Discrete choice models_TT.ppt

Discrete choice models
CHAPTER 5
1
Lectured by: Dr. Tewodros Tefera

Choice models and preferences
• Choice modelling is the preferred model for studies on consumer
preferences
• Choice models are closely related stated preference theory
• Stated preference survey: consumers state their choices among a
potential set of alternatives (e.g. different brands, different product
characteristics, different stores)
• Options can include both real and hypothetical market alternatives
• Choice models start from stated preferences to go back to their
determinants
• The alternative to stated preference is revealed preference
• where consumers are not asked directly what they prefer or choose
but their actual choices and determinants are observed indirectly,
for example considering what they purchase in different situations
2

Stated vs. revealed preference
• Example
• A customer finds that the price of her favourite washing powder
in her usual supermarket has doubled
• Will she buy less washing powder, like a smaller pack?
• Or would she move to a different brand?
• Would she go back home without buying washing powder at all?
• It could be difficult to define a model which explains choices using
revealed preference, i.e. observing behaviors at the checkout till
• if the customer decides not to buy washing powder at all, how
would it be possible to infer this choice simply from a look at the
products in her shopping trolley?
• if the customer buys an alternative brand with exactly the same
size and price as before the price increase would a revealed
preference model capture that consumer decision?
3

Stated vs. revealed preference
•Revealed preference allows one to model these
behaviors, but only after an expensive collection of
information on the frequency quantity and brands of
washing powder purchases
•Stated preference alternative
• a survey where the consumer is asked to choose between a
set of alternative choices which differ by brand, pack size
and price
• Provided that the survey is designed in an appropriate way
(not necessarily easy) the collected data open the way to a
more effective model
4

Choice models
• Consumer models are usually targeted on the average
behaviour
• With revealed preference one might apply a regression
model; where we purchased the quantity is the dependent
variable and price and other explanatory variables are on
the right-hand side.
• With stated preference models a discrete choice variable is
on the left-hand side of the equation
• Example
• the choice whether to purchase washing powder or not (binary
dependent variable);
• choice among a set of alternative brands (categorical DV)
5

Why regression does not work
• With binary or categorical dependent variables standard regression
analysis is not appropriate
• Example
• binary dependent variable y coded to be zero for non-purchases and one for purchases
• X is a continuous metric variable
• Problems
• After least square estimation predictions of y using the value of x would
produce many other values than zero and one including values below zero and
values above one
• Different coding for the binary dependent variable (e.g. one and two, or zero
and ten) would lead to very different estimates for the a and b coefficients
which makes the interpretation of the regression parameters difficult
• The above model does not meet the assumptions of the regression model
since multivariate normality of the dependent variable for any value of the
explanatory variables is broken
6
with
0 for non purchases
1 for purchases
y x
y
a b 
  

 


Discrete choice models
• Discrete choice models generalize the regression model for the
situations where y is a non-metric variable
• a binary (0-1) variable or
• an ordinal variable (like a questionnaire item assuming the values completely
disagree, disagree, neither, agree, completely agree) or
• a categorical variable (for example a nominal variable recording the preferred
holiday destination).
• The right-hand side variable is generally assumed to be metric
• Binary and categorical variables on the right-hand side can be
translated into dummies and used as explanatory variables like in
regression analysis
• Non-metric dependent variables violate the normality and the
homoskedasticity assumptions of regression; an alternative
approach is used to estimate discrete choice models
7
i i
y x
a b 
  

Binary choice model
• Y can assume the discrete values zero or one
• To model y as a function of x one can exploit a latent variable (as for
SEM)
• y assumes either value zero or one depending on the threshold value d of a
metric and continuous latent variable z
• The regression model is rewritten as
• the dependent variable y is one when a latent continuous variable z
is above the threshold d and zero otherwise
• The model is completed by a regression equation linking the latent
variable to the explanatory variable
8
0 if
1 if
i i
i i
y z
y z
d
d
 


 

i i i
z x
a b 
  

The auxiliary regression
• The above model has a metric and continuous dependent variable
• After some assumptions the distribution of  is known
• Problems:
1) z is not observed and
2) d is unknown
• Problem 2 is easily resolved: as long as the intercept a appears in the
regression equation, one may arbitrarily choose d (the easiest way is to fix it
at zero) and the only result which will change is the estimate of the intercept
a
• Problem 1 requires one to create z for each observation as a function of y,
taking into account the information which we have, that is the proportions of
zero and one for the y variable
• It is necessary to make an assumption on the probability distribution for this
latent variable and how it is linked to y, i.e. a link function between y and z
must be specified
i i i
z x
a b 
  
9

The link functions
•The link function specifies the relationship between z and
y through the expected value of the appropriate
distribution function for the generic observation yi
•For example, with binary data, one can assume that the
probabilities of each observation yi follow a binomial
distribution
•there are a number of transformations of y which create a
z variable compatible with the binomial distribution
10

The logistic transformation
• Probabilities that y=1 (on the vertical axis) concentrate around zero for values of
x below a certain threshold, then go quickly towards 1 when x is above the
threshold.
• The function fits well with the need for approximating the probabilities of a
binary outcome as a function of the explanatory variable.
• The logistic transformation of y into z is obtained by applying the logit link
function to the expected value of y.
11

Logistic regression
• The logit transformation is the link function for logistic regression
• The logit transformation is the log of the odds that y=1 relative to y=0
• The logit link allows to transform the binary variable y into a
continuous variable z
• The final equation is a regression model with a continuous variable on
the left-hand side
• The only difference from the standard regression model is that the
distribution of the error is not normal but logistic.
• Estimation of a and b can be obtained by maximum likelihood which
works with any known probability distribution of the errors and
returns the maximum likelihood estimates (the most probable values
for the parameters)
12

Types of discrete choice models
• Logistic regression: at least one of the explanatory variables is
metric and continuous
• Logit model: all of the variables on the right-hand side are non-
metric (binary or categorical)
• This is a conventional distinction; often the two terms are used
interchangeably
• In a logit model with a categorical or binary x variable, the
coefficient b is mathematically related to the odds ratio (with
respect to the baseline category of x) of having a positive outcome
• For example, if the dependent variable is one when the
consumer buys a specific brand and x measures whether the
consumer has kids or not, one can compute with eb the odds
ratio of buying the brand for consumers with kids as compared
to consumers without kids.
13

Probit model
• The Probit model is also applied to binary dependent variables but
with different assumptions on the link function and the error
distribution
• The link function (called probit) is the inverse of the standard
normal cumulative distribution function
• This link function guarantees that the distribution of the model
which is finally estimated is still normal
• The choice between the probit and the logit distribution depends
on the type of dependent variable
• if the dependent variable can be reasonably assumed to be a
proxy for a true underlying variable which is normally distributed
then the probit model should be chosen
• if the dependent variable is considered to be a truly qualitative
and binomial character then logit modelling should be preferred
• generally the two models lead to very similar results, unless
cases are concentrated to the tails of the distributions in which
case the logit link function should be chosen
14

Generalizations
•ordered logit (ordered probit) models
• the dependent variable is not binary but categorical and the
categories are ordered
•multinomial logit (multinomial probit)
• The dependent variable is categorical but categories cannot be
ordered
•multivariate logit (multivariate probit)
• Several discrete choice models are estimated simultaneously
(there are multiple dependent variable)
15

Conjoint analysis
•Very popular research technique in marketing closely
associated with stated preference analysis
•Mainly exploited for the development of new products
and the modification of product characteristics
•Conjoint analysis is not a model or an estimation
technique but rather a methodology for constructing the
data collection instrument when the final objective is
choice modeling
16

Marketing applications of conjoint analysis
• The most common application in consumer research is the analysis of consumer
evaluations of different combinations of product attributes
• Example
• A car manufacturer needs to take some decision about some options to be
provided for car configuration
• range of colours
• model of car stereo
• presence of air conditioning, etc.
• Rather than asking consumers about their evaluation of these attributes on a
one-by-one basis, conjoint analysis starts by creating potential combinations
of the product attributes
• E.g.
• Combination 1: red car, with an mp3 stereo player and no air-conditioning,
• Combination 2: red car, but with a standard CD player and air-conditioning,
etc.
• Respondents choose among these alternative potential products defined by
the combination of attributes
• From the final choice, conjoint analysis elicits the relevance of each attribute
17

Conjoint analysis
•When several attributes are considered simultaneously
the number of potential combinations is quite high
•Conjoint analysis creates many different choice sets each
one containing a limited number of options
•Conjoint analysis is based on the statistical control of
• the way choices are allocated in the sample
• the distribution of attributes
•Hence, the collected data enable inference on preferences
and evaluations for the individual attributes
18

Theoretical basis for conjoint analysis
• The underlying theory for conjoint analysis is based on the
economic concept of utility
• each individual has a specific set of preferences for bundles of products (and
attributes)
• individuals take decisions in a way to maximize the level of satisfaction from
consumption (the utility level)
• By observing many individuals it is possible to go back from stated
choices to preferences
• Conjoint analysis is inspired by scientific experimental designs and
the terminology reflects this association
• Attributes are called factors (e.g. car colour)
• The different values factors can assume are the levels (red, blue, yellow, etc.)
19

Factors and choice sets
• An additional factor could be the price of the car
• By including price levels in the choice set it becomes possible to
evaluate how much consumers would be willing to pay for the car
they prefer
• Among the potential set of choices there are some nonsense
choices e.g. including all car options but setting a very low price
• Nonsense choices can be excluded by the researcher who has
control on the overall choice set
• Questionnaire
• Respondents must choose from the preferred combination of attributes or
• Respondents must rank all possible choices according to their preferences
• Conjoint analysis is a decompositional method (recall
multidimensional scaling techniques),as it starts from an overall
evaluation to infer preferences for the individual product attributes
20

Theories attributes and choice
•The experimental design and the modelling of preferences
depend on theories which link the evaluation of single
attributes to the final choice
• part-worth model: assumes that total utility of a choice is equal
to the sum of utilities of the attributes of that specific choice
• vector linear model: applicable when all attributes are measured
on a metric (continuous) scale, assumes a linear relationship
between the utility of individual attributes and total utility
• ideal point model: assumes that the consumer has an ideal level
for all factors and the total utility depends upon the distance
between the actual levels and the ideal levels
21

Experimental design
• The key problem of conjoint analysis is the large number of
alternative combinations of attributes which arise when there are
many factors and levels
• E.g. a product with six attributes, each with three levels potentially allows for
729 different combinations
• It would be unrealistic to assume that respondents are able to
choose among so many alternatives
• This problem can be solved by an appropriate experimental design
• Objective: understand the relationship between the factors and the
potential choice with a number of observations as small as possible
• The experimental design sets the criteria to obtain the preference
information from an aggregation of respondents (full factorial designs:
all potential products are compared (729 in the example))
• fractional factorial designs: exploits the experimental design to
reduce the number of choices, still guaranteeing that the sample
will produce meaningful aggregate results
22

Types of conjoint analyses
•Traditional conjoint analysis
• each respondent is faced with the whole set of attributes
• it requires either a full factorial design or a fractional factorial design
• all attributes appear in the choice set of each respondent (although not for all
levels)
• becomes inapplicable as the number of factors or levels increases
•Adaptive conjoint analysis
• these design issues are dealt with
• each respondent only deals with a sub-set of potential choices
• these sub-set can be defined in different ways. For example: respondents
could be asked to rank the factors first, then the ranking is exploited to adapt
data collection
• Computer software learns from the earlier responses and builds the data-sets
accordingly
23

Choice-based conjoint (1)
• The decomposition of the observed choices into weights and
preferences for single attributes is generally obtained for an
aggregate of consumers or for homogeneous groups of consumers
• Several techniques can be employed for this purpose
• The evolution of discrete choice models has given relevance to a
specific type of adaptive conjoint analysis, choice-based conjoint
• Choice-based conjoint gives the respondent the possibility of
evaluating all attributes, not in a single (often too complex) choice,
but rather within a sequence of smaller choice sets where the
possibility of choosing none of the alternatives is also given
24

Example
• Example
• Car colour: red or blue
• Air conditioning: yes or no
• Single choice set
• red with air conditioning (AC)
• red without AC
• blue with AC
• blue without AC
• Choice-based conjoint
• first choose among
• red with AC
• blue without AC
• none of them
• then choose between
• blue with AC
• blue without AC
• none of them
• These choices are related and with a smaller set of choices it is possible to compare all
attributes
25

Choice-based conjoint (2)
• The advantages of choice-based conjoint are apparent with
complex cases
• Respondents do not need to compare too many stimuli at once,
• They face a more realistic choice among a limited set of
alternatives
• With many factors and levels, each respondent can be asked to
face a limited number of choice sets
• The sufficient condition is that an homogeneous group of
respondents (i.e. respondents that are similar in terms of
characteristics that can influence the choice) is confronted with
the whole range of alternatives, then the estimation technique
will do the rest
26

Estimation and models
• The experimental design is at the core of a successful choice-based
conjoint
• There is an evolving research effort to guarantee the quality of the
analysis
• Once the data has been collected the natural estimation technique is the
multinomial logit
• choices represent the categorical dependent variable and the attribute
levels are the explanatory variables
• There are computer packages specifically developed for conjoint analysis
• SPSS Conjoint module
• deals with the experimental design
• provides estimates based on an orthogonal decomposition of the
design matrix
• In SAS/STAT, the TRANSREG procedure is a useful support to define the
experimental design
27

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