contributions of factors from the evaluation made by the president, with their partial correlation coefficients. It can be seen that the president ranked mainly according to effort made toward assignments and achievement. The second method which we propose is to obtain the part worth values as the minimizer of the quadratic fractional function (11). We consider S2(b) as the criterion of goodness of fit S2(b)=
(z-i)‘(z-i)
with
= (z-i)‘(z-~)
(0)‘(U)
(Lc)‘(“-c)’ il + 22 -I- *. . + in
(11)
n C=iyb=
constant vector.
ei + iz + . ** + in I I3 n: S~RSSis the criterion of goodness of fit of monotone regression, whereas S2(b) will be the criterion of goodness of fit for obtaining b by minimizing the DISTANCE, defined as difference or ORDER. If F(X) = (Db - Z)‘(Db
- Z) - X(Db - C)‘(Db
- C)
= (1 - X)b’D’Db - 2b’D’(Z - XC) + Z’Z - XC’C = (1 - X)b’D’Db - 2b’D’Z + 2Xb’D’C + Z’Z, for the parameter
X such that 0 I X L 1, then the following relation holds (see [IS]).
(12)
Factors in Conjoint Analysis
267
Table 4. The result of the president’s evaluation of esch department in the company. (It suggests each factor may depend on the order of his evaluation.)
1. favorable 1 1. Fully Achieved
2. unfavorable
2. average
I
Mds
10
7
8
P
8
9
9
1. excellent
AP
5
6
6
2. average
FYl
7
10
10
El
11
11
11
3. unsatisfactory
2. Unachieved
I 2. unfavorable i
3. unsatisfactory
(b) The Result of This First Method
(a) The Result of MONANOVA The number of iterations = 100 >
Part
Part
Partial
Worth
Worth
Correlation
Factor 1
Factor 1
1. fully achieved
-0.156
1. fully achieved
2. unachieved
-1.531
2. unachieved
Factor 2 1. favorable
-0.925
1. favorable
2. unfavorable
-0.003
2. unfavorable
1. excellent
2. average
0.202
2. average
-0.249
3. unsatisfactory
-1.430
-0.361
2.239
Constant
Constant coefficient
Rank correlation coefficient
1. Let Fx 4 min{F(X)
1b}.
If F,* = 0
at X = A*,
then A* = min {S2(b)
1b}
Further, the minimizer of F(X*) is also an optimal solution of S2(b). See
0.628
Factor 3 1.541
3. unsatisfactory
PROOF.
0.767 -0.637
1. excellent
Rank correlation
0.830
Factor 2
Factor 3
THEOREM
1.174 -1.409
the Appendix.
0.859
H. NOGUCHI AND H. ISHII
268
Since 0 I X I: 1, F(X) is a convex function of b, the minimizer bA of F(X) is the solution of aF(X) = 2(1db
X)D’Db
- 2D’Z -t 2XD’C = 0,
that is, the solution of (1 - X)D’Db
+ XD’C = D’Z.
(13)
+ Z’Z = (Z - Dbx)‘Z.
(14)
Now, Fx = -b’xD’Z Setting X = A* for
Fx = 0 and substituting X* into bx, we obtain the part worth values b* as
the solution to equations (13) and (15), (Z - Db)‘Z
= 0,
(15)
from Theorem 1. For the example in Table 2, -0
1
o-
10
0
1
1 0 0 10001’ 0 0 1
1
0
1
0
0
l_
o D=
-0 cc =
1
0
0
1
2(bll + b12 + bls) + 3(bzl+ 6
b22)
From (13), bx is calculated as follows: 2 bllx = b1sX+ -7 1-X 4 b12X= brsX + 1 -X’
(17)
b 21x = -; - b13X - -?2(1 - X) ’ b22x = ; - b13X-
5 2(1 - X)’
Substituting thii bx into F(X), we obtain
(18) The condition to minimiie Fx implies X* = 0, and so b* is b;, = bT3 + 2,
’
bT2 = bT3 + 4,
b;, = 2 - bz3,
b;, = 1 - b;3.
(1%
Thii is the indefinite solution b* expressed in terms of a parameter b:3. If we assume that the total score of each factor of all houses equals zero to obtain a definite solution of b, then the solution will be that given in Table 3. This example is a particular case for which X’ = 0. The next example is a case for which X* # 0. The example is simply that of Table 2 except the preference rankings of A and B have been reversed. The results are shown in Table 5. In this example, the order relations of Z and Z are not consistent, and so we introduce Z1, an order preserving transformation of Z, for example z1A = zlB =
(aA + &J) (5.667 + 5.333) = = 5.500, 2 2
Factors in Conjoint Analysis
269
Table 5. The result of the first method. House
Zl
Zl
i
A
5.611
5.500
5.667
B
5.389
5.500
5.333
C
3.611
3.667
3.667
D
3.389
3.333
3.333
E
1.611
1.667
1.667
F
1.389
1.333
1.333
Z
Type
Partial Correlation
0.999
Coefficient
0.316
constant = 3.5 Consistent
Spearman’s
Statistic
rank
correlation coefficients
between 21 and Z
Stress = m
0.943
and bI fi (D*‘D*)-lD*‘Zl.
S2(b)
= 0.083
Again, we can calculate bx from (13), giving the result of the second
method as follows:
2 hx
=
h3X
+
-
b12X =
h3X
+
-
1-X’ 4 1-X’
b21X =
2 -
(20)
11
7 h3X
-
-7
6(1 - X)
b We substitute bx into F(X)
(21) The condition Fx = 0 implies
Consequently, we obtain b* as
b;, = bT3+ 2.165,
bi2 = bi3 + 4.330,
b;l = 1.515 - b&,
b& = 1.555 - bi3.
(22)
Again, we assume that the total score of each factor of all houses equals zero, so that the solution is as given in Table 6. Table 6. The result of the second method. i
Z
A
5.840
5
B
5.480
6
House Type
Consistent
Statistic
between 21 and Z
pear-man s ran correlation coefficients
C
3.680
4
D
3.320
3
E
1.520
2
F
1.160
1
Stress = JKiiZ S2(b) = 0.076
H. NOGUCHI AND H. ISHII
270
In the second method, the criterion is a minimization of S2(b); therefore, it is natural that the value of S2(b) should become smaller than that in the first method. In summary, the first step is to obtain definite solutions by either the first or the second method. The second step is to compare the contributions of factors to the total evaluation by using the partial correlation coefficients. Through these steps, it is possible to check the reliability of the values obtained in MONANOVA;
for this particular reason, we would like to recommend these
methods. Also, as we can see from the criterion S2(b) of the second method, the fitness between Z and 2 is better than that of ordinary regression analysis under the condition that
21 + z2 +
. . . +
zn = lOO(%).
Therefore, this method has an advantage when Z is obtained as a popularity voting ratio, not as preference ranking data.
5. CONCLUSION There is also a method, called “LINMAP”,
which is classified as a variation of conjoint anal-
ysis; this is one of the methods for measuring definite solutions. This method is effective only when formularization of preference model functions is possible. However, conjoint analysis is more commonly adopted in other cases when such formularization is impossible. Consequently, MONANOVA has been widely applied in these other cases. On the other hand, it is well known that solutions obtained through MONANOVA are not definite, i.e., stable. Other researchers had, in the past, pointed out the problems of the method in terms of reliability and stability, as we mentioned in Section 3. Nevertheless, we believe that a more elaborate method has not appeared yet so as to easily obtain definite solutions. Therefore, we have proposed two more elaborate methods to replace conventional MONANOVA is this paper. One is the method that combines regression analysis with order preserving transformations; as the example shows, this method proves to be more stable in obtaining reliable solutions, compared with MONANOVA. The other is for determining a minimizer of S2(b) defined as a distance difference in place of Stress (an order difference defined). In this latter method, the solutions obtained are more applicable, especially when the value Z is applied for a preference voting ratio, than those obtained through regression analysis. Whichever method is applied, we have transformed the part worth values b into partial correlation coefficients, so as to be easily adopted in statistics, and tried to obtain the contributions of factors to the total evaluations so as to compare them with constant stability. In the future, we would like to seek wider applications of part worth values b and to investigate suitable conditions for such applications.
APPENDIX PROOF
A
OF THEOREM
1 [17]
First, note that R(b) and U(b) are convex functions of b. Hence, F(X) is a convex function of b. Since FA = minF(X) = min{R(b) -dF(X) = 2(1bb
- AU(b) 1b),
X)D’Db - 2D’Db + 2XDC,
Factors in Conjoint
Analysis
271
with fixed A, we obtain Fx, the minimum of F(A), by finding bx from v A” > X’, we have F{w+(I-~)Y)
= min (R(b) - {CA’+ (1 - t)A”} U(b)] = R(bt) - {tA’ + (1 - t)X’} U(bt) = t {R(b,) - X’U(b,)} + (1 - t) {R(b,) - X”U(bt)} 2 tmin {R(b) - A/U(b)} + = tFx, +
and
for
= 0. Now,
(1 -
t)Fp
Fx, = min {R(b) - X/U(b)} =
(1 - t)
,
R (b’) - X'U(b’) > R (b’) - X"U (b’) (24)
2 min (R(b) - X”U(b)} = Fp. Hence,.from (23) and (24), Fx is monotone decreasing and a concave function of A. ~ lirflm Fx
*
(23)
min {R(b) - X”U(b)}
= +oo,
xli’p,Fx +
and so there exists a unique A* such that Fi = 0; that F (A’, b;) = R(b;)
-
Furthermore,
= -00,
is,
X*U(b;) = 0.
Since 0=
R (b;) - X*U(b:) 5 R(b) - X*U(b),
and x*
=
for any b,
R(b) -R (bi) < _ U(bi)
- U(b)’
we have that bi is an optimal solution of S2( b).
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Rae, Conjoint measurement for quantifying judgement data, Journal of Marketing Research 3, 355-363, (1971). 2. H. Asano, Collection of Marketing Research Techniques for Developing New Pruducts, Japan Management Association, (1988). 3. H. Noguchi and T. Isogai, Conjoint Analysis, Studies in the Humanities and Social Science, XL, pp. 111-148, Faculty of General Education, Osaka University, (1991). 4. R.M. Johnson, Pahwise nonmetric multidimensional scaling, Psychometrika 38, 11-18, (1973). 5. R.M. Johnson, A simple method for pairwise monotone regression, Psychometrika 40, 163-168, (1975). 6. J.B. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis, Psychometrika 29, l-28, (1964). 7. J.B. Kruskal, Analysis of factorial experiments by estimating monotone transformations of data, Journal of
the Royal Statistical Society B27, 251-263, (1965). 8. P.E. Green et al., Conjoint analysis in consumer research: Issue and outlook, Journal of Consumer 5, 103123, (1978). 9. V. Srinivasan et al., Linear programming technique for multidimensional analysis of preferences, Psychometrika 38, 337-369, (1973). 10. Y. Takane et al., An individual differences additive model: An alternating least squares method with optimal scaling features, Psychometrika 45, 183-209, (1980). 11. H. Asano, A Monte Carlo simulation of conjoint analysis. An empirical study on the recovery of structure, NRC Marketing Bulletin 3, 89-97, (1982). 12. H. Katahira, Constructing a perceptual map by rank order logit analysis: A more efficient method of nonmetric MDS, In Contemporary Research in Marketing, Proceedings of the lph Annual Conference of the European Marketing Academy, (Edited by K. Moller et al.), pp. 791-808, (1985). 13. K. Ogawa, An approach to simultaneous estimation and segmentation in conjoint analysis, Marketing Science 6, 66-81, (1986). 14. LA. Van der Lans et al., Constrained part worth estimation in conjoint analysis using the self-explicated utility model, Int. J. Res. Marketing 9, 325-344, (1992). 15. C. Hayashi, One dimensional quantification and multidimensional quantification, Annals of the Japan Association for Philosophy of Science 3, 115-120, (1968). 16. I. Ibaraki, H. Ishii et al., Algorithms for quadratic fractional programming problems, Journal of the Opemtions Research Society of Japan 19, 228-244, (1976). 17. H. Dinkelbach, On nonlinear fractional programming, Management Science 13, 492-498, (1967).
Mathematical
and Computer Modelling 31 (2000) 261-271 www.elsevier.nl/locate/mcm
Methods for Determining the Statistical Part Worth Value of Factors in Conjoint Analysis H. NOGUCHI AND H. ISHII Faculty of Engineering, Osaka University Suita, Osaka 565, Japan Abstract-MONANOVA is one type of conjoint analysis used for measuring the part worth value of factors to the total evaluation, exclusively using preference ranking data of a group of commercial products designed by presorted factors. Its criterion, called Stress, is the same as that of monotone regression in MDS. Consequently, part worth values obtained from MONANOVA do not necessarily lead to definite solutions but give an approximate comparison of each factor’s contribution to the total evaluation of the products. In this paper, we would like to discuss two problems with MONANOVA: namely, its reliability and stability. We would then lie to propose two possible solutions to these problems: the first combines regression and monotone methods; the second employs quadratic fractional programming. With these, we hope to obtain each factor’s contribution to the total evaluation as a partial correlation coefficient and to demonstrate that one can compare the factor’s contribution to the total evaluation with constant stability.@ 2000 Elsevier Science Ltd. All rights reserved.
Keywords-Conjoint analysis, MONotone ANalysis Of VAriance, Partial correlation coefficient, Monotone regression, Quadratic fractional programming.
1. INTRODUCTION Conjoint analysis is a scaling method originally developed in mathematical psychology. It is also used for measuring each factor’s contribution to the whole evaluation of products made from some presorting of the factors. Consequently, in the field of marketing, conjoint analysis has been attracting the world’s attention since the 1980s as the method to preestimate the values of presorted factors from the total evaluation data. The first application of conjoint analysis in industry was made by Green and Rao [l] in 1971. They analyzed data from consumer’s preference ranking, investigating the question, “Which combination of ads would be the most effective?” through printing eight types of advertisements in five different magazines. In Japan, Asano [2] and Noguchi and Isogai [3] have been researching the applications of conjoint analysis in the field of industry. However, even with MONANOVA (MONotone ANalysis Of VAriance) as the representative method of conjoint analysis, one cannot always measure theoretically the definite part worth values from the consumer’s preference ranking data, since the method is based upon the criterion The authors wish to express sincere acknowledgments anonymous referees.
for the helpful comments and suggestions provided by
08957177/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: SO8957177(00)00095-9
Typeset by
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262
H. NOGUCHI AND H. ISHII
called Stress (defined result,
the obtained
This particular been widely
undesirability
tried to measure Therefore,
definite
coefficients”
to the total
in MDS, MultiDimensional
cannot
data.
values directly
be applied
is well known;
users to specify
ranking
by default
As a
methods.
the method
to a certain
degree,
there are very few researchers
by elaborating
it is possible
nevertheless,
values of factors
However,
Scaling).
in statistical
has
simply
who have
on MONANOVA.
to apply the scores in statistics
and that it is also possible to consistently
estimate
through
contributions
“partial of factors
evaluation.
In Section
2, we briefly
3, through
part worth values. illustrate
preference
we hope to show that
correlation
Regression”
of MONANOVA
used, since it enables
from the consumers’
In Section
as in “Monotone
scores from MONANOVA
survey
MONANOVA
MONANOVA,
In Section
we explain
4, we propose
them with examples.
as a representative
Finally,
the problems
more elaborate
in Section
method
of conjoint
with reliability
methods
analysis.
and stability
of the
based on MONANOVA
5, we summarize
and
our results.
2. MONANOVA We can classify the various between
a scaling
methods
of preference
of conjoint
analysis,
data and a criterion
Table 1. Classification
Scaling of Data
as in Table 1, according
of goodness
of methods
Criterion Goodness
in conjoint
of
of fit. analysis.
Method
of Fit
paired rank comparison Method
ordinal scale
of
Amount stress
metric
method of least squares
ordinal scale
to paired comparison
maximum
Analysis
analysis
= TRADEOFF Monotone
ordinal scale
transform
of Conjoint
Trade-off
and pairwise sign consistent
Total
to the relation
analysis
[4,5] of variance
= MONANOVA
likelihood
Multiple
regression
Logarithm
estimation
[6,7] analysis
likelihood
= LOGIT
method
[8]
Linear programming Method
of
Individual
minimize violation ordinal scale
Difference
of
techniques
multidimensional
and model space
analysis of preference = LINMAP Weighted
nominal scale ordinal scale
least squares type
the following
notation,
and then introduce
[9]
additive model
based on the alternating least squares = WADDALS
We require
for
both data space
[lo]
MONANOVA.
Y = [?h&,...,Ynl’:
ordinal scale of a consumer’s preference for products where n is the number of products, and ’ means transpose,
z = [%I, 22,. . . ) z,]’ :
order preserving
O-l design matrix products,
transformation
to indicate
of y,
each level of factors
of (1)
Factors
m:
in Conjoint
the number
Analysis
263
of levels of each factor,
b = [bI, b2,. . . , b,]‘:
the part worth values to be estimated,
&=Db:
an additive
We use the goodness
conjoint,
model.
called Stress, as defined
of fit criterion,
by Kruskal
[6,7],
(z-i)‘(z-2) stress =
(2)
(“-q’p-2). \;
In MONANOVA,
the method
forms i relation that it
minimizes
obtains
to y while increasing
monotonically.
Stress with the restriction
may be rather
weak and incomplete).
that
with the restriction
Here, zi will be the expected
zi has a monotonic
relationship
Z
that
value of yi
with yi (though
Now, if
= (Z - Db)‘(Z
R(b)
Stress is minimized
b so that
U(b) = (Db - q’
- Db),
numerator
Db) ,
(Db -
(3)
of S2,
denominator of S2,
(4
then we wish to minimize SE
J
R(b)
(5)
0) ’
with respect, to b. The partial
as
1
‘U’/2
g=db=ij
- [
= -iD’
derivative
$(Z
of S with respect
- R1i2 (U,,‘]
- Db) + (Db - a)
= -;
1,
to b is
[;D’(Z
- Db) + SD’(Db
- a)]
(6)
where a = Db and since = -2D’Z + 2D’Db = -2D’(Z - Db), db XJ d x = x [b’D’Db - 2b’D’a + a’a] = 2D’(Db We derive the gradient
vector gk from (6) using the following bk+l
dS b&k
and IX step width,
where the initial
- a).
(8)
algorithm:
where
= bk - agk,
gk= 2%
iterative
(7)
’
k=O,1,2
)...,
value bo is chosen freely. This operation
(9) stops at the point
when either (a) a predetermined value for Stress is reached, or (b) the maximum iteration determines the minimum.
3. ON THE RELIABILITY STABILITY OF THE PART
AND WORTH
We show an example for which we can easily discuss the reliability and stability of the value of the part worth b.
H. NOGUCHI
264 Table 2. Consumer’s
AND H.ISHII
preference example of six real estate options.
Factors 2. Forms
1. Styles
TYPW of Houses
Eclectic Japanese
Totally t- Japanese
I
I
z
Totally
and Western
Western
A
0
1
0
1
B
0
1
0
0
1
50
C
1
0
0
1
0
40
6a
0
D
1
0
0
0
1
3@l
E
0
0
1
1
0
20
F
0
0
1
0
1
l@l
(blz)
(b13)
@21)
(b22)
(‘w)
qq. . . m
consumer’s
preference order
In Table 2, six different kinds of real estate from A to F are given. These houses consist of two factors (styles and forms) for the house. When a consumer selects them, let us suppose that his preference is in alphabetical order (A 2 B 2 C 2 D 2 E 2 F) according to the order of their scores. By adopting MONANOVA,
let us try to obtain the commercial values of each piece of real
estate by analyzing the two factors (styles and forms) from the consumers’ preference ranking. These commercial values correspond to the part worth value b. We shall consider two numerical examples with each part worth of the two factors having the scores as given in the following tables. Numerical
Example
1
Part Worth
Total Worth’s
Numerical
Score
Example
bll
bi2
bal
bzz
1
5
-3
4
0
A
B
C
D
E
9
2
2
5
2
1
1
1
Stress
= 0
F 2
-3.
2
Part Worth
bll
blz
b13
-4
-2
-6
8
7
0
-4
6
5
0
2
-2
2
4
-2
A Total Worth’s
5
b13
Score
6
0
B 2
5
b-21
C 2
4
2
b22
4
3
2
1
D
E
3
2
2
Stress = 0
F > -
1
In the case that the value b is obtained as in Numerical Example 1, the value Z will be equivalent to the total value. The value Z is not always coincident to the value Z; however, Z will increase as the consumers rank their preference higher; then the orders of Z and Z become parallel. As a result, Kruskal’s Stress becomes 0. Each factor’s contribution to the total evaluation
Factors
can be obtained
by subtracting
the minimum
of styles is bi2 - his = 5 - (-3) contribution
Analysis
265
value from the maximum.
= 8, and that
Example
case will be completely
2, a number
to each other,
of forms to the total
the contribution
of possible
values
so the result
evaluation
will be largely
(1) Which
determined
of the two results
(2) How stable Even though
Stress
will be four, and that
by forms.
of styles.
The value
Example
becomes:
of forms is four times as high as that
the consumers
for b are given.
to the Z score, as in Numerical
equivalent
of Z and Z are equivalent
Z in each
1. The point
gains
= 0. In this case, the
of styles will be one.
Consequently,
Thus,
the preference
In the cases above, two questions
of
remain.
to the initial
vector given?
we have the restrictions
(a) the reappearance
of the order relation
model of the part worth
(c) the minimization
Z and Z,
between
b for Z, and
of S,
the solution for the part worth values are not necessarily unique. utilize the contributions of b for statistical use. In the literature, has been formally
discussed
by Asano
Moreover, it is impossible to the stability of such solutions
[l l], Katahira
[12], Ogawa
der Lans et al. [14]. They considered using the preference order y as a condition unique solution and proposed a method to derive a unique solution in some special neither
the
is more reliable?
are they with respect
(b) an additive
as the above
Hence, the contribution
of forms is bzi - b22 = 4 - 0 = 4. As a result,
of styles is twice as high as that of forms.
In Numerical
contribution
in Conjoint
obtain
a general
statistically. In the next section,
method
based on MONANOVA
we propose
two more elaborate
4. The first method order preserving
PROPOSED
has the objective
function
[13], and Van to obtain a cases. They
nor discuss how to deal with the value b methods
based on MONANOVA.
METHODS (10) as a criterion
of goodness
of fit and uses an
transformation Minimize
c
(Z - 2)’
-+ 0.
The criterion (10) is of least squares type. After we transform using a full rank one-dimensional quantification method similar b = (D*‘D*)-1
(10) the matrix D to the matrix D* to that of Hayashi [15], we obtain
D*‘Z
(where D* is the matrix obtained by removing the first category’s column of each factor after the second factor from D in order to prevent [(D’D)] = 0). We calculate Z by using this b. We then investigate whether the order relation of Z is consistent with that of Z. If it is consistent, this b is the expected value. If the order relations for Z and Z are not consistent,
we transform
Z to Z1 using
an order
preserving transformation. We obtain the value of bi as (D*‘D*)-lD*‘Zr as before. This bl then hopefully gives an expected value with consistent order relations for Zi and Z. We tested out several cases when the order relations for Z and Z were not consistent. We found that the order relation becomes most stable only after an order preserving transformation of Z as Zr . In the case for which the order relation for Z is still not consistent even after transforming to Z1, it is possible to apply the order preserving transformation repeatedly to find the optimum estimate. After such transformations are repeated, it is obvious that the value b will converge to the vector 0. Finally, the part worth values for each factor were centered around zero (so that cj bij = 0), and the partial correlation coefficient between each factor and the consumer’s preference Z was
266
H.
NOGUCHI AND H. ISHII
obtained. The partial correlation coefficient is simply the correlation between Djbj and Z where Dj is the matrix of columns of D for the J‘th factor, and bj is the vector of part-worth values for the jth factor. The statistical null hypothesis that a particular factor is uncorrelated with the preference scores can then be tested using the respective coefficient. We summarize these results for Example 2 in Tables 2 and 3. Table 3. The results of calculation from the example of Table 2. Factor 2
Factor 1
The Value of b
b13
bzl
bzz
TYP_ of Houses
i
Z
-2
0.5
-0.5
A
6
6
==+ B
5
5
c
4
4
constant = 3.5
D
3
3
Spearman’s rank
E
2
2
correlation coefficients
F
1
1
bll
blz
0
2
Range of a Same Factor
12 - (-2)1 = 4
10.5 - (-0.5)1 = 1
Partial Correlation Coefficient
0.956
0.293
Consistent Statistic between 2 and Z
Stress = 0
1.000
In Table 4, we show another example for the evaluation of a business by its president. Table 4 shows the evaluation ranking of eleven departments of an enterprise made by its president. The following evaluation factors have been preset: (1) Achievement (1. fully achieved, 2. unachieved), (2) Economic Environment of the Year (1. favorable, 2. unfavorable), (3) Effort Made Toward Assignments (1. excellent, 2. average, 3. unsatisfactory). The results for the contributions of the factors to the total evaluation of these departments as prepared by the president are given in (a) and (b). The outcomes when conventional MONANOVA is applied are given in (a), while those when our first proposed method is applied are given in (b). Table 4 shows that the result of our first method is better, in the order-relation with Z, than that of MONANOVA. Also, it is possible to consistently compare the contributions of factors from the evaluation made by the president, with their partial correlation coefficients. It can be seen that the president ranked mainly according to effort made toward assignments and achievement. The second method which we propose is to obtain the part worth values as the minimizer of the quadratic fractional function (11). We consider S2(b) as the criterion of goodness of fit S2(b)=
(z-i)‘(z-i)
with
= (z-i)‘(z-~)
(0)‘(U)
(Lc)‘(“-c)’ il + 22 -I- *. . + in
(11)
n C=iyb=
constant vector.
ei + iz + . ** + in I I3 n: S~RSSis the criterion of goodness of fit of monotone regression, whereas S2(b) will be the criterion of goodness of fit for obtaining b by minimizing the DISTANCE, defined as difference or ORDER. If F(X) = (Db - Z)‘(Db
- Z) - X(Db - C)‘(Db
- C)
= (1 - X)b’D’Db - 2b’D’(Z - XC) + Z’Z - XC’C = (1 - X)b’D’Db - 2b’D’Z + 2Xb’D’C + Z’Z, for the parameter
X such that 0 I X L 1, then the following relation holds (see [IS]).
(12)
Factors in Conjoint Analysis
267
Table 4. The result of the president’s evaluation of esch department in the company. (It suggests each factor may depend on the order of his evaluation.)
1. favorable 1 1. Fully Achieved
2. unfavorable
2. average
I
Mds
10
7
8
P
8
9
9
1. excellent
AP
5
6
6
2. average
FYl
7
10
10
El
11
11
11
3. unsatisfactory
2. Unachieved
I 2. unfavorable i
3. unsatisfactory
(b) The Result of This First Method
(a) The Result of MONANOVA The number of iterations = 100 >
Part
Part
Partial
Worth
Worth
Correlation
Factor 1
Factor 1
1. fully achieved
-0.156
1. fully achieved
2. unachieved
-1.531
2. unachieved
Factor 2 1. favorable
-0.925
1. favorable
2. unfavorable
-0.003
2. unfavorable
1. excellent
2. average
0.202
2. average
-0.249
3. unsatisfactory
-1.430
-0.361
2.239
Constant
Constant coefficient
Rank correlation coefficient
1. Let Fx 4 min{F(X)
1b}.
If F,* = 0
at X = A*,
then A* = min {S2(b)
1b}
Further, the minimizer of F(X*) is also an optimal solution of S2(b). See
0.628
Factor 3 1.541
3. unsatisfactory
PROOF.
0.767 -0.637
1. excellent
Rank correlation
0.830
Factor 2
Factor 3
THEOREM
1.174 -1.409
the Appendix.
0.859
H. NOGUCHI AND H. ISHII
268
Since 0 I X I: 1, F(X) is a convex function of b, the minimizer bA of F(X) is the solution of aF(X) = 2(1db
X)D’Db
- 2D’Z -t 2XD’C = 0,
that is, the solution of (1 - X)D’Db
+ XD’C = D’Z.
(13)
+ Z’Z = (Z - Dbx)‘Z.
(14)
Now, Fx = -b’xD’Z Setting X = A* for
Fx = 0 and substituting X* into bx, we obtain the part worth values b* as
the solution to equations (13) and (15), (Z - Db)‘Z
= 0,
(15)
from Theorem 1. For the example in Table 2, -0
1
o-
10
0
1
1 0 0 10001’ 0 0 1
1
0
1
0
0
l_
o D=
-0 cc =
1
0
0
1
2(bll + b12 + bls) + 3(bzl+ 6
b22)
From (13), bx is calculated as follows: 2 bllx = b1sX+ -7 1-X 4 b12X= brsX + 1 -X’
(17)
b 21x = -; - b13X - -?2(1 - X) ’ b22x = ; - b13X-
5 2(1 - X)’
Substituting thii bx into F(X), we obtain
(18) The condition to minimiie Fx implies X* = 0, and so b* is b;, = bT3 + 2,
’
bT2 = bT3 + 4,
b;, = 2 - bz3,
b;, = 1 - b;3.
(1%
Thii is the indefinite solution b* expressed in terms of a parameter b:3. If we assume that the total score of each factor of all houses equals zero to obtain a definite solution of b, then the solution will be that given in Table 3. This example is a particular case for which X’ = 0. The next example is a case for which X* # 0. The example is simply that of Table 2 except the preference rankings of A and B have been reversed. The results are shown in Table 5. In this example, the order relations of Z and Z are not consistent, and so we introduce Z1, an order preserving transformation of Z, for example z1A = zlB =
(aA + &J) (5.667 + 5.333) = = 5.500, 2 2
Factors in Conjoint Analysis
269
Table 5. The result of the first method. House
Zl
Zl
i
A
5.611
5.500
5.667
B
5.389
5.500
5.333
C
3.611
3.667
3.667
D
3.389
3.333
3.333
E
1.611
1.667
1.667
F
1.389
1.333
1.333
Z
Type
Partial Correlation
0.999
Coefficient
0.316
constant = 3.5 Consistent
Spearman’s
Statistic
rank
correlation coefficients
between 21 and Z
Stress = m
0.943
and bI fi (D*‘D*)-lD*‘Zl.
S2(b)
= 0.083
Again, we can calculate bx from (13), giving the result of the second
method as follows:
2 hx
=
h3X
+
-
b12X =
h3X
+
-
1-X’ 4 1-X’
b21X =
2 -
(20)
11
7 h3X
-
-7
6(1 - X)
b We substitute bx into F(X)
(21) The condition Fx = 0 implies
Consequently, we obtain b* as
b;, = bT3+ 2.165,
bi2 = bi3 + 4.330,
b;l = 1.515 - b&,
b& = 1.555 - bi3.
(22)
Again, we assume that the total score of each factor of all houses equals zero, so that the solution is as given in Table 6. Table 6. The result of the second method. i
Z
A
5.840
5
B
5.480
6
House Type
Consistent
Statistic
between 21 and Z
pear-man s ran correlation coefficients
C
3.680
4
D
3.320
3
E
1.520
2
F
1.160
1
Stress = JKiiZ S2(b) = 0.076
H. NOGUCHI AND H. ISHII
270
In the second method, the criterion is a minimization of S2(b); therefore, it is natural that the value of S2(b) should become smaller than that in the first method. In summary, the first step is to obtain definite solutions by either the first or the second method. The second step is to compare the contributions of factors to the total evaluation by using the partial correlation coefficients. Through these steps, it is possible to check the reliability of the values obtained in MONANOVA;
for this particular reason, we would like to recommend these
methods. Also, as we can see from the criterion S2(b) of the second method, the fitness between Z and 2 is better than that of ordinary regression analysis under the condition that
21 + z2 +
. . . +
zn = lOO(%).
Therefore, this method has an advantage when Z is obtained as a popularity voting ratio, not as preference ranking data.
5. CONCLUSION There is also a method, called “LINMAP”,
which is classified as a variation of conjoint anal-
ysis; this is one of the methods for measuring definite solutions. This method is effective only when formularization of preference model functions is possible. However, conjoint analysis is more commonly adopted in other cases when such formularization is impossible. Consequently, MONANOVA has been widely applied in these other cases. On the other hand, it is well known that solutions obtained through MONANOVA are not definite, i.e., stable. Other researchers had, in the past, pointed out the problems of the method in terms of reliability and stability, as we mentioned in Section 3. Nevertheless, we believe that a more elaborate method has not appeared yet so as to easily obtain definite solutions. Therefore, we have proposed two more elaborate methods to replace conventional MONANOVA is this paper. One is the method that combines regression analysis with order preserving transformations; as the example shows, this method proves to be more stable in obtaining reliable solutions, compared with MONANOVA. The other is for determining a minimizer of S2(b) defined as a distance difference in place of Stress (an order difference defined). In this latter method, the solutions obtained are more applicable, especially when the value Z is applied for a preference voting ratio, than those obtained through regression analysis. Whichever method is applied, we have transformed the part worth values b into partial correlation coefficients, so as to be easily adopted in statistics, and tried to obtain the contributions of factors to the total evaluations so as to compare them with constant stability. In the future, we would like to seek wider applications of part worth values b and to investigate suitable conditions for such applications.
APPENDIX PROOF
A
OF THEOREM
1 [17]
First, note that R(b) and U(b) are convex functions of b. Hence, F(X) is a convex function of b. Since FA = minF(X) = min{R(b) -dF(X) = 2(1bb
- AU(b) 1b),
X)D’Db - 2D’Db + 2XDC,
Factors in Conjoint
Analysis
271
with fixed A, we obtain Fx, the minimum of F(A), by finding bx from v A” > X’, we have F{w+(I-~)Y)
= min (R(b) - {CA’+ (1 - t)A”} U(b)] = R(bt) - {tA’ + (1 - t)X’} U(bt) = t {R(b,) - X’U(b,)} + (1 - t) {R(b,) - X”U(bt)} 2 tmin {R(b) - A/U(b)} + = tFx, +
and
for
= 0. Now,
(1 -
t)Fp
Fx, = min {R(b) - X/U(b)} =
(1 - t)
,
R (b’) - X'U(b’) > R (b’) - X"U (b’) (24)
2 min (R(b) - X”U(b)} = Fp. Hence,.from (23) and (24), Fx is monotone decreasing and a concave function of A. ~ lirflm Fx
*
(23)
min {R(b) - X”U(b)}
= +oo,
xli’p,Fx +
and so there exists a unique A* such that Fi = 0; that F (A’, b;) = R(b;)
-
Furthermore,
= -00,
is,
X*U(b;) = 0.
Since 0=
R (b;) - X*U(b:) 5 R(b) - X*U(b),
and x*
=
for any b,
R(b) -R (bi) < _ U(bi)
- U(b)’
we have that bi is an optimal solution of S2( b).
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