THE CONCEPT OF CONJOINT ANALYSIS
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THE CONCEPT OF CONJOINT ANALYSIS
Scott M. Smith, Ph.D.
Marketing managers are faced with numerous difficult tasks directed at assessing
future profitability, sales, and market share for new product entries or modifications
of existing products or marketing strategies. These specific tasks include:
1. Predicting the profitability and/or market share for proposed new product
concepts given the current offering of competitors.
2. Predicting the impact of new competitor products on profits or market share if
we make no change in our competitive position.
3. Predicting customer switch rates either from our current products to new
products we offer (cannibalism), or from our competitors’ products to our new
products (draw).
4. Predicting the differential response of items 13 by key market segments
purchasing our product.
5. Predicting competitive reaction to our strategies of introducing a new product.
Specifically, should a new product be introduced, and if so, what is the optimal
design configuration for this new product? Further, should pricing or other
attributes of our current products be modified in response to the competition).
6. Predicting the impact of situational variables on customer preference.
7. Predicting the differential response to alternative advertising strategies and/or
advertising themes.
8. Predicting the customer response to alternative pricing strategies, specific
price levels, and proposed price changes.
9. Predicting competitive response to distribution strategies studying such diverse
problems as determining the optimal channel of distribution, number or type of
outlets, vendor selection, or sale person quotas.
Each of the identified management problems may be addressed and solved using
the conjoint analysis methodology. In addition, a conjoint based competitive strategy
may be implemented by modifying the marketing mix, i.e., new product/concept
identification, pricing, advertising and distribution. This competitive strategy may
focus on new segments or product repositioning.
In addition to product and corporate strategy, conjoint research has been applied to
family decision making; Tourism, tax analysis; time management; direct foreign;
and medicine.
How does Conjoint Analysis Work?
Conjoint analysis involves the measurement of psychological judgments (such as
consumer preferences, or acceptance) or perceived similarities or differences
between choice alternatives.
The name "Conjoint analysis" implies the study of the joint effects. In marketing
applications, we study the joint effects of multiple product attributes on product
choice.
Alternative Conjoint Analysis Methodologies
Stimulus Construction: Two Factor at a Time; Full Factorial design, Fractional
Factorial Design, Self Explicated, Adaptive Conjoint Analysis, Hierarchical Bayes
Estimation.
Data Collection: Two Factor at a Time Tradeoff Analysis; Full Profile Concept
Evaluation; Feature and Level Evaluation; Combinatorial Approaches.
Model Type: Compensatory and NonCompensatory Models; Part Worth Function;
Vector Model; Mixed Model; Ideal Point Model;
Measurement Scale: Rating Scale; Paired Comparisons; Constant Sum; Rank
Order
Estimation Procedure: Metric and NonMetric Regression; MONANOVA; PREFMAP;
LINMAP; Nonmetric Tradeoff; Multiple Regression; LOGIT; PROBIT; Hybrid; TOBIT;
Discrete Choice Self Explicated Additive Models
Simulation Analysis: Maximum Utility; Average Utility (BradleyTerryLuce);
LOGIT; PROBIT
CONJOINT MEASUREMENT
As consumers or decision makers we often think in terms of concepts, objects or
solutions, rather than relative numerical values.
Conjoint measurement (as distinguished from conjoint analysis) permits the use of
rank or rating data, when evaluating pairs of attributes or attribute profiles (rather
than single attributes). Based on this rank or rating input, the conjoint measurement
procedures are applied to identify a mathematical function of the m brand attributes,
which (1) is interval scaled (produces a set of interval scaled output), (2) best
corresponds to the set of subjective evaluations (ordinal judgments) of the brand
alternatives made by the respondent, and (3) is either a categorical or polynomial
function in the attributes for the rank order data.
The conjoint measurement model assumes that (1) the set of objects being
evaluated is at least weakly ordered (may contain ties), (2) each object evaluated
may be represented by an additive combination of separate utilities existing for the
individual attribute levels, and (3) the derived evaluation model is interval scaled and
comes as close as possible to recovering the original rank order [non metric] or
rating [metric] input data.
The power of conjoint measurement to convert rank or scaled evaluations into
interval scaled output has resulted in much methodological advancement, including
multidimensional scaling and conjoint analysis.
Several different implementations of conjoint measurement are evidenced in conjoint
analysis algorithms and computer programs. These implementations reflect both
algorithmic differences and alternative approaches to data collection and
measurement. The most noticeable are categorical conjoint measurement, monotone
ANOVA models, OLS regression methods and linear programming methods. For
purposes of this tutorial, only OLS regression methods are discussed.
The Ordinary Least Squares regression approach to conjoint analysis offers a simple,
yet robust method of deriving alternative forms of respondent utilities (partworth,
vector, or ideal point models). The attractiveness of the OLS model is in part a result
of the ability to scale respondent choices using rating scales, rather than rankings.
The ability to implement designs having larger numbers of attributes and levels
(through fractional factorial designs) has made this methodology the de facto
standard for conjoint analysis.
The objective of OLS conjoint analysis is to produce a set of additive partworth
utilities (vector or ideal point utilities may also be estimated) that identify each
respondent's preference for each level of a set of product attributes. In application,
the OLS model solves for utilities using a dummy matrix of independent variables.
Each independent variable indicates the presence or absence of a particular attribute
level. The dependent variable is the respondent's evaluation of one of the profiles
described by the independent variables. This model is expressed:
zi = f(yi1 i2 ... im) =B1i1(x1i1 ) + B2i2(x2i2 ) + ,..., + Bmim(xmim )
where:
B = the beta weights estimated in the regression
X = the matrix of dummy values identifying the levels of the factorial design, and
y = the ranking or rating evaluations of the respondent.
The first step in the analysis is to develop either a full or fractional factorial design. A
full profile approach is demonstrated in Figure 12 for our sixattribute example. The
use of fractional factorial designs permits the estimation of a parameter for the main
effect of each attribute included in the analysis. This design, when analyzed, would
produce estimates of individual respondent utilities for each of the 18 attribute
levels. The utilities are additive.
The Measurement of Preference
The measurement of preference is an established part of consumer research that is
based in expectancy value models of attitude theory and measurement. In conjoint
analysis, we examine the preference for a set of brands or other choice alternatives
that are described by an inventory of attributes.
The domain of preference research in conjoint analysis is both broad and multi
faceted. It extends to such diverse issues as how many attributes should be
measured; the influence of the number of attribute levels; the appropriateness of
measuring choice behavior rather than rating or ranking choice alternatives; the
advantages of constructing individualized rather than generic attribute sets. A second
line of preference research has focused on the appropriateness of alternate scaling
methodologies for the measuring of preferences.
Modeling the decision process itself is a third major area of research, and includes
the appropriateness of alternative decision models (compensatory, conjunctive,
disjunctive, Elimination by Aspect, etc.) that may be used either singularly or in
combination to predict preference; the form of the utility preference model estimated
for a given attribute (part worth, linear, or ideal point); and the type of simulation
models used to estimate choice preferences.
While we are tempted to engage in an extended discussion of each of these topics,
we
are constrained by the introductory nature of this discussion. We will limit this
section to a basic discussion of modeling the form of the utility model.
Preference Models
Utility preference models are the mathematical formulations that define the utility
levels for each of the attributes. In practice, the attributes are modeled as either a
piecewise linear (partworth), linear, or curvilinear function.
The PartWorth Model
The partworth model is the simplest of the utility estimation models. This model
represents attribute utilities by a piecewise linear curve. This curve is formed by a
set of straight lines that connect the point estimates of the utilities for the attribute
levels (Figure 21).
The partworth function is defined as:
t
sj = Σ fp Yjp
p=1
where:
sj = Preference for the stimulus object at level j,
fp = the function representing the part worth of each of the
j different levels of the stimulus object, Yjp for the pth attribute.
Yjp = the level of the pth attribute for the jth stimulus object.
The part worth model reflects a utility function that defines a different utility (part
worth) value for each of the j levels of a given attribute. Because of design
considerations, most conjoint studies constrain the number of levels to be less than
5, though in actuality, the number of levels varies from 2 to 9 or more.
The implications of specifying a given preference model (partworth, linear, or ideal
point) extend beyond the actual shape of the preference curve being modeled. Each
preference model requires that a different number of parameters be estimated. The
part worth model requires that a distinct dummy variable column within the design
matrix define each level of an attribute. As would be expected, a total of j1 dummy
variables are required to estimate j levels.
The Vector Model
The Vector model is represented by a single linear function that assumes preference
will increase as the quantity of attribute p increases (preference decreases if the
function is negative). Preference for the jth attribute is defined as:
t
sj = Σ Wp Yjp
p=1
where:
Wp = the individual's weights assigned to each of the p attributes. One weight is
derived for each attribute.
Yjp = Level of the pth attribute for the jth Stimulus
The vector model for the attribute with four levels would appear as a straight line,
with the levels on the line. The vector model requires that a single parameter be
estimated for each variable treated as a vector. In contrast to the partworth model,
the vector model defines the attribute variable not as a series of dummy variables,
but as a single linear variable where the values are the measured values or levels
associated with the attribute.
The Ideal Point Model
The ideal point function is implemented as a curvilinear function that defines an
optimum or ideal amount of an attribute. The ideal point model is appropriate for
many qualitative attributes, such as those associated with taste or smell. Too much
sweetness may be less than optimal, while just the right amount is highly preferred.
The ideal point model establishes an inverse relationship between preferences and
the weighted distance (dj2) between the location of the jth stimulus and the
individual's ideal point, Xp. The ideal point model is expressed as:
t
d 2
j = Σ Wp(Yjp  Xp)2
p=1
Where:
Yjp = Level of the jth Stimulus with respect to the individual's ideal point, Xp.
Xp = The Individual's ideal point, p, and
Wp = the individual's weights assigned to each of the p attributes. One weight is
derived for each attribute.
Yjp = Level of the pth attribute for the jth Stimulus
The idealpoint model for the attribute with three levels would appear as a curve with
the center of the curve higher than either end, with the highest point being the ideal
quantity of the attribute.
Mathematically, the implications of specifying each of the models ultimately extend
to the number of parameters that must be estimated. The vector model treats the
variable Yjp as a continuous (interval scaled) variable, such that only t parameters
(j=1,...,t) must be estimated.
For the ideal point model, 2t parameters must be estimated (Wp and Xp), and for
the part worth model, (q1)t parameters must be estimated, where q is specified to
the number of levels for each of the t attributes.
Stimulus Construction: The Basis for Conjoint Analysis
Stimulus construction in conjoint analysis focuses on the related problems of
determining which attributes to present to the respondent, and how (in terms of
what kinds of conjoint model) the attributes are presented. Because these problems
are not independent of the conjoint model employed, we will consider the tradeoff
and full profile conjoint methodologies and their associated models.
The sample case, provided by Paul Green and Catherine Schaeffer, identifies 30
students recruited from an MBA level Marketing Research class to answer questions
about student apartments. The questionnaire and associated materials are found in
Appendix B. The apartments considered were described by six attributes or factors,
each with 3 "levels":
(1) Walking Time to Classes: (10, 20, 30 minutes)
(2) Noise Level of Apartment House: (Very Quiet, Average, Extremely Noisy)
(3) Safety of Apartment Location: (Very Safe, Average, Very Unsafe)
(4) Condition of Apt: (Newly Renovated Throughout, Renovated Kitchen, Poor
Condition)
(5) Size of Living/Dining Area: (24 x 30, 15 x 20, 9 x 12)
(6) Monthly Rent Including Utilities: ($225, $360, $540)
The Full Profile and Fractional Factorial Models
Full profile descriptions are an attempt to represent real world decision alternatives
in a realistic manner. Like real world alternatives, full profile descriptors present an
integrated multiattribute concept (Green, 1974).
The first full profile designs took the form of nonmetric additive models and were
initially applied to complete blockfull factorial designs. Because the full factorial
designs expand the number of profiles in exponential fashion, the number of factors
and levels considered in these early studies were small. The sample data illustrates
this problem, where a 36 design results in 729 possible unique profiles can be
produced from the set of 6 factors that we are investigating. The full profile approach
is illustrated by the following two sample profiles:
CARD #1
CARD #2
Walking Time To Class
10 MINUTES
20 MINUTES
Noise Level of Apartment
VERY QUIET
AVERAGE NOISE LEVEL
Safety of Apartment Location
VERY SAFE
VERY UNSAFE
Condition of Apartment
RENOVATED
KITCHEN RENOVATED
Size of Living/Dining Area
24 BY 30 FEET
15 BY 20 FEET
Monthly Rent With Utilities
$540
$360
For the respondent to evaluate 729 profiles is an unmanageable task. It is fortunate
that fractional factorial statistical designs may be invoked to greatly reduce the data
collection task. In the example case of six factors each with 3 levels, the use of a
fractional factorial design reduces the 36 = 729 possible profiles to only 18 profiles
(Figure 13).
It is from this reduced set of profiles that we estimate the set of choice utilities
associated with each of the individual factors and their associated levels. It is
noteworthy that while the 18 trial design is sufficient to estimate main effects,
interaction effects between factors can not be estimated with this small number of
profiles. The estimation of interaction between specific variables requires that
additional variables be added to the design matrix.
Figure 13: Stimulus Combinations
Card
Rent
Time
Noise Renovation
Dining
Safety
1
$540
10 Min.
V. Quiet All
24 x 30
V. Safe
2
$360
20 Min.
Average Kitchen
15 x 20
V. Unsafe
3
$225
30 Min.
E. Noisy None
9 x 12
Average
4
$540
10 Min.
Average Kitchen
9 x 12
Average
5
$360
20 Min.
E. Noisy None
24 x 30
V. Safe
6
$225
30 Min.
V. Quiet All
15 x 20
V. Unsafe
7
$225
10 Min.
E. Noisy Kitchen
24 x 30
V. Unsafe
8
$540
20 Min.
V. Quiet None
5 x 20
Average
9
$360
30 Min.
Average All
9 x 12
V. Safe
10
$360
10 Min.
E. Noisy All
15 x 20
Average
11
$225
20 Min.
V. Quiet Kitchen
9 x 12
V. Safe
12
$540
30 Min.
Average None
24 x 30
V. Unsafe
13
$360
10 Min.
V. Quiet None
9 x 12
V. Unsafe
14
$225
20 Min.
Average All
24 x 30
Average
15
$540
30 Min.
E. Noisy Kitchen
15 x 20
V. Safe
16
$225
10 Min.
Average None
15 x 20
V. Safe
17
$540
20 Min.
E. Noisy All
9 x 12
V. Unsafe
18
$360
30 Min.
V. Quiet Kitchen
24 x 30
Average
Again, the objective is to find a set of partworths for the separate factor levels so
that when these are appropriately added, one can find a total utility for each
combination.
Conjoint Analysis Steps
The steps of the full profile analysis follow:
1. The respondent is given a set of stimulus profiles (constructed along factorial
design principles in the full profile case). In the twofactor approach, pairs of factors
are presented, each appearing approximately an equal number of times.
2. The respondents rank or rate the stimuli according to some overall criterion, such
as preference, acceptability, or likelihood of purchase.
3. In the analysis of the data, partworths are identified for the factor levels such
that each specific combination of partworths equals the total utility of any given
profile. A set of partworths is derived for each respondent.
4. The goodnessoffit criterion relates the derived ranking or rating of stimulus
profiles to the original ranking or rating data.
5. A set of objects are defined for the choice simulator. Based on previously
determined partworths for each respondent, each simulator computes an utility
value for each of the objects defined as part of the simulation.
6. Choice simulator models are invoked which rely on decision rules (first choice
model, average probability model or logit model) to estimate the respondent's object
of choice. Overall choice shares are computed for the sample.
Results of Conjoint Analysis
The OLS conjoint analysis results for one respondent in the example of Table 13
would appear as:
0.00 5.00 10.00 0.00 6.67 13.33 0.00 23.33 26.67 0.00 5.00 10.00
0.00 10.00 25.00 0.00 38.33 51.67
This set of derived utility values can be used to obtain a total utility for each of the
18 combinations in Figure 13. For example, to find the utility of the first
combination in Figure 13, we simply add the part worths of the respective levels
identified by combination 1:
Dimension Value
Respondent
Utility
Walking Time to Class 10 Min.
10.00
Noise Level of
Very Quiet
13.33
Apartment
Safety of Apartment
Very Safe
26.67
Location
Renovations
All Renovated
10.00
Size of Living Room
24x30
25.00
and Dining Room
Monthly Rent
$540
0.00
Total
85.00
The total part worths sum to 85.00. The utility of the combination with the highest
value is described as an apartment that rents for $250, is 10 minutes from campus,
is very quiet, is all renovated, has a dining/living room that is 24 x 30, and is in a
neighborhood judged to be very safe. It is possible to construct total utility values for
each of the 729 possible combinations of the six attributes.
Computing Factor Importance
The estimation of utilities for each of the factors permits the estimation of average
factor importance in addition to the estimation of average utility levels for each of
the factors.
The importance of each of the i factors is estimated as a function of the range of the
average observed utilities for the levels of each the factors. Importance is computed
as:
(Maxu i  Minu i)
Ii = Σ 
(Maxu i  Minu i)
For the six factors graphed in Figure 16, the utilities with their associated
importance are:
Lowest 
Highest
Relative
Utility
Utility
Importance
Time to Class
1.72 4.65 2.93
11.27
Neighborhood Noise 1.61 4.54 2.93
11.27
Neighborhood Safety 1.42 8.70 7.28
28.00
Condition of Apt.
.54 6.27 5.73
22.04
Dining/Living Size 2.21 3.21 1.00
3.85
Rental Amount
1.49 7.62 6.13
23.58
 
Sum of Importance:
26.00
100.00
Choice Simulators
The final stage of the conjoint analysis is the choice simulator. The purpose of the
choice simulator is to estimate percent of respondent choice for specific factor
profiles entered into the simulator. Most often, the current competitors in the market
are defined by identifying specific levels of the choice attributes. The simulator
estimates choice share for the current market. Next, the data set identifying the
competitors is supplemented with new products that are being considered for
introduction into the market. The simulator responds by assigning choice shares for
each of the items. The increase or decrease in brand shares is noted, as is the source
of that share increase or decrease.
The most common simulator models include the first choice model, the average
choice (BradleyTerryLuce) model, and the Logit model. The First choice model
identifies the product with the highest utility as the product of choice. This product is
selected and receives a value of 1. Ties receive a .5 value. After the process is
repeated for each respondent's utility set, the cumulative "votes" for each product
are evaluated as a proportion of the votes or respondents in the sample.
The BradleyTerryLuce model estimates choice probability in a different fashion. The
choice probability for a given product is based on the utility for that product divided
by the sum of all products in the simulated market.
The logit model uses an assigned choice probability that is proportional to an
increasing monotonic function of the alternative's utility. The choice probabilities are
computed by dividing the logit value for one product by the sum for all other
products in the simulation. These individual choice probabilities are averaged across
respondents. In summary, while the literature shows the maximum utility (first
choice model) to provide the best overall validation, choice behavior has a strong
probabilistic component. We have not measured this component adequately, but
instead attributed lack of validity to "noise", our inability to model information search
and overload effects, and measurement error.
References
Green, P. E. and V. Srinivasan (1978), "Conjoint Analysis in Consumer Research"
Issues and Outlook", Journal of Consumer Research, Vol. 5, (September), pp 103
123.
Luce, R. D. and J. W. Tukey. "Simultaneous Conjoint Measurement: A New Type of
Fundamental Measurement," Journal of Mathematical Psychology, 1 (February 1964),
pp 127.
Scott M. Smith, Ph.D.
Marketing managers are faced with numerous difficult tasks directed at assessing
future profitability, sales, and market share for new product entries or modifications
of existing products or marketing strategies. These specific tasks include:
1. Predicting the profitability and/or market share for proposed new product
concepts given the current offering of competitors.
2. Predicting the impact of new competitor products on profits or market share if
we make no change in our competitive position.
3. Predicting customer switch rates either from our current products to new
products we offer (cannibalism), or from our competitors’ products to our new
products (draw).
4. Predicting the differential response of items 13 by key market segments
purchasing our product.
5. Predicting competitive reaction to our strategies of introducing a new product.
Specifically, should a new product be introduced, and if so, what is the optimal
design configuration for this new product? Further, should pricing or other
attributes of our current products be modified in response to the competition).
6. Predicting the impact of situational variables on customer preference.
7. Predicting the differential response to alternative advertising strategies and/or
advertising themes.
8. Predicting the customer response to alternative pricing strategies, specific
price levels, and proposed price changes.
9. Predicting competitive response to distribution strategies studying such diverse
problems as determining the optimal channel of distribution, number or type of
outlets, vendor selection, or sale person quotas.
Each of the identified management problems may be addressed and solved using
the conjoint analysis methodology. In addition, a conjoint based competitive strategy
may be implemented by modifying the marketing mix, i.e., new product/concept
identification, pricing, advertising and distribution. This competitive strategy may
focus on new segments or product repositioning.
In addition to product and corporate strategy, conjoint research has been applied to
family decision making; Tourism, tax analysis; time management; direct foreign;
and medicine.
How does Conjoint Analysis Work?
Conjoint analysis involves the measurement of psychological judgments (such as
consumer preferences, or acceptance) or perceived similarities or differences
between choice alternatives.
The name "Conjoint analysis" implies the study of the joint effects. In marketing
applications, we study the joint effects of multiple product attributes on product
choice.
Alternative Conjoint Analysis Methodologies
Stimulus Construction: Two Factor at a Time; Full Factorial design, Fractional
Factorial Design, Self Explicated, Adaptive Conjoint Analysis, Hierarchical Bayes
Estimation.
Data Collection: Two Factor at a Time Tradeoff Analysis; Full Profile Concept
Evaluation; Feature and Level Evaluation; Combinatorial Approaches.
Model Type: Compensatory and NonCompensatory Models; Part Worth Function;
Vector Model; Mixed Model; Ideal Point Model;
Measurement Scale: Rating Scale; Paired Comparisons; Constant Sum; Rank
Order
Estimation Procedure: Metric and NonMetric Regression; MONANOVA; PREFMAP;
LINMAP; Nonmetric Tradeoff; Multiple Regression; LOGIT; PROBIT; Hybrid; TOBIT;
Discrete Choice Self Explicated Additive Models
Simulation Analysis: Maximum Utility; Average Utility (BradleyTerryLuce);
LOGIT; PROBIT
CONJOINT MEASUREMENT
As consumers or decision makers we often think in terms of concepts, objects or
solutions, rather than relative numerical values.
Conjoint measurement (as distinguished from conjoint analysis) permits the use of
rank or rating data, when evaluating pairs of attributes or attribute profiles (rather
than single attributes). Based on this rank or rating input, the conjoint measurement
procedures are applied to identify a mathematical function of the m brand attributes,
which (1) is interval scaled (produces a set of interval scaled output), (2) best
corresponds to the set of subjective evaluations (ordinal judgments) of the brand
alternatives made by the respondent, and (3) is either a categorical or polynomial
function in the attributes for the rank order data.
The conjoint measurement model assumes that (1) the set of objects being
evaluated is at least weakly ordered (may contain ties), (2) each object evaluated
may be represented by an additive combination of separate utilities existing for the
individual attribute levels, and (3) the derived evaluation model is interval scaled and
comes as close as possible to recovering the original rank order [non metric] or
rating [metric] input data.
The power of conjoint measurement to convert rank or scaled evaluations into
interval scaled output has resulted in much methodological advancement, including
multidimensional scaling and conjoint analysis.
Several different implementations of conjoint measurement are evidenced in conjoint
analysis algorithms and computer programs. These implementations reflect both
algorithmic differences and alternative approaches to data collection and
measurement. The most noticeable are categorical conjoint measurement, monotone
ANOVA models, OLS regression methods and linear programming methods. For
purposes of this tutorial, only OLS regression methods are discussed.
The Ordinary Least Squares regression approach to conjoint analysis offers a simple,
yet robust method of deriving alternative forms of respondent utilities (partworth,
vector, or ideal point models). The attractiveness of the OLS model is in part a result
of the ability to scale respondent choices using rating scales, rather than rankings.
The ability to implement designs having larger numbers of attributes and levels
(through fractional factorial designs) has made this methodology the de facto
standard for conjoint analysis.
The objective of OLS conjoint analysis is to produce a set of additive partworth
utilities (vector or ideal point utilities may also be estimated) that identify each
respondent's preference for each level of a set of product attributes. In application,
the OLS model solves for utilities using a dummy matrix of independent variables.
Each independent variable indicates the presence or absence of a particular attribute
level. The dependent variable is the respondent's evaluation of one of the profiles
described by the independent variables. This model is expressed:
zi = f(yi1 i2 ... im) =B1i1(x1i1 ) + B2i2(x2i2 ) + ,..., + Bmim(xmim )
where:
B = the beta weights estimated in the regression
X = the matrix of dummy values identifying the levels of the factorial design, and
y = the ranking or rating evaluations of the respondent.
The first step in the analysis is to develop either a full or fractional factorial design. A
full profile approach is demonstrated in Figure 12 for our sixattribute example. The
use of fractional factorial designs permits the estimation of a parameter for the main
effect of each attribute included in the analysis. This design, when analyzed, would
produce estimates of individual respondent utilities for each of the 18 attribute
levels. The utilities are additive.
The Measurement of Preference
The measurement of preference is an established part of consumer research that is
based in expectancy value models of attitude theory and measurement. In conjoint
analysis, we examine the preference for a set of brands or other choice alternatives
that are described by an inventory of attributes.
The domain of preference research in conjoint analysis is both broad and multi
faceted. It extends to such diverse issues as how many attributes should be
measured; the influence of the number of attribute levels; the appropriateness of
measuring choice behavior rather than rating or ranking choice alternatives; the
advantages of constructing individualized rather than generic attribute sets. A second
line of preference research has focused on the appropriateness of alternate scaling
methodologies for the measuring of preferences.
Modeling the decision process itself is a third major area of research, and includes
the appropriateness of alternative decision models (compensatory, conjunctive,
disjunctive, Elimination by Aspect, etc.) that may be used either singularly or in
combination to predict preference; the form of the utility preference model estimated
for a given attribute (part worth, linear, or ideal point); and the type of simulation
models used to estimate choice preferences.
While we are tempted to engage in an extended discussion of each of these topics,
we
are constrained by the introductory nature of this discussion. We will limit this
section to a basic discussion of modeling the form of the utility model.
Preference Models
Utility preference models are the mathematical formulations that define the utility
levels for each of the attributes. In practice, the attributes are modeled as either a
piecewise linear (partworth), linear, or curvilinear function.
The PartWorth Model
The partworth model is the simplest of the utility estimation models. This model
represents attribute utilities by a piecewise linear curve. This curve is formed by a
set of straight lines that connect the point estimates of the utilities for the attribute
levels (Figure 21).
The partworth function is defined as:
t
sj = Σ fp Yjp
p=1
where:
sj = Preference for the stimulus object at level j,
fp = the function representing the part worth of each of the
j different levels of the stimulus object, Yjp for the pth attribute.
Yjp = the level of the pth attribute for the jth stimulus object.
The part worth model reflects a utility function that defines a different utility (part
worth) value for each of the j levels of a given attribute. Because of design
considerations, most conjoint studies constrain the number of levels to be less than
5, though in actuality, the number of levels varies from 2 to 9 or more.
The implications of specifying a given preference model (partworth, linear, or ideal
point) extend beyond the actual shape of the preference curve being modeled. Each
preference model requires that a different number of parameters be estimated. The
part worth model requires that a distinct dummy variable column within the design
matrix define each level of an attribute. As would be expected, a total of j1 dummy
variables are required to estimate j levels.
The Vector Model
The Vector model is represented by a single linear function that assumes preference
will increase as the quantity of attribute p increases (preference decreases if the
function is negative). Preference for the jth attribute is defined as:
t
sj = Σ Wp Yjp
p=1
where:
Wp = the individual's weights assigned to each of the p attributes. One weight is
derived for each attribute.
Yjp = Level of the pth attribute for the jth Stimulus
The vector model for the attribute with four levels would appear as a straight line,
with the levels on the line. The vector model requires that a single parameter be
estimated for each variable treated as a vector. In contrast to the partworth model,
the vector model defines the attribute variable not as a series of dummy variables,
but as a single linear variable where the values are the measured values or levels
associated with the attribute.
The Ideal Point Model
The ideal point function is implemented as a curvilinear function that defines an
optimum or ideal amount of an attribute. The ideal point model is appropriate for
many qualitative attributes, such as those associated with taste or smell. Too much
sweetness may be less than optimal, while just the right amount is highly preferred.
The ideal point model establishes an inverse relationship between preferences and
the weighted distance (dj2) between the location of the jth stimulus and the
individual's ideal point, Xp. The ideal point model is expressed as:
t
d 2
j = Σ Wp(Yjp  Xp)2
p=1
Where:
Yjp = Level of the jth Stimulus with respect to the individual's ideal point, Xp.
Xp = The Individual's ideal point, p, and
Wp = the individual's weights assigned to each of the p attributes. One weight is
derived for each attribute.
Yjp = Level of the pth attribute for the jth Stimulus
The idealpoint model for the attribute with three levels would appear as a curve with
the center of the curve higher than either end, with the highest point being the ideal
quantity of the attribute.
Mathematically, the implications of specifying each of the models ultimately extend
to the number of parameters that must be estimated. The vector model treats the
variable Yjp as a continuous (interval scaled) variable, such that only t parameters
(j=1,...,t) must be estimated.
For the ideal point model, 2t parameters must be estimated (Wp and Xp), and for
the part worth model, (q1)t parameters must be estimated, where q is specified to
the number of levels for each of the t attributes.
Stimulus Construction: The Basis for Conjoint Analysis
Stimulus construction in conjoint analysis focuses on the related problems of
determining which attributes to present to the respondent, and how (in terms of
what kinds of conjoint model) the attributes are presented. Because these problems
are not independent of the conjoint model employed, we will consider the tradeoff
and full profile conjoint methodologies and their associated models.
The sample case, provided by Paul Green and Catherine Schaeffer, identifies 30
students recruited from an MBA level Marketing Research class to answer questions
about student apartments. The questionnaire and associated materials are found in
Appendix B. The apartments considered were described by six attributes or factors,
each with 3 "levels":
(1) Walking Time to Classes: (10, 20, 30 minutes)
(2) Noise Level of Apartment House: (Very Quiet, Average, Extremely Noisy)
(3) Safety of Apartment Location: (Very Safe, Average, Very Unsafe)
(4) Condition of Apt: (Newly Renovated Throughout, Renovated Kitchen, Poor
Condition)
(5) Size of Living/Dining Area: (24 x 30, 15 x 20, 9 x 12)
(6) Monthly Rent Including Utilities: ($225, $360, $540)
The Full Profile and Fractional Factorial Models
Full profile descriptions are an attempt to represent real world decision alternatives
in a realistic manner. Like real world alternatives, full profile descriptors present an
integrated multiattribute concept (Green, 1974).
The first full profile designs took the form of nonmetric additive models and were
initially applied to complete blockfull factorial designs. Because the full factorial
designs expand the number of profiles in exponential fashion, the number of factors
and levels considered in these early studies were small. The sample data illustrates
this problem, where a 36 design results in 729 possible unique profiles can be
produced from the set of 6 factors that we are investigating. The full profile approach
is illustrated by the following two sample profiles:
CARD #1
CARD #2
Walking Time To Class
10 MINUTES
20 MINUTES
Noise Level of Apartment
VERY QUIET
AVERAGE NOISE LEVEL
Safety of Apartment Location
VERY SAFE
VERY UNSAFE
Condition of Apartment
RENOVATED
KITCHEN RENOVATED
Size of Living/Dining Area
24 BY 30 FEET
15 BY 20 FEET
Monthly Rent With Utilities
$540
$360
For the respondent to evaluate 729 profiles is an unmanageable task. It is fortunate
that fractional factorial statistical designs may be invoked to greatly reduce the data
collection task. In the example case of six factors each with 3 levels, the use of a
fractional factorial design reduces the 36 = 729 possible profiles to only 18 profiles
(Figure 13).
It is from this reduced set of profiles that we estimate the set of choice utilities
associated with each of the individual factors and their associated levels. It is
noteworthy that while the 18 trial design is sufficient to estimate main effects,
interaction effects between factors can not be estimated with this small number of
profiles. The estimation of interaction between specific variables requires that
additional variables be added to the design matrix.
Figure 13: Stimulus Combinations
Card
Rent
Time
Noise Renovation
Dining
Safety
1
$540
10 Min.
V. Quiet All
24 x 30
V. Safe
2
$360
20 Min.
Average Kitchen
15 x 20
V. Unsafe
3
$225
30 Min.
E. Noisy None
9 x 12
Average
4
$540
10 Min.
Average Kitchen
9 x 12
Average
5
$360
20 Min.
E. Noisy None
24 x 30
V. Safe
6
$225
30 Min.
V. Quiet All
15 x 20
V. Unsafe
7
$225
10 Min.
E. Noisy Kitchen
24 x 30
V. Unsafe
8
$540
20 Min.
V. Quiet None
5 x 20
Average
9
$360
30 Min.
Average All
9 x 12
V. Safe
10
$360
10 Min.
E. Noisy All
15 x 20
Average
11
$225
20 Min.
V. Quiet Kitchen
9 x 12
V. Safe
12
$540
30 Min.
Average None
24 x 30
V. Unsafe
13
$360
10 Min.
V. Quiet None
9 x 12
V. Unsafe
14
$225
20 Min.
Average All
24 x 30
Average
15
$540
30 Min.
E. Noisy Kitchen
15 x 20
V. Safe
16
$225
10 Min.
Average None
15 x 20
V. Safe
17
$540
20 Min.
E. Noisy All
9 x 12
V. Unsafe
18
$360
30 Min.
V. Quiet Kitchen
24 x 30
Average
Again, the objective is to find a set of partworths for the separate factor levels so
that when these are appropriately added, one can find a total utility for each
combination.
Conjoint Analysis Steps
The steps of the full profile analysis follow:
1. The respondent is given a set of stimulus profiles (constructed along factorial
design principles in the full profile case). In the twofactor approach, pairs of factors
are presented, each appearing approximately an equal number of times.
2. The respondents rank or rate the stimuli according to some overall criterion, such
as preference, acceptability, or likelihood of purchase.
3. In the analysis of the data, partworths are identified for the factor levels such
that each specific combination of partworths equals the total utility of any given
profile. A set of partworths is derived for each respondent.
4. The goodnessoffit criterion relates the derived ranking or rating of stimulus
profiles to the original ranking or rating data.
5. A set of objects are defined for the choice simulator. Based on previously
determined partworths for each respondent, each simulator computes an utility
value for each of the objects defined as part of the simulation.
6. Choice simulator models are invoked which rely on decision rules (first choice
model, average probability model or logit model) to estimate the respondent's object
of choice. Overall choice shares are computed for the sample.
Results of Conjoint Analysis
The OLS conjoint analysis results for one respondent in the example of Table 13
would appear as:
0.00 5.00 10.00 0.00 6.67 13.33 0.00 23.33 26.67 0.00 5.00 10.00
0.00 10.00 25.00 0.00 38.33 51.67
This set of derived utility values can be used to obtain a total utility for each of the
18 combinations in Figure 13. For example, to find the utility of the first
combination in Figure 13, we simply add the part worths of the respective levels
identified by combination 1:
Dimension Value
Respondent
Utility
Walking Time to Class 10 Min.
10.00
Noise Level of
Very Quiet
13.33
Apartment
Safety of Apartment
Very Safe
26.67
Location
Renovations
All Renovated
10.00
Size of Living Room
24x30
25.00
and Dining Room
Monthly Rent
$540
0.00
Total
85.00
The total part worths sum to 85.00. The utility of the combination with the highest
value is described as an apartment that rents for $250, is 10 minutes from campus,
is very quiet, is all renovated, has a dining/living room that is 24 x 30, and is in a
neighborhood judged to be very safe. It is possible to construct total utility values for
each of the 729 possible combinations of the six attributes.
Computing Factor Importance
The estimation of utilities for each of the factors permits the estimation of average
factor importance in addition to the estimation of average utility levels for each of
the factors.
The importance of each of the i factors is estimated as a function of the range of the
average observed utilities for the levels of each the factors. Importance is computed
as:
(Maxu i  Minu i)
Ii = Σ 
(Maxu i  Minu i)
For the six factors graphed in Figure 16, the utilities with their associated
importance are:
Lowest 
Highest
Relative
Utility
Utility
Importance
Time to Class
1.72 4.65 2.93
11.27
Neighborhood Noise 1.61 4.54 2.93
11.27
Neighborhood Safety 1.42 8.70 7.28
28.00
Condition of Apt.
.54 6.27 5.73
22.04
Dining/Living Size 2.21 3.21 1.00
3.85
Rental Amount
1.49 7.62 6.13
23.58
 
Sum of Importance:
26.00
100.00
Choice Simulators
The final stage of the conjoint analysis is the choice simulator. The purpose of the
choice simulator is to estimate percent of respondent choice for specific factor
profiles entered into the simulator. Most often, the current competitors in the market
are defined by identifying specific levels of the choice attributes. The simulator
estimates choice share for the current market. Next, the data set identifying the
competitors is supplemented with new products that are being considered for
introduction into the market. The simulator responds by assigning choice shares for
each of the items. The increase or decrease in brand shares is noted, as is the source
of that share increase or decrease.
The most common simulator models include the first choice model, the average
choice (BradleyTerryLuce) model, and the Logit model. The First choice model
identifies the product with the highest utility as the product of choice. This product is
selected and receives a value of 1. Ties receive a .5 value. After the process is
repeated for each respondent's utility set, the cumulative "votes" for each product
are evaluated as a proportion of the votes or respondents in the sample.
The BradleyTerryLuce model estimates choice probability in a different fashion. The
choice probability for a given product is based on the utility for that product divided
by the sum of all products in the simulated market.
The logit model uses an assigned choice probability that is proportional to an
increasing monotonic function of the alternative's utility. The choice probabilities are
computed by dividing the logit value for one product by the sum for all other
products in the simulation. These individual choice probabilities are averaged across
respondents. In summary, while the literature shows the maximum utility (first
choice model) to provide the best overall validation, choice behavior has a strong
probabilistic component. We have not measured this component adequately, but
instead attributed lack of validity to "noise", our inability to model information search
and overload effects, and measurement error.
References
Green, P. E. and V. Srinivasan (1978), "Conjoint Analysis in Consumer Research"
Issues and Outlook", Journal of Consumer Research, Vol. 5, (September), pp 103
123.
Luce, R. D. and J. W. Tukey. "Simultaneous Conjoint Measurement: A New Type of
Fundamental Measurement," Journal of Mathematical Psychology, 1 (February 1964),
pp 127.
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