Apple Inc. v. Samsung Electronics Co. Ltd. et al

Filing 991

Administrative Motion to File Under Seal Documents Re Apples Opposition To Samsungs Motion To Exclude Opinions Of Certain Of Apple Experts filed by Apple Inc.. (Attachments: #1 Declaration Of Cyndi Wheeler In Support Of Apples Administrative Motion To File Under Seal Documents Re Apples Opposition to Exclude Apple Experts Opinions, #2 [Proposed] Order Granting Apples Administrative Motion To File Under Seal, #3 Apples Opposition To Samsungs Motion To Exclude Opinions Of Certain Of Apples Experts, #4 Declaration Of Mia Mazza In Support Of Apples Opposition To Samsungs Motion To Exclude Opinions Of Certain Of Apples Experts, #5 Exhibit Mazza Decl. Ex. D, #6 Exhibit Mazza Decl. Ex. F, #7 Exhibit Mazza Decl. Ex. G, #8 Exhibit Mazza Decl. Ex. J, #9 Exhibit Mazza Decl. Ex. K, #10 Exhibit Mazza Decl. Ex. L, #11 Exhibit Mazza Decl. Ex. R, #12 Exhibit Mazza Decl. Ex. S, #13 Exhibit Mazza Decl. Ex. T, #14 Exhibit Mazza Decl. Ex. U, #15 Exhibit Mazza Decl. Ex. V, #16 Exhibit Hauser Decl. Ex. B, #17 Exhibit Hauser Decl. Ex. C, #18 Exhibit Hauser Decl. Ex. D, #19 Exhibit Hauser Decl. Ex. E, #20 Exhibit Musika Decl. Ex. S, #21 Exhibit Musika Decl. Ex. T, #22 Exhibit Musika Decl. Ex. U, #23 [Proposed] Order Denying Samsungs Motion To Exclude Opinions Of Apples Experts)(Jacobs, Michael) (Filed on 5/31/2012) Modified on 6/3/2012 attachment #1 Sealed pursuant to General Order No. 62 (dhm, COURT STAFF).

Exhibit E

11 Conjoint Choice Experiments:
General Characteristics and Alternative
Model Specifications
Rinus Haaijer and Michel Wedel

11.1

Introduction

Conjoint choice experimentation involves the design of product profiles on the
basis of product attributes specified at certain levels, and requires respondents to
repeatedly choose one alternative from different sets of profiles offered to them,
instead of ranking or rating all profiles, as is usually done in various forms of
classic metric conjoint studies. The Multinomial Logit (MNL) model has been the
most frequently used model to analyze the 0/1 choice data arising from such conjoint choice experiments (e.g., Louviere and Woodworth 1983; Elrod, Louviere
and Davey 1992). One of the first articles describing the potential advantages of a
choice approach for conjoint analysis was by Madanski (1980). His conclusion
was that conjoint analysts could adopt the random utility model approach to explain gross trends or predilections in decisions instead of each person’s specific
decision in each choice presented. The real breakthrough for conjoint choice came
with the Louviere and Woodworth (1983) article in which they integrated the
conjoint and discrete choice approaches.
The MNL model is the standard model for analyzing discrete choices, and can
be derived from utility maximization (McFadden 1976). However, the MNL
model does not accommodate heterogeneity of consumer choice behavior and
potentially suffers from the Independence of Irrelevant Alternatives (IIA) property, which may be too restrictive in many practical situations. Latent class or
mixture MNL models have been developed to accommodate heterogeneity (Kamakura, Wedel and Agrawal 1994). The Multinomial Probit (MNP) model does
not suffer from IIA and deals with heterogeneity, but this model has some practical limitations related to identification, prediction and obtaining the choice probabilities. Haaijer et al. (1998) were the first to use a special specification of the
MNP model for conjoint choice experiments.
In this chapter we review the alternative approaches to analyze conjoint choice
experiments. But before doing that, we briefly describe in section 13.2 the general
elements in conjoint analysis and the „classic” conjoint analysis approaches. Next,
in section 13.3, the conjoint choice approach is discussed more extensively and an
overview is given of recent conjoint choice applications in the marketing literature. Section 13.4 gives several approaches that can be used to estimate a conjoint
choice experiment, including the MNL, the Latent Class MNL, and MNP models.
These various models will be illustrated using an application to a conjoint choice

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experiment on coffee makers. Finally, section 13.5 compares the results of the
various models and gives further discussion and conclusions.

11.2

General Concepts and Classic Conjoint Analysis

In marketing one wants to know which characteristics of products or services are
important to consumers, for reasons of product optimization, new product design,
price setting, market segmentation and competitive positioning amongst others. A
technique, originally developed in the early 60's by Luce and Tukey (1964), that
could eventually be applied to answer that question, is conjoint analysis. In conjoint analysis products or services are defined on a limited number of relevant
attributes or characteristics each with a limited number of levels. These products,
called profiles, have to be evaluated by respondents, who rank or rate them (as
described in this section) or choose their most preferred ones from smaller choice
sets (see section 13.3). As an introduction to conjoint choice experiments, in this
section we describe briefly the general characteristics of conjoint analysis and the
„classic” conjoint approaches, including ranking and rating conjoint. For a more
extensive review see, e.g., Green and Srinivasan (1978, 1990), Louviere (1988) or
Carroll and Green (1995).
The conjoint methodology is a decompositional approach to analyze consumer
preferences. Product profiles are constructed from the product attributes, each
defined at a certain number of levels, using factorial or fractional factorial designs
(the latter to reduce the number of profiles and respondent burden in evaluating
them). Respondents give an overall „score” to each product profile and the analyst
has to find out what the preference contributions are for each separate attribute
and level. Here it is commonly assumed that the overall utility of a profile is constructed by adding the preferences for the attribute-levels. This implies a compensatory preference model, in which a „low” score on a certain attribute can be compensated by a „high” score on another attribute. In conjoint experiments the contribution of an attribute (level) to the total utility is called a „part-worth”, and the
total utility of a profile in a compensatory, additive preference model is equal to
the sum of the part-worths: U = s X s s , where U is the utility of the profile,
X s the value of attribute-level s and s is the weight parameter of attribute-level
s. The part-worths can be computed from X s s . More complex constructions are
possible, such as a multiplicative model for the overall utility or interaction effects
in the utility function.
Based on the analysis of the observed data several marketing questions can be
answered (e.g., Vriens 1994) such as: 1) What is the (relative) importance of attributes and levels?, 2) What is the overall utility of specific profiles?, and 3) Are
their individual differences?. Cattin and Wittink (1982) identified five different
purposes for conjoint analysis in commercial applications: new product or concept
identification, pricing, market segmentation, advertising and distribution. Later,
competitive analysis and repositioning were added to this list (Wittink and Cattin
1989). Because conjoint analysis can be used for so many purposes, it has become

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201

a very popular marketing technique, with many applications in (commercial) marketing research (Cattin and Wittink 1982; Wittink and Cattin 1989; Wittink,
Vriens and Burhenne 1994).
In a conjoint study several steps have to be taken. First of all, the attributes and
the levels for each attribute have to be selected. Based on these attributes and
levels the set of possible profiles can be constructed. However, it is easy to see
that the total number of possible profiles can be very large even for a relative
small number of attributes and levels. When there are for instance 3 attributes with
4 levels and 2 with 3 levels 576 ( 43 32 ) different profiles can be constructed,
which is clearly a too large number for respondents to rank or rate. Therefore,
fractional factorial designs can be used to limit the total number of profiles in the
analysis, while the main effects and first order interaction effects can still be
estimated independently in many of these designs. The design one uses, and
therefore the total number of profiles in the analysis, depends on how many
interaction terms one wants to be able to estimate. In principle all kind of
attributes, including price and brand, can be used in a conjoint study. However,
the inclusion of brand as an attribute may lead to complications since it may
represent implicit attributes such as quality (e.g., Oliphant et al. 1992; Struhl
1994). Having price as a separate attribute, orthogonal to the other attributes, may
lead to unrealistic profiles, and care must be taken that no unrealistic price-brand,
or price-attribute, combinations appear in the design. The selection of the number
of levels of the attributes may also have some important implications. When all
attributes have the same number of levels, the (absolute) values of the estimated
part-worths give an indication of the (relative) importance of the attributes.
However, it is not always possible to have the same number of levels for all
attributes, since some attributes may be binary (e.g., a Yes/No or Present/Absent
attribute) while others may have (many) more levels (e.g., „Brand”). Furthermore,
Wittink et al. (1991) showed that when an attribute has more levels it becomes
more important. They called this the „Number of Levels Effect”, an effect that has
led to a substantial stream of research in its own.
Second, the evaluation task has to be selected. Above we mentioned ranking
and rating tasks, but many more data collecting methods are available that all fall
within the class of („classic”) conjoint analysis (see, e.g., Vriens (1995) for a
detailed description of these methods), such as the full profile method (Green and
Rao 1971), the tradeoff matrix method (Johnson 1974), the paired comparison
method, Adaptive Conjoint Analysis (ACA) (Johnson 1985), or Hybrid Conjoint
(Green, Goldberg and Montemayor 1981; Green 1984). All of these approaches
can be used to obtain individual (segment or aggregate) level part-worths. Individual-level results are obtained using the observed „scores” of a respondent on the
profiles and the characteristics of these profiles, and are often derived with regression-type procedures applied to each subject’s data. Subject characteristics or
classification procedures may be used, however, for segmentation purposes, where
respondents that perform similar on the conjoint task are put together in segments,
which may be described using the subject characteristics.
Third, one has to choose the way the profiles are presented to the respondent
and the way the data are collected (cf., e.g., Vriens 1995). The presentation of

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Rinus Haaijer and Michel Wedel

profiles can be done verbally, as a (printed) list of attributes and levels, with the
use of pictorials, computer aided designs or actual products. Data collection can be
done with a personal interview, a mailed questionnaire, over the telephone, or with
a computer assisted procedure. Of course, some combinations of profile presentation and data collection are more suitable than others and some are not (always)
possible. For instance, the construction of actual products is only possible for a
very limited number of product categories because of the costs involved to actually produce all profiles in the experiment. See Vriens (1995) for a more extensive
discussion on these issues.
Green and Srinivasan (1978) classified estimation methods for conjoint analysis in three categories. First, they described methods that assume that the dependent variable is, at most, ordinally scaled. In that case estimation methods like
MONANOVA (Kruskal 1965), PREFMAP (Carroll 1972), or LINMAP (Srinivasan and Shocker 1973a/b; Pekelman and Sen 1974) can be used. Second, when it
is assumed that the dependent variable is interval scaled, OLS regression techniques can be used. Third, for the paired comparison data in a choice context, the
binary Logit or Probit model can be used. Tho se models arise as special cases of
the models that we discuss more extensively later in this chapter.
In order to test the predictive ability of conjoint analysis, respondents most often have to evaluate a so-called holdout task after the main task. This task is usually similar to the main task, but the set of profiles differs. The responses on these
holdout tasks are not used for estimation purposes but for prediction. The idea is
of course that the estimated model should predict the holdout results as well as
possible. Especially when no „real-life” data are available, the holdout task is
simple way to test the predictive validity of a conjoint model. When no separate
holdout task is obtained, predictive power can be tested by using the results of part
of the respondents for estimation purposes to predict the results of the remaining
respondents. However, this latter approach is only viable at the aggregate level.
The results of classic conjoint analyses are often used to predict choice or market share (Cattin and Wittink 1982). For instance, one may be interested to know
what the predicted market shares of a specific product modification would be, or
how the introduction of a new or modified product may affect the market shares of
existing products in the market. To answer these kind of questions, market simulations have to be performed. In order to do this the individual level estimates have
to be converted to choices to predict actual market behavior of the respondents.
Many choice rules are possible, but one often-used method to achieve this employs the first-choice rule, where it is simply assumed that respondents choose the
product with the highest utility. However, this approach may be inadequate because a deterministic rule is used to predict a probabilistic phenomenon (e.g.,
Louviere and Timmermans 1990). With the first-choice rule, the situation that an
alternative has a probability of being selected over another alternative of 51% is
treated the same as the situation that an alternative has a probability of 99% of
being selected, which clearly present very different sets of preferences.
DeSarbo and Green (1984) listed five reasons why choice predictions constructed from the results of ranking or rating conjoint may not be accurate. They
stated that (classic) conjoint studies are subject to incompleteness with respect to

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203

profiles, because the profile is never equal to the product, incompleteness with
respect to model specification, because most often only main effects and some
two-way interactions are estimated, and incompleteness with respect to situation,
because conjoint assumes equal effects for marketing control variables across
suppliers. Furthermore, they mentioned the artificiality of the conjoint analysis,
caused by the fact that the amount of information in reality may be different from
that in a conjoint experiment, and the instability of tastes and beliefs of consumers, because they may change over time. All of the above may be reasons that
choice predictions are not accurate. However, DeSarbo and Green (1984) mention
that aggregate market predictions from conjoint analysis can be quite good.

11.3

Conjoint Choice Experiments

11.3.1

Conceptual

Conjoint choice analysis has some advantages as compared to conventional conjoint analysis. There are no differences in response scales between individuals,
choice tasks are more realistic than ranking or rating tasks, respondents can evaluate a larger number of profiles, choice probabilities can be directly estimated, and
ad hoc and potentially incorrect assumptions to design choice simulators are
avoided (Carroll and Green 1995). Several other authors point out similar (as well
as some additional) advantages of the choice approach relative to the conventional
approach (e.g., Louviere 1988; Elrod, Louviere and Davey 1992; Sawtooth Software Inc. 1995; DeSarbo, Ramaswamy and Cohen 1995; Cohen 1997; Vriens,
Oppewal and Wedel 1998).
In the classic conjoint approaches described in the previous section, all profiles
are presented to the respondent, while in the choice approach the total set of profiles is divided into several choice sets and respondents have to choose their most
preferred alternative from each choice set. To set the scale of utilities between
choice sets a base alternative often is added to each choice set. An advantage of
the choice approach is that this base alternative not only can be one particular
product profile, but it can also be a so-called „no-choice” option (see Haaijer,
Kamakura and Wedel 2001 for a detailed discussion on the base alternative in
conjoint choice experiments). In this case the choice probabilities can possibly be
interpreted as market shares of the various profiles. The probability for the „nochoice” then might be interpreted as an indicator for the overall preference for the
product category under research (e.g., Louviere and Woodworth 1983; Oppewal
and Timmermans 1993). A disadvantage of including a no-choice alternative in
the design is that respondents choosing that alternative provide no information on
the alternatives and attributes and hence some information is lost (Elrod, Louviere
and Davey 1992). Another potential problem with the no-choice option is the
reason why respondents choose it. A reason could be that their preferred brand or
price level is not in the choice set (or in general because of the presence -or absence- of a specific level of any attribute). Furthermore, a reason to choose the nochoice could be that respondents are not interested at all to do the task. Finally,

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Rinus Haaijer and Michel Wedel

they may find the choice too difficult and choose the no-choice if they decide not
to spend more time on the choice task and avoid the difficult choice. In those cases
one needs to be careful how to interpret the estimated no-choice probability. Johnson and Orme (1996) claim, after analyzing several conjoint choice experiments,
that there is no evidence that the latter explanations may be true.

11.3.2

Design

The approach Louviere and Woodworth (1983) developed involved constructing
conjoint choice experiments with the use of 2 J designs when there are J possible
alternatives, obtained by generating all possible combinations of attribute levels. If
there are, for instance, two attributes each with two levels, four alternatives can be
constructed. The 2 J design used then contains all combinations of the four alternatives present or absent in the choice set. From the full 2 J design an orthogonal
main effects experimental design is selected such that a relatively small number of
choice sets remain for estimation purposes. A disadvantage of 2 J fractional factorial designs is that when there are many alternatives (J), this approach will result
in large tasks for respondents where choice sets can contain (too) many alternatives. A more general version of the 2 J fractional factorial design can be used
when each choice set contains a fixed number of alternatives (M) and each alternative has S attributes with each L levels. In that situation a LM S main effects,
orthogonal, fractional factorial experimental design can be used to create joint
combinations of attribute levels (e.g., Louviere and Woodworth 1983; Steenkamp
1985; Louviere and Timmermans 1990). In case the number of levels is not equal
for all attributes a L M S design still can be used, where L now represents the
maximum number of levels present in the study. Columns in the design representing attributes with fewer levels can be constructed by converting the columns with
L levels to columns representing attributes with fewer levels.
The actual coding of levels in the choice designs can be done in several ways.
For numerical attributes (e.g., price) actual values can be used in the design, which
leads to so-called linear attributes. However, most of the time some dummy specification is used. This specification can involve „regular” dummy coding (e.g., „1”
if a level is present and „0” if it is not present) or so-called effects-type coding. In
the situation of 3 levels of an alternative, with effects-type coding, the first level is
coded, e.g., as [1 0] , the second as [0 1] and the third as [-1 - 1] . For attributes with 2 levels the codes are +1 and -1 respectively. This way of coding has as
advantage, when all attributes are coded this way and each level appears with
equal frequency in the design, that the sum of the part-worths for each level is
equal to zero, so that the total model is centered around zero. Combinations of
different ways of coding are possible.
A specific characteristic of conjoint choice experiments is that one needs two
designs, instead of one design in the classic conjoint approach, to set up the ex-

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205

periment. One design is needed to construct the profiles, like in the classic conjoint approach, but an additional design is needed to put these profiles in various
choice sets. It is beyond the scope of this chapter to discuss extensively how efficient designs for conjoint (choice) experiments should be constructed, but the key
elements are described briefly. For much more detail the interested reader is referred to, e.g., Addelman (1962), Louviere and Woodworth (1983), Steenkamp
(1985, in Dutch), Kuhfeld, Tobias and Garratt (1994) or Huber and Zwerina
(1996). In principle one wants the main effects and interaction effects to be orthogonal in the design, however, Kuhfeld, Tobias and Garratt (1994) show that
orthogonal designs are not always more efficient than non-orthogonal designs,
hence a trade off has to be made between these two concepts. Furthermore, they
show that the efficiency of a given design is affected by the coding of quantitative
factors, even though the relative efficiency of competing designs is unaffected by
coding (Kuhfeld, Tobias and Garratt 1994, p. 549).
The range of levels for quantitative factors should be as large as possible to
maximize efficiency. However, the levels should of course not be implausible.
Huber and Zwerina (1996) describe four properties that characterize efficient
choice designs. They mention level balance, orthogonality, minimal overlap and
utility balance. Level balance means that each level of an attribute appears with
equal frequency. However, level balance and orthogonality are often conflicting.
Choice sets should have minimal overlap since alternatives that have the same
level of an alternative provide no information on the preference for that attribute.
Hence, the probability that an attribute level repeats itself in each choice set
should be as low as possible. Level balance, orthogonality and minimal overlap
are used to construct optimal utility-neutral designs. The efficiency of such design
can be improved by balancing the utilities of the alternatives in each choice set.
This is important since choice sets that generate extreme probabilities are less
effective at constraining the parameters of the choice model than are moderate
ones (Huber and Zwerina 1996, p. 308), although they do have a big positive
impact on the log-likelihood of a choice model. So, a high likelihood may go
together with imprecise parameter estimates for choice sets with more extreme
probabilities. One possible way to achieve more utility balanced designs is simply
by re-labeling the levels of the attributes, which has as advantage that it does not
affect orthogonality, in contrast to swapping techniques. One problem not solved
yet is how efficient designs can be obtained when a base alternative (such as a nochoice) is present in the experiment (Huber and Zwerina 1996), another is that
efficient designs for the MNP model have not been developed yet, although recently design procedures for the related mixed logit model have been proposed
(Sándor and Wedel 1999).
Another issue that plays a role is the type of design to use in the analysis: a design with fixed, randomized or individualized choice sets. With a fixed choice set
approach each respondent (or each group of respondents, in a slightly more general fixed approach) receives exactly the same choice sets at exactly the same
stage of the choice task. In a randomized experiment each respondent (or group of
respondents) also receives the same choice sets but in a different order to compensate for learning and fatigue effects that are expected to average out in this way. In

206

Rinus Haaijer and Michel Wedel

an individualized experiment each respondent receives his own choice sets. An
advantage of individualized choice sets is that it can be tested whether preferences, or attribute importance, change in later stages of the choice experiment
(Johnson and Orme 1996), since for each choice set (i.e., the 1st, 2nd, ..., last for
each respondent) estimates can be obtained in this situation. A study by Johnson
and Orme (1996) showed, when comparing several conjoint choice experiments,
that the importance of brand decreases throughout a conjoint choice experiment,
while that of price increases. A disadvantage of using individualized choice sets is
that no choice frequencies can be computed for alternatives in each choice set,
since each set is only evaluated by one respondent. Another disadvantage of individualized choice sets is that comparison and clustering becomes more difficult
(Oliphant et al. 1992), which would be possible when all (groups of) respondents
receive the same choice sets. In that latter case, respondents that show similar
choice patterns can be grouped together in segments. This problem is however
alleviated by mixture model approaches to conjoint choice experiments, as detailed below. Depending on what type of analysis, the fixed, randomized or individualized approach can be the preferred choice.

11.3.3

Applications

In this section an overview is given of recent applications of conjoint choice experiments in the marketing literature. Some of the studies listed here have been
already discussed briefly in the previous section. This overview is not intended to
be complete, but the aim is to give an impression of possible applications of conjoint choice. In particular, we will show for each study several characteristics of
the conjoint experiment. Table 1 lists the studies we present in this overview (it
was not possible to retrieve all information for all studies) .
Table 1 shows that the range of products investigated in conjoint choice experiments is rather wide. The products range from fast-moving consumer goods,
like toothpaste, to durable products, like houses and cars. The same holds for the
number of choice sets presented to the respondents and the number of alternatives
in the choice sets. In the various applications, respondents had to choose from 3 to
32 choice sets containing 2 to 8 alternatives. The profiles in these choice sets were
defined on 2-12 attributes. In the Oppewal, Louviere and Timmermans (1994)
study 33 attributes where used, but their aim was to reduce this number using
Hierarchical Information Integration.
The number of respondents used in the various studies also shows a wide range
from 64 up to almost 1000 respondents. There seems to be more agreement about
the type of base alternatives to use in a conjoint choice study. Most of the studies
listed in Table 1 used „none”, „own” or „other” base alternatives and only a few
used a fixed profile as base. Most of the studies that did use a fixed base alternative assume a specific situation (for instance like „given that you are going on a
holiday, what would be your most preferred trip”) and are less interested in obtaining market shares, which is the major advantage of including some sort of „nochoice” base alternative. This may be the main reason to include such a base alter-

Conjoint Choice Experiments

207

native in the other applications. Note that for the studies with only two alternatives
no base alternative was used. The number of levels of the attributes used in the
studies also shows a rather consistent pattern. Most studies use attributes with 2-4
levels. In situations that more levels are used for an attribute, this most often is a
brand-attribute.
Table 1:

Conjoint Choice Applications

Authors

Product/
Product category

Attri- Choice Alterbutes sets
natives*

Respondents

Base

Levels /
Design

Elrod, Louviere and Rental apartments
Davey (1992)
Oliphant et al. (1992) Insurance

4

27

3

115

Own

24 34

9

20

5

149

None

42 28

Oppewal and
Shopping centers
Timmermans (1992)
Chrzan (1994)
Mail orders
Fashion access.
Consumer fashion

4

8/16

3 /4

?

Other

44

5
?
10

8
16
16

3
6
3

605
300
876

None
Other
Other

25
8 42 34
4 3 28

Oppewal, Louviere
and Timmermans
(1994)
Allenby, Arora and
Ginter (1995)
Allenby and Ginter
(1995)

Shopping centers

33

3

3 /4

396

None

45 23+48 25

Batteries

3

12/24

2

65

-

33

Credit cards

7

13-17

2

946

-

42 33 22

Activity packages

4

5/6

3

221

Fixed

34

Food

2

16

8

600

Fixed

?

Houses

4

16

3

278

None

44 (48)

Flower exhibits

3

?

3

64

Fixed

3 22

Tourist Portfolio

12

12

3

Fixed

312

Toothpaste

5

32

5

184

None

?

Coffee makers

5

4/8

5/3

185

Fixed

33 22

Cars

6

9

4

200

None

8 33 22

Dellaert, Borgers and
Timmermans (1995)
DeSarbo, Ramaswamy and Cohen
(1995)
Timmermans and
Van Noortwijk
(1995)
Dellaert, Borgers and
Timmermans (1996)
Dellaert, Borgers and
Timmermans (1997)
Moore, Gray-Lee and
Louviere (1998)
Vriens, Oppewal and
Wedel (1998)
Wedel et al. (1998)
*: Base included

660

The information in Table 1 shows that conjoint choice experiments can be and
have been used for a wide range of possible applications. In almost any situation
in which consumers have to choose between several options the conjoint approach
can be used to determine which attributes of the product are important for the

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Rinus Haaijer and Michel Wedel

respondent. In this case „product” can be some fast moving product like toothpaste, a durable like a car or a house, a service such as tourist attractions, and even
products that are not actually bought by respondents, like „shopping centers”. In
most of the recent applications the number of attributes, levels and choice alternatives used in the design is rather low (attributes and alternatives around 4, levels
around 3), although there are some exceptions. The number of choice sets that can
be presented to respondents showed a wide range. A recent study by Sawtooth
Software (Johnson and Orme 1996) showed that given the rather short response
times in conjoint choice experiments, many choice sets can be offered to respondents even without reducing the quality of the choices. With modern computer
assisted data collecting methods for conjoint choice the response times can be
obtained very easily, and can actually be used to improve estimation of partworths, see Haaijer, Kamakura and Wedel (2000).

11.4

Conjoint Choice Models

11.4.1

Introduction

In this section we discuss several approaches to analyze conjoint choice experiments. First of all, the standard MNL approach is discussed in section 13.4.3.
Second, section 13.4.4 describes the Latent Class MNL model. Section 13.4.5
provides two MNP models, one in which choice sets are assumed independent,
and one where the choices from one individual are treated as correlated. But first
we specify the general structure of conjoint choice models in this section and
describe the data we will use as application in section 13.4.2.
In a conjoint choice model each respondent has to choose one alternative from
each of several choice sets. These choice sets are constructed by dividing the set
of profiles over K choice sets. In this chapter we assume that each choice set contains the same number of alternatives, without losing generality. In order to formulate models for conjoint choice experiments, we start from random utility maximization (McFadden 1976). The utility of alternative m in choice set k for individual
j is defined as:
(1)

U jkm = X km + e jkm ,

where X km is a (1xS) vector of variables representing characteristics of the mth
choice alternative in choice set k, is a (Sx1) vector of unknown parameters, and
e jkm is the error term. Note that we assume that the X-matrix in (1) does not depend on j, because in conjoint choice experiments no individual characteristics
appear in the analysis in general. Note, however, that when an individualized
design is used X does depend on j, but we omit this index here for convenience.

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209

For each individual j, it is assumed that the alternative with the highest utility
is chosen and a variable y jkm is observed which is for each choice set k defined
as:
(2)

1 when U jkm > U jkn
y jkm =

n

m
, n = 1, .., M .

0 when n

m : U jkn > U jkm

As mentioned in section 13.3.1, in conjoint choice experiments a base
alternative is often used in each choice set k to scale the utility over choice sets.
This base alternative cannot only be a regular profile, it also can be specified as a
no-choice alternative („None of the above”) or an „own-choice” alternative („I
keep my own product”). This kind of base alternative, however, presents the
problems of how to include it in the design of the choice experiment, and in what
way to accommodate it in the choice model. Regular choice alternatives are most
often coded in the design matrix with effect-type or dummy coding. Since the nochoice alternative does not possess any of the attributes in the design, it is often
coded simply as a series of zero’s, which makes the fixed part of its utility zero in
each choice set. However, the utility level of the no-choice alternative still has to
be specified when effect-type coding is used, since the zeros of the no-choice act
as real levels in that case and this potentially leads to biased estimates. The nochoice alternative can be specified in two ways. The first specification is to
include a no-choice constant in the design matrix X in (1). This introduces one
additional parameter in the model to estimate. Note that when brand dummies (or
other attribute specific dummies) are used for each level of the attribute, no
additional parameter is needed since in that case the utility level of the no-choice
is already set by those dummies. However, the total number of parameters to
estimate is equal in both cases. The second specification is to formulate a nested
MNL model, in which it is assumed that subjects first choose between the nochoice and the other choice alternatives in the choice set, and in a second stage
make their choice among the alternatives when they decide to make a „real”
choice. This also introduces one additional parameter in the model, the
dissimilarity coefficient of the Nested MNL model. Which of these representations
for the no-choice option is preferable is discussed in Haaijer, Kamakura and
Wedel (2001).

11.4.2

The Data

The various models that will be introduced in the next sections to analyze the
above conjoint choice structure will be illustrated with an application, which is a
replication of part of an application reported by Haaijer et al. (1998), with coffeemakers as the product category. The five attributes, and their levels, for the coffeemakers are listed in Table 2. Using a factorial design, sixteen profiles were constructed. Data were collected from 185 respondents, divided into two groups that
received different choice sets based on the same sixteen profiles. Respondents had

210

Rinus Haaijer and Michel Wedel

to choose from eight sets of three alternatives. Each choice set included the same
base alternative, which is a fixed regular alternative in this experiment. Furthermore, eight holdout profiles were constructed, which were divided into four holdout sets with three alternatives, where the same base alternative was used as in the
estimation data. These holdout sets were offered to all respondents. The estimation
and holdout designs were coded using effects-type coding.
For all models in subsequent sections we will obtain parameter estimates. Furthermore, to compare model performance, we report the log-likelihood value, AIC
and BIC statistics, and Pseudo R2 value (e.g., McFadden 1976) relative to a nullmodel in which the probabilities in a choice set are equal for all alternatives. The
AIC criterium (Akaike 1973) is defined as: AIC = - 2 ln L + 2n and the BIC
criterium (Schwarz 1978) is defined as: BIC = - 2 ln L + n ln (O) , where lnL
is the log-likelihood in the optimum, n the total number of estimated parameters in
the model, and O the number of independent observations in the experiment.
Table 2:

Attributes and Levels of Coffee-Makers

Attribute
Level

Brand

Capacity

Price
(Dfl)

Special
Filter

Thermosflask

1

Philips

6 cups

39,-

Yes

Yes

2

Braun

10 cups

69,-

No

No

3

Moulinex

15 cups

99,-

11.4.3

Multinomial Logit

The most popular discrete choice model is the Multinomial Logit (MNL) model. It
follows when the assumption is made that the error term in (1), e jkm , is independently and identically distributed with a Weibull density function. A Weibull density function for a random variable Y is defined as (see, e.g., McFadden 1976):
(3)

P( Y

-y

y ) = exp-exp .

This distribution belongs to the class of double negative exponential
distributions as are, e.g., the Type I extreme value distribution and the Gumbell
distribution, which are sometimes also used to specify the MNL model. The MNL
model treats observations coming from different choice sets for the same
respondent as independent observations. Therefore, in estimating the MNL model,
100 respondents choosing from 10 choice sets yields the same computational
burden as 1000 respondents choosing from 1 choice set. In the standard MNL
model, with one choice observation for each individual, the choice probabilities

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211

have a simple closed form. The choice probabilities in the conjoint MNL approach
can be obtained through a straightforward generalization of this standard model.
The probability p km that alternative m is chosen from set k is in this case
simply equal to (cf., e.g., Maddala 1983, p. 60-61; Ben-Akiva and Lerman 1985;
Swait and Louviere 1993):
(4)

p km =

exp( X km

)

.

M

exp( X kn

)

n=1

The standard log-Likelihood for discrete choice models is in the conjoint
context for the MNL model extended by adding a summation over choice sets:
J

(5)

K

M

L MNL =

y jkm ln ( p km ) .
j=1 k =1 m=1

The simple and easy to calculate form of the choice probabilities (4) in the
MNL model has much contributed to its popularity in conjoint choice
experiments. However, there is a serious limitation to the use of this model that is
related to the Independence of Irrelevant Alternatives (IIA) property. This
property arises from the assumption of independent random errors and equal
variances for the choice alternatives, and implies that the odds of choosing one
alternative over another alternative must be constant regardless of whatever other
alternatives are present (e.g., Louviere and Woodworth 1983; Ben-Akiva and
Lerman 1985), which may often be too restrictive in practical situations. If it is
assumed that the IIA property holds and the MNL model is used, predicting the
choice probabilities of new alternatives can simply be done by inserting the
attribute values of these new alternatives in the closed form expressions for the
choice probabilities (4). Green and Srinivasan (1978) stated that in consumer
behavior contexts the IIA property might not be a realistic assumption, especially
when some of the alternatives are close substitutes (cf. McFadden 1976). When
covariances across alternatives are incorrectly assumed to be zero, the estimates
for the effects of explanatory variables are inconsistent (Hausman and Wise 1978;
Chintagunta 1992). When the IIA property does not hold, other models that avoid
IIA, should be used instead of the standard MNL model, however, at the cost of
computational complexity. One of the most general of these models is the
Multinomial Probit (MNP) model, which is discussed in section 13.4.4.
When the IIA assumption is true, the parameters of the Logit model can be estimated when the sufficient condition is satisfied that the alternatives are independent across choice sets (Louviere and Woodworth 1983). So, choices between
alternatives must be pairwise independent across choice sets. The alternatives in a
conjoint choice experiment are obtained by using an orthogonal, fractional factorial main effects design (Louviere and Woodworth 1983; Louviere and Timmermans 1990). A constant base alternative is useful, because it preserves the design
orthogonality of the attribute vectors of conjoint alternatives (Louviere 1988;

212

Rinus Haaijer and Michel Wedel

Elrod, Louviere and Davey 1992). However, in the case of the Logit model, design orthogonality does not imply information orthogonality, for which the parameters would be uncorrelated. When similarities across alternatives are incorrectly assumed to be zero, the estimates for the effects of marketing variables are
incorrect (e.g., Chintagunta 1992).
The expression for the choice probabilities (4) may be expanded to accommodate ranking data, which is particularly useful in conjoint analysis (McFadden
1986; Kamakura, Wedel and Agrawal 1994). However, the assumptions needed to
translate rankings into choices may not hold, especially when individuals use
elimination and nesting strategies the IIA property does not hold (Louviere 1988).
Also, the use of brand names in the conjoint design may result in correlations
between the utilities of the alternatives, violating the IIA property. In order to be
able to test for IIA, design plans that allow as many relevant two-way interactions
as possible to be tested can be used (Louviere and Woodworth 1983).
Table 3:

MNL Estimation Results
Attribute (level)
1 Brand (1)
2 Brand (2)
3 Capacity (1)
4 Capacity (2)
5 Price (1)
6 Price (2)
7 Filter (1)
8 Thermos (1)
Statistics
Ln-likelihood
AIC
BIC
Pseudo R2

Est.
0.040*
-0.329*
-1.015*
0.494*
0.313*
0.372*
0.340*
0.312*

S.e.
.013
.013
.014
.010
.018
.013
.070
.010

-1298.706
2613.412
2655.810
0.201

*: p < 0.05.

The MNL specification was used to analyze the data set described in section
13.4.2. In Table 3 the parameter estimates and fit statistics are listed. With the
effects-type coding used, the part-worth for the last level of each attribute can be
constructed by taking the sum of the estimates of the other levels of that attribute
and change the sign. The results show that respondents prefer a high capacity to a
low capacity, a low price level over a high price level, and that they prefer the
presence of a special filter and thermos flask to the absence of those attributes.
Finally, the third brand is the most attractive brand and the second the least. The
pseudo R2 has a value of 0.201, which for this kind of choice data is a reasonable
value. The estimates in Table 3 were used to predict the holdout sets. This resulted
in a predicted log-likelihood of -754.860 (Pseudo R2 =0.072). This shows that the
MNL model does not a very good job in predicting the holdout sets in this applica-

Conjoint Choice Experiments

213

tion, since the obtained Pseudo R2 has a value much lower as those resulting from
the estimation sample.
The standard MNL model described in this section assumes that all respondents act similar in their choice behavior. However, several groups of respondents
may exist that show different choice behavior. The next section describes the
Latent Class MNL model that can be used to obtain segments of respondents.

11.4.4

Latent Class MNL

Next to its disadvantages related to the IIA assumption, the MNL model also suffers from the problem that it treats all subjects in the sample as homogeneous, and
does not deal with heterogeneity. The MNL model cannot be estimated at the
individual level, and thus subject-specific part-worths cannot be obtained (e.g.,
Elrod, Louviere and Davey 1992). The issue of subject heterogeneity has received
a lot of attention in the marketing literature and has become a topic of much research (cf. Wedel et al. 1999). Basically, there are two ways to accommodate
heterogeneity. In this section we deal with one, where one specifies a discrete
distribution of the response coefficients
across the population, that is, one
j
postulates that groups of respondents exist with different part-worths. This leads to
latent class or finite mixture discrete choice models, which have been applied to
conjoint choice experiments by Kamakura, Wedel and Agrawal (1994), and DeSarbo, Ramaswamy and Cohen (1995). In the subsequent sections we deal with
Multinomial Probit models that specify a continuous distribution of heterogeneity.
Finite mixture models connect very well to marketing theories of market segmentation (Wedel and Kamakura 1997) and have enjoyed considerable success. Managers seem comfortable with the idea of market segments, and the models appear
to do a good job of identifying segments from conjoint choice data.
Kamakura, Wedel and Agrawal (1994) developed a unifying mixture regression model for segmentation of choice data. Their approach assumes that choices
are based on random utility maximization. The observed choice variables y jkm ,
are assumed to be independent multinomial, and to arise from a population that is
a mixture of Q unobserved segments, in proportions 1 ,
, Q . We do not
know in advance from which segment a particular subject arises. The probabilities
q are subject to the following constraints.

Q

(6)

q

1

q

1

q

0 q

1,..., Q

Given segment q, the choice probability for profile m for choice set k is:
(7)

Pkm|q = Prob[U km |q U kn |q n = 1,

, M, n

m ],

214

Rinus Haaijer and Michel Wedel

where U km|q is the random utility derived from alternative m at k in segment s.
Consumers are assumed to maximize their utility over the entire choice set,
M }. As before, the random utility for segment q is
U km|q = max { U kn|q n = 1,
assumed to be a function of the attributes:

U km |q = X km

(8)

q

+

kmq

If the random components,

.

kmq

, are assumed to be independent and

identically Weibull distributed, the choice probabilities for segment q are:
(9)

P km |q =

exp[ U km |q ]

.

M

exp [ U kn |q ]
n=1

The idea behind the mixture model is that if the probability conditional upon
knowing the segments have been formulated, the unconditional probability of
observing the K choices is obtained as:
Q

(10)

Pj=

K

M

PY jkm .
km|q

q
q=1

k =1 m=1

As additional features of the model, the prior probabilities of segment membership can be reparameterized according to a concomitant variables model and
the model can deal with rank-ordered data (Kamakura, Wedel and Agrawal 1994).
Wedel et al. (1998) extended this mixture model for conjoint choice experiments
by assuming that the brand can be decomposed into latent dimensions and segment-specific ideal points along those dimensions. The utility function further
contains a linear combination of the attribute-level dummies. This model integrates conjoint analysis and multidimensional scaling, which makes it especially
suited for product positioning.
The mixture regression model for conjoint choice experiments was applied to
the coffee-maker data for Q = 1 up to 6 segments; the BIC statistic indicated
Q = 4 segments as optimal. The four- segment solution was run ten times from
different starts to overcome problems of local optima. Table 4 gives the estimated
parameters for all 4 segments.
Table 4 shows that the four segments differ in their preferences for the attributes. The first segment (49,7% of the sample) wants a coffee-machine that contains as many features as possible, for a as low price as possible, but it does not
matter what brand it is, since the brand parameters are not significant different
from zero for this segment. So, this seems to be a price-sensitive segment. The
second segment (17,4%) does not want a low capacity machine and prefers one
with a thermos-flask, but this segment seems more quality seeking, since its partial
utility for the lowest price level is negative, while that for the highest price level

Conjoint Choice Experiments

215

has the highest partial utility. This segment also has no preference for a particular
brand. The third segment (13,5%) also likes only one feature (a special filter in
this case), but does not want to pay a high price for it. This segment in addition
has a preference for one of the brands (brand 2). Finally, the fourth segment
(19,4%) finds the brand of the coffee-machine most important. It has a high preference for the first and third brand, and in addition for the presence of a special
filter. So, this segment seems to be a real brand-seeking segment.
Table 4:

LCMNL Estimation Results

Attribute
(level)
Brand (1)
Brand (2)
3 Capacity (1)
4 Capacity (2)
5 Price (1)
6 Price (2)
7 Filter (1)
8 Thermos (1)
1
2

Segm. 1
Est.
0.005
-0.224
-2.681*
1.302*
1.263*
0.698*
0.630*
0.461*

Segment Size

0.497

Statistics
Ln-likelihood
AIC
BIC
Pseudo R2

Segm. 2
S.e.
.179
.163
.289
.152
.284
.213
.132
.139

Est.
0.289
-0.321
-0.956*
0.627*
-0.934*
0.150
0.134
0.876*
0.174

Segm. 3
S.e.
.188
.179
.247
.175
.346
.193
.133
.167

Est.
-0.051
0.683*
-1.191*
0.578*
1.431*
0.605
1.753*
0.289
0.135

Segm. 4
S.e.
.316
.295
.481
.266
.359
.312
.340
.159

Est.

S.e.
*

0.920
-1.673*
-0.001
0.355
-0.322
-0.064
0.390*
-0.053

.202
.346
.165
.203
.271
.218
.133
.167

0.194

-1040.271
2115.542
2336.058
0.360

*: p < 0.05.

From the results of an Latent Class analysis different marketing strategies can be
developed for the various segments, provided that these are big enough to make it
profitable to develop a specific strategy. Table 4 shows that the smallest segment
(segment 3) still contains 13.5% of the respondents, which may be big enough to
target. The estimates in Table 4 were used to predict the likelihood of the holdout
choice sets. This gives a predicted log-likelihood for the LCMNL model of 708.832 (Pseudo R2 =0.128). Comparing this to the predictive fit of the MNL
model (Pseudo R2 =0.072) we see a substantial improvement. Thus the LCMNL
model improves upon the MNL model by accommodating heterogeneity, by providing actionable information on market segments, and by providing better holdout predictive performance.
Although the LCMNL model accounts for consumer heterogeneity, it still
treats choices made by the same respondent as independent. In the next section
two versions of the MNP model will be developed. One in which it is still assumed that choice sets are independent, but choice alternatives within a choice set
may be correlated, and one that in addition relaxes the independence of choice
sets.

216

11.4.5

Rinus Haaijer and Michel Wedel

Multinomial Probit

Introduction
In the previous section we saw how latent class MNL model can be used to account for heterogeneity. The MNP model also presents a way to deal with continuous heterogeneity distributions of the part-worths across consumers. Specifically, the parameters are specified to follow a normal distribution, which by some
has been argued to be a better representation of heterogeneity than a discrete mixing distribution (Allenby and Rossi 1999). The continuous heterogeneity distribution has several advantages: It has been argued to characterize the tails of heterogeneity distributions better and may predict individual choice behavior more accurately than finite mixture models, since the tails may have a substantial impact on
the predictive performance of the models. It provides a parsimonious representation of heterogeneity and flexibility with regard to the appropriate choice of the
distribution of heterogeneity (see, e.g., Arora, Allenby and Ginter 1998).
In the conjoint choice context, the Multinomial Probit model (MNP) offers the
major advantage of allowing correlations among the repeated choices that consumers make from the multiple choice sets next to allowing correlation of random
utilities of alternatives within choice sets. This follows since the MNP model
relaxes the assumption of independence of the error terms in random utility models (e.g., Daganzo 1979; Kamakura 1989), and thereby alleviates IIA. Factors such
as learning, boredom, or anchoring to earlier choice tasks may distort the measurement of preferences when these are assumed independent, like in the MNL
model, and these effects should be tested and/or accounted for (McFadden 1986).
Several studies showed that subjects’ utilities for alternatives may indeed depend
on the choice context (e.g., Huber, Payne and Puto 1992, Simonson and Tversky
1992; Nowlis and Simonson 1997), where „context” is defined as the particular set
of alternatives evaluated. Since the design of conjoint choice analysis involves
only a subset of all possible profiles (constructed by fractional factorial designs)
and choice sets that vary in composition (constructed by blocking designs), context effects are likely to occur in those experiments. Simonson and Tversky (1992)
distinguished local contrast effects and background contrast effects. Local contrast
effects are caused by the alternatives in the offered set only, while background
contrast effects are due to the influence of alternatives previously considered. In a
conjoint choice experiment, local contrast effects may occur due to the composition of a particular choice set in terms of the attribute levels of the profiles, affecting attribute importance, inducing correlations among the utilities of profiles in the
choice set and leading to a violation of IIA (Simonson and Tversky 1992). On the
other hand, background contrast effects may occur in conjoint choice experiments
if the attribute importance of profiles in a particular choice set are influenced by
tradeoffs among profiles in previous choice sets. In this case covariance among
the random utilities of alternatives in different choice sets may occur. This violates
the assumption of independence of choices among alternatives in different sets, as
assumed in the MNL model. For a more extensive discussion on context effects
see, e.g., Tversky (1972), Huber, Payne and Puto (1982), Huber and Puto (1983),
Simonson (1989), or Simonson and Tversky (1992).

Conjoint Choice Experiments

217

Timmermans and Van Noortwijk (1995) explicitly modeled context effects in
conjoint choice experiments by including cross effects in the design matrix such
that the utility of an alternative depends on its own and other alternatives’ attributes. However, they only model context effect within choice sets (i.e., the local
context effects) and not between choice sets (i.e., the background context effects).
Haaijer et al. (1998) applied the MNP model to conjoint choice experiments, accounting for both the local and the background context effects described above
through a specific covariance structure. They showed that it is important to account for both types of context effects. Two sections below give a MNP model
that deals with heterogeneity, IIA and local context effects, and a MNP model that
in addition accounts for background context effects, respectively.
First we specify the MNP model in general, starting again from the utility
function. Assume again that there are J respondents, each receiving the same H
profiles which are divided into K smaller sets with M alternatives each. A base
alternative that is common to all sets is added to the profiles and scales the utility
levels between choice sets. This base alternative can be a no-choice alternative or
a regular profile. The other profiles are unique to their particular choice set, so that
H = K(M1)+ 1 . The utilities of the alternatives for individual j are contained in
the latent unobservable vector u j , which satisfies:
(11)

uj= X

j

+ ej ,

where X is a (HxS)-matrix containing the attributes of the alternatives,

j

is a

(Sx1) vector of random weights, and e j is the vector containing the random component of the utilities.
In the MNP model is it assumed that e j is distributed as:
(12)

e j ~ N H (0 ,

e

),

independent between individuals; e is a (HxH) positive definite covariance
matrix. In the MNP model not only the -parameters in (11) have to be estimated
but also the parameters in the covariance matrix e .
A potential problem of the MNP model is that of identification. A model is
identified when there is only one set of estimates that maximizes the likelihood.
When different parameter estimates give the same results, interpretation of the
estimates becomes difficult. Bunch and Kitamura (1989) demonstrated that nearly
half of the published applications of MNP are based on non-identified models. It is
easy to see that when the covariance matrix in (12) of the MNP model is multiplied with a factor and all -estimates in (11) with a factor
, that this leads to
the same results. So, at least one parameter in the MNP model must be fixed to
scale the model and to identify the other parameters. Often, one of the variance
parameters is used for this purpose, but this is not sufficient, however. In the stan-

218

Rinus Haaijer and Michel Wedel

dard MNP model with one choice set (K=1), only M(M - 1)/2 - 1 of the

M(M + 1)/2 covariance parameters in are identified (Dansie 1985; Bunch
1991; Keane 1992). So, M + 1 restrictions must be imposed on the -matrix in
this situation.
Furthermore, in conjoint choice experiments, (holdout) predictions are often
required. Another problem, besides identification, of the general MNP formulation
is that these predictions for new profiles, not included in the conjoint design, cannot be made with the covariance matrix in (12) because in predicting choice probabilities for alternatives not included in the design of the experiment, estimates of
the covariances of these new profiles are required and those are not available (cf.,
e.g., Pudney 1989 p.115; Elrod and Keane 1995; Haaijer et al. 1998).
In order to arrive at an MNP model for conjoint choice experiments that is
both identified and that allows for predictions of new profiles, restrictions have to
be imposed on the covariance matrix. We allow for heterogeneity in the attribute
level coefficients by specifying
j in equation (11) as (cf., e.g., Hausman and
Wise 1978; Daganzo 1979; Ben-Akiva and Lerman 1985):
(13)
with
(14)

j

j

=

+

~ N S (0 ,

j

,

) , independent of e j . Then

u j ~ N H (X ,

),

with:
(15)

=

e

+ X

X .

The specification that enables the prediction of new alternatives that we use
assumes e = I H , and for reasons of parsimony and identification we parameterize

as a matrix of rank one:

=

, with an S-vector of parameters,

where S is the number of columns in the X-matrix. The number of parameters in
now is equal to the number of -parameters. Especially when the number of columns (S) in X or the number of profiles (H) is large, this specification for
is
very parsimonious compared to a full random coefficients model or general Probit
results in an increase in the number of
model. A more general specification for
covariance parameters so that identification often becomes a problem. So, we now
have:
(16)

= IH + X

X .

Conjoint Choice Experiments

219

This random coefficients model may account for heterogeneity, violations of
IIA, and local and background context effects potentially caused by all attributes
in the conjoint design. See Haaijer et al. (1998) and Haaijer (1999) for a more
extensive discussion of this specification and its characteristics. Rossi, McCulloch
and Allenby (1996) developed a related random coefficients Bayesian MNL
model.
Like for the MNL and LCMNL model, estimates for the parameters are obtained for the MNP model by maximization of the likelihood (see below) over
and the parameters in the covariance matrix. However, when there are more than
three alternatives in a choice set the choice probabilities cannot be evaluated numerically in the MNP model (cf., e.g., McFadden 1976; Maddala 1983; Kamakura
1989; Keane 1992). Simulation techniques have been developed that solve this
problem of the MNP model. To obtain the estimates in the MNP models in the
next two subsections, the Simulated Maximum Likelihood (SML) method is applied using the SRC simulator. A discussion of simulation techniques is beyond
the scope of this chapter, for an extensive discussion see, e.g., Hajivassiliou
(1993).
Multinomial Probit with independent choice sets
A straightforward way to apply the MNP model is to use it in the similar way as
the MNL model. In this case we take an individual's utilities to be independent
between the choice sets, and thus account for local, but not for background context
effects. We then have JK independent observations, and the log-likelihood is again
a straightforward generalization of the standard likelihood of choice models,
where a summation over choice sets is introduced, similar as in the MNL model of
section 13.4.3 Letting p km denote the fraction of individuals choosing alternative
m in set k, the log-likelihood is (in a slightly different notation as the loglikelihood (5) of the MNL model) equal to:
K

M

L MNP csi = J

(17)

p km ln (

km

),

k =1 m=1

where km is the probability that alternative m is chosen in set k. Note again that
in conjoint choice models consumer characteristics or other individual specific
variables are usually not included, hence km does not depend on j and each
individual has the same probability of choosing any specific alternative, since we
assumed that they all receive the same choice sets. This model is called the
choice-set-independent MNP model (MNPcsi). For this MNP model, the assumption of utility maximization results in an expression for km that involves an
(M-1)-dimensional integral:
0

(18)

km

= P( u kn - u km

0

n

m

k

~
) = P( u km

0 ) = d km (t ) dt

220

Rinus Haaijer and Michel Wedel

~
where k is the set of profiles in choice set k and d km (.) is the density of u km .
This specification accounts for local contrast effects only, since it allows utilities
within choice sets to be correlated.
Table 5:

MNPcsi Estimation Results
Attribute
(level)
1 Brand (1)
2 Brand (2)
3 Capacity (1)
4 Capacity (2)
5 Price (1)
6 Price (2)
7 Filter (1)
8 Thermos (1)
Statistics
Ln-likelihood
AIC
BIC
Pseudo R2

Est.
-0.106
-0.179
-1.166*
0.587*
0.326
0.378
0.354*
0.269

S.e.
.203
.222
.123
.090
.497
.269
.098
.152

Est.
0.717
0.107
0.585*
-0.001
-0.084
0.482
0.298
0.173

S.e.
.386
.665
.270
.224
.599
.473
.372
.259

-1279.100
2590.201
2674.997
0.213
*: p < 0.05.

In Table 5 the coffee-maker data results are listed for the structural parameters
and the covariance parameters , as well as the fit-statistics. It shows that the
results of the MNPcsi model are a somewhat disappointing. Although it produces a
better log-likelihood than the MNL model, due to the high standard errors only a
few parameters are significantly different from zero with a p-value of 5%. The
results indicate that the capacity and presence of a special filter are the only relevant attributes. Furthermore, the one significant covariance parameter (belonging
to the first capacity level) is responsible for the increased fit of the MNPcsi model
with respect to the MNL model, but no clear conclusions can be drawn from these
results. In addition, the AIC and BIC statistics indicate that the LCMNL has better
fit than the MNPcsi model. The estimates of Table 5 were use to predict the holdout sets. The predicted log-likelihood is equal to -784.677 (Pseudo R2 = 0.035),
which is worse from the LCMNL model and even worse than those of the MNL
model.
The results of the MNPcsi model indicate that allowing for heterogeneity and
correlation of utilities within choice sets may help to improve model fit in terms of
the log-likelihood value. However, this application also showed that the discrete
(LCMNL) representation of heterogeneity seems to do better than the continuous
(MNP) one. In the next subsection the MNP model that in addition allows for
correlations between choice sets is developed.

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221

Multinomial Probit with dependent choice sets
The MNP specification in the previous subsection only allowed for correlations
within choice sets and not between. In this section we assume that utilities of the
same individual are not independent over choice sets, but rather that utilities of
alternatives in different choice sets are correlated. In this case a total probability
has to be obtained for the complete choice observation of an individual. A simple
example illustrates this. Assume we have two choice sets with each three alternatives (so, H=5). For each individual we observe two choices, one from each set.
Consider an individual j choosing the second alternative from the first set and the
base alternative from the second choice set. The resulting joint probability for this
example is equal to (b represents the base alternative):
(19)

2b

= P ( u j12 > u j11 , u j12 > u j1b , u j2b > u j21 , u j2b > u j22 ) .

This probability can be expressed involving a four-dimensional integral. In the
general case, a K-vector of choices is observed for each individual, and we have to
consider M K arrays containing the multiple choices from different choice sets.
Each array corresponds to a joint probability, involving an (H-1)-dimensional
integral that describes the probability of observing the array of choices from all
choice sets (cf. Hausman and Wise 1978; Papatla 1996). In this case, the form of
the probabilities for the MNP model becomes somewhat complicated, and we omit
the formal presentation of these probabilities since the notation provides no additional insight.
The log-likelihood for this MNP approach is equal to:
MK

(20)

L MNP = J

pl ln (

l

),

l=1

where l indexes the K-dimensional choice arrays, p l denotes the observed fractions of the choice arrays, and l denotes the choice probabilities expressed as
functions of the model parameters. This specification accounts for both the local
and background contrast effect, because the choice probabilities, as in (19), depend on all profiles in the design or, alternatively, with heterogeneity of the parameters across choice sets. This is not the case with models that treat the choice
sets as independent, such as the MNL model, LCMNL model and the MNP model
of the previous subsection.
Table 6 lists the parameter estimates and fit statistics. After estimation, all eigenvalues of the final Hessian were positive, indicating that the model is identified
(Bekker, Merckens and Wansbeek 1994).
Table 6 shows the same pattern of -estimates as in the MNL model. However,
the fit of the MNP model is much better than that of the MNL model. This is
caused by the estimated covariance parameters. The log-likelihood of the MNP
model is somewhat lower than that of the LCMNL model, and consequently also
the Pseudo R2 is lower. The AIC statistic would favor the LCMNL model over the

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Rinus Haaijer and Michel Wedel

MNP model, but the BIC statistic, which penalties the likelihood more severely,
indicates the MNP model as best. Table 6 shows that most attribute levels are
potentially responsible for correlations between and within choice sets. Note,
however, that because of the effects-type coding some effects may cancel out (see
Haaijer (1999), chapter 5.2.4 for a discussion), so one has to investigate not only
the estimated parameters but in addition the estimated covariance matrix (16).
Table 6:

MNP Estimation Results
Attribute
(level)
1 Brand (1)
2 Brand (2)
3 Capacity (1)
4 Capacity (2)
5 Price (1)
6 Price (2)
7 Filter (1)
8 Thermos (1)
Statistics
Ln-likelihood
AIC
BIC
Pseudo R2

Est.
-0.029
-0.240*
-1.075*
0.565*
0.432*
0.244*
0.355*
0.393*

S.e.
.101
.078
.092
.060
.116
.082
.038
.054

Est.
*

0.417
-0.387*
0.850*
-0.348*
-0.562*
-0.145
0.023
-0.206*

S.e.
.096
.099
.094
.083
.139
.100
.058
.071

-1086.622
2205.245
2256.770
0.332
*: p < 0.05.

Interestingly, the covariance matrix of the MNP model reveals alternatives with
near zero covariances with all other alternatives. This indicates that these are (almost) independent of the other alternatives. After constructing the matrix the estimates reveal what attribute (levels) are responsible for correlations within and
between choice sets.
The estimates of the MNP model were again used to predict the holdout sets.
For the MNP model the predicted log-likelihood is equal to -679.075 (Pseudo R2
=0.165). This shows that it is very important to account for both kind of correlations, or context effects, which not only results in an improved model fit, with
relatively few covariance parameters, but also in an improved holdout predictive
fit, which is better than that of the MNL, MNPcsi, and LCMNL models.

11.5

Discussion and Conclusion

In the above sections we showed several models that can be used to analyze conjoint choice experiments. We discussed the standard MNL model, the Latent Class

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223

MNL model and two versions of an MNP model with a specific covariance structure. The performance of the models was illustrated with an application. It is interesting to compare the results of the various models, although we realize that this
comparison is based on only this one application, so some care must be taken with
respect to drawing conclusions.
When we compare the estimated log-likelihood values, the LCMNL model
gives the best result, followed by the MNP model. At some distance the MNPcsi
model and MNL model follow. However, since the number of parameters in the
LCMNL model is much larger than that in the MNP model (35 and 16 respectively), the BIC-statistic, that compensates for the number of parameters and observations in the model, ranks the MNP model as best. The AIC-statistic, that only
puts a penalty on the number of parameters, still lists the LCMNL model as best.
If we compare the predictive power of the models the MNP model comes out as
best, followed by the LCMNL, MNL and MNPcsi models. Based on these results
one could conclude that the using the MNL model for (conjoint) choice experiments may not be a good choice. It does not account for heterogeneity and correlated choice alternatives within and between choice sets. This results in an inferior
model fit and predictive fit compared to models that do account for these elements. On the positive side, however, we saw that the estimates for the structural
parameters do not differ much between the models. This is in line with findings by
Börsch-Supan et al. (1990), who also found that differences in model fit, with
respect to the log-likelihood value, is often caused by the error structure while the
structural parameters are relatively left unaffected, although a misspecified covariance matrix not only affects the standard errors of the covariance parameters but
also of the structural parameters. This is exactly what may have happened in the
MNPcsi model. So, in terms of finding the „right” structural parameters the MNL
model may do a reasonable job, but when these results are used to predict the
performance of new alternatives, or holdout choice sets, the MNL model falls
short to the LCMNL and MNP model. Of course, further research should be done
to test the generalizability of these findings. Haaijer et al. (1998) tested three data
sets, including the one in this chapter, and found in all three cases that the MNP
model outperforms the Independent Probit model (which has similar characteristics as the MNL model) on estimation fit and holdout predictive power. Haaijer,
Kamakura and Wedel (2000) support these findings for two other data sets.
Several authors have compared continuous and discrete specifications of heterogeneity (e.g., Lenk, DeSarbo, Green and Young 1996; Vriens, Wedel and
Wilms 1996; Allenby, Arora and Ginter 1998; Allenby and Rossi 1999). These
comparisons were made on scanner panel data rather than on conjoint choice data.
Nevertheless, the conclusion from these studies is that for predictive purposes
continuous (MNP) specifications may be preferable over discrete (LCMNL) specifications. From a substantive angle, the MNP-type of specification, particularly
when applied in conjunction with the Gibbs sampler, that allows for individual
level parameters to be estimated, seems preferable in direct marketing applications, where such individual level estimates are of great use in targeting individuals. However, advantages of the discrete model specification accrue in situations

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Rinus Haaijer and Michel Wedel

where managers are interested in targeting market segments (see Wedel and Kamakura 1997 for an extensive discussion).
Several other issues need to be further investigated. The findings of this chapter that local and background context play a role in respondents’ choices should be
studied more closely. The importance of these effects in relation to the number of
attributes, choice sets, levels of attributes and alternatives could be given additional attention. The attribute level effect found in the literature could be particularly related to context effects. Furthermore, the involvement of respondents and
their knowledge on the product category, as well as the product category itself,
could also influence the importance of these context effects. The influence of all
these factors on the choice, and hence ultimately on the parameters of interest,
should be minimized or at least be accounted for. Research should be done on the
optimal design for conjoint choice experiments analyzed with Probit models, since
optimal designs are not yet available. The results obtained from such an optimal
conjoint experiments should lead to managerial more insightful and precise information on the product in question. In addition, one needs research on many product categories, designs, etcetera to investigate how generalizable findings are.
Furthermore, the performance of the Simulated Maximum Likelihood method
explored in this chapter as optimization methods for the MNP models should be
compared with Bayesian estimation using the Gibbs sampler. An interesting avenue for further research is in combining the Latent Class and MNP approaches,
thus having the advantages of the predictive performance of the continuous and of
the managerial appeal of segments of the discrete heterogeneity representation. It
would be interesting to test the performance of such a Latent Class MNP model.
In this chapter we already saw that the MNP model leads to better prediction results as compared to the LCMNL model. In addition accounting for different segments in an MNP context could further improve predictive performance and enhance managerial appeal, although the number of parameters to estimate may
become a limiting factor, with respect to estimation time as well as their identification.
In any case, we may conclude that the MNL (or IP) model is no longer the preferred choice for analyzing conjoint choice experiments.

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