The application of discrete choice models in marketing

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.9, 2014
The Application of Discrete Choice Models In Marketing
Decisions
Ateq Al-ghamedii1 and Melfi Alrasheedi 2
1 Department of Statistics, Faculty of Science, King Abdulaziz University
Jeddah, Saudi Arabia. Email: drateq@gmail.com
2 Department of Quantitative Methods, School of Business, King Faisal University
Hofof, Saudi Arabia. Email: kfu.me@hotmail.com
53
ABSTRACT
The paper is devoted to the analysis of logit models and their application in the market. A theoretical basis for
logit models is determined. Equations for logit probabilities are derived and methods are applied in order to
analyze real market situations. The real data set is analyzed to estimate 2 logit models, as well as a probit model.
Obtained results are compared with experimentally calculated logit probabilities.
Keywords: Decisions; discrete choice model; logit and probit models; simulation; statistical modeling
1. INTRODUCTION
Scientific research, as well as ordinary life situations, often involves choices and decisions. There are many
directions of application for choice models, such as transport demand (Domencich & McFadden, 1975;
McFadden, 1974), market research (Malhotra, 1984; Huang & Rojas, 2010), adoption decisions (Adeogun et al.,
2008), and so on. In particular, the analysis of demand for different goods is a popular direction. Statistics and
probability theory are of great assistance in this type of situation.
In principle, the decision makers (to be referred to as consumers in this paper) can be people, households or
firms and the alternatives might be products or courses of action. It should be emphasized that choosing of one
alternative implies not choosing any other alternatives. In addition, the number of alternatives is finite (Train,
2003).
Discrete choice models are usually derived under an assumption of utility-maximizing behaviour (Train, 2003).
This means that the decision maker n makes the decision j if utility of choice is the largest:
U U i j nj ni  ,  . This utility is composed of two parts (Train, 2003):
nj nj nj U V  , (1)
where nj  contains the factors that affect the utility, but are not included in nj V . The problem with this type of
discrete choice model is that nj  is not seen by the researcher. Decomposition (1) is fully general, since nj  is
defined as a difference between the true utility nj U and the part of utility nj V captured by the researcher (Train,
2003). Each characteristic of nj  , such as its distribution, depends on researcher specifications. The aim of the
researcher is to estimate the parameters of this distribution.
In this paper we are going to examine the application of a logit model for discrete choice analysis. The goal of
the paper is to estimate the coefficients of the logistic regression. The chosen estimation method for this purpose
is a maximum likelihood estimator (MLE) (Myung, 2002), which is the most common method for a model
estimation.
The estimated logit coefficients will allow for calculation of the probability of making a particular decision. Such
an obtained probability distribution would play a key role in the analysis of the demand for new goods on the
market.

Vol.4, No.9, 2014
In addition to the standard logit model, which is used for the calculation of first choice probability for a given
good among the choice set, the ordered probability can be useful. Such probability of the rankings can be
obtained from the exploded or ordered logit model (Johnson et al., 2010; Kumar & Kant, 2007). The most
important features of both logit and exploded logit models, as well as the methods of their estimation, are
discussed in the next section.
1
 X  (2)
54
2. ANALYSIS OF LOGIT MODELS
Logit or logistic distribution has a closed form of cumulative distribution function
1 exp( X)
( )
 
X
p
logit
probit
Figure 1: Logit and probit cumulative distributions
One can see from Figure1 that the logit cumulation distribution function (CDF) is similar to the common normal
CDF; however, the tails of logit are heavier. The main advantage of the logit distribution is the simplicity of the
CDF, while the normal CDF involves an unevaluated integral.
In order to derive the choice probabilities for the logit model, we analyze N consumers and J possible choices.
The utility of each choice is known only for the customer, but not for the researcher. This utility nj U can be
presented as in (1). The first term in (1), nj V , is usually decomposed into two parts  nj x and j k . Term  nj x
is vector of variables that relate to alternative j as faced by decision maker j, and j k is a constant that is specific
to alternative j. The researcher treats nj  as random. The joint density of the random vector  n   n1... nJ
is denoted as f ( n ) . In this notation the probability that the consumer n chooses alternative j is
P PU U i j PV U i j P V V i j nj nj ni nj nj ni ni ni nj nj ni                ,  (3)
This probability is a cumulative distribution; namely, the probability that each random term ni nj   is below
the observed quantity nj ni V V . Using the density ( ) n f  , this cumulative probability can be rewritten as
         nj ni nj nj ni n n P I   V V i j f ( )d (4)

Vol.4, No.9, 2014
Here I () is indicator function, which is equal 1 if equation in the brackets is true. One should emphasize here
that the choice probability nj P depends only on the differences ni nj   and Vnj Vni .
The logit model is obtained by assuming that each  nj is an independent and identically distributed extreme
value to each additional  nj
value. Such specific distributions arise in the case of modeling of rare events. The
generalized extreme value distribution (GEV) PDF is defined as
 
kz kz k
exp  (1  ) (1  ) 
0
 , ( ) /
exp( )
 (8)
55





 
   

  
 

exp( exp( )) 0
1
1
( )
1/ 1 1/
z x
z z k
f x
k k
(5)
where k, , are shape, scale and location parameters. Various values of shape parameter k yield to extreme
value type I, II, III distributions. Specifically, the three cases k  0,k  0,k  0
correspond to Gumbel,
Frechet and reversed Weibull distributions, respectively. GEV distributions are used in weather predictions,
extreme floods and snowfalls, market crashes, and so on (Train, 2003). The main peculiarity of such
distributions is that in the case where one generates N data sets from the same distribution, and creates a new
data set that includes the maximum values from these N data sets, the resulting data set can only be described by
one of the three above-mentioned distributions (Fisher & Tippett, 1928).
In the case of the logit model, the density of unobserved utility nj is Gumbel or type I extreme value (Persson
& Rydén, 2007)
( ) exp( )exp( exp( )) nj nj nj f      (6)
and CDF
( ) exp( exp( )) nj nj F     (7)
The differences between two terms are distributed logistic differences. That is  nji  nj  ni follows the
logistic distribution


nji
1 exp( )
( )
nji
F nji


 
At this stage, our goal is to derive the choice probabilities. We use (4) for the probability of choice:
Pnj  P ni  nj Vnj Vnii  j P ni  nj Vnj Vnii  j (9)
If nj  is considered given, then (9) is the cumulative distribution for each ni  , which is defined as

Vol.4, No.9, 2014
exp(exp( nj Vnj Vni ))
As far as we can use the assumption of  independence, the CDF over all i  j is the product of individual
CDFs:
Pnj | nj exp( exp( nj Vnj Vni )) (10)
nj nj nj nj ni d V V P      ))) exp( exp(( ) exp( )) exp( exp( |        
exp( )
56


   
i j
nj  are not given; therefore, the choice probability is the integral of (9) over all values of  nj , weighted by its
density
nj nj nj
i j
(11)
After algebraic manipulations, one can obtain the closed form for the needed choice probability



i j
ni
nj
nj
V
V
P
exp( )
(12)
It is evident that the choice probability is between 0 and 1 and that the normalization condition is fulfilled. This
model will be estimated with MLE in the next section.
Here,  j and  j is the average random coefficient and its standard deviation. These two parameters are
important for the simulation of logit model and their influence will be tested in the next section.
3. SIMULATION AND DISCUSSION
In this section, the analysis of a real dataset is accomplished. A market research firm collected this data as a part
of research study to evaluate the demand for a new good. The consumers were asked to respond to several choice
experiments. The experiment required that each consumer completely rank the goods that are preferred over
outside goods. There were 10 price levels and 8 brands in the dataset. The price index “0” and brand index “0”
corresponds to a hypothetical outside good.
In each choice experiment the customer ranks only the goods that are preferred over outside goods. For instance,
if for the given brand j with price level j p the utility 0 U U j  , then the rank for this brand is assigned.
Several choice experiments are performed with 170 independent customers. Therefore, in cases where when the
current brand is ranked
Unj Un0
, (13)
where n0 U is utility of the outside good for the
th n customer. In order to analyze the input dataset, it would be
useful to calculate some quantitative dependencies. The histogram of prices frequencies is shown in Figure 2. It
illustrates the number of experiments in which the particular price value was used.

Vol.4, No.9, 2014
-2 0 2 4 6 8 10
0 2 4 6 8 10
57
Frequency
Price
500 1000 1500 2000
Figure 2: Price frequencies
One can see from Figure 2 that the histogram is relatively flat, except for the number of experiments with the
outside good as an input. Figure 3 illustrates the frequency of different rank assignments.
Frequency
Ranks
2000 4000 6000
Figure 3: Frequency of assigned ranks
If we recall to the initial conditions of the discrete choice experiment, one can make a conclusion that the outside
good was often preferred to the examined brands. In other words, condition (16) was often not held.
Figure 4 shows the distribution of how many times the customers did not choose any products. One can see that
if we remove the first column, the distribution will be close to normal.

Vol.4, No.9, 2014
20 30 40 50 60 70
Rank count – “9”
58
10 20 30
Frequency
Figure 4: Frequency of assigned ranks
One can calculate the experimental probability for the given brand to be ranked first for each price. This
probability Pr is defined as follows:
B
N


B
N
p
1
Pr( ) (14)
where
1
NB is the number of times when the given brand with price was 1st ranked,

NB - is the total number of
experiments with given brand from price p.
1
2
3
4
5
6
0.8
7
0
2
4
6
8
10
0.6
0.4
0.2
0
0
0.1 0.2 0.3 0.4
brand
price
Pr
Figure 5: Experimental probabilities to be 1st ranked
Figure 5 illustrates the calculated probabilities for all brands. One can observe the highest probabilities for
brands with indexes 1 and 4. 2D plots of Pr(p) for these brands are shown in Figure 6.

Vol.4, No.9, 2014
0.4
0.35
0.3
0.25
b p
 
j p
log  
59
1 2 3 4 5 6 7 8 9 10
0.5
0.45
0.4
0.35
0.3
0.25
0.2
1 2 3 4 5 6 7 8 9 10
0.2
Figure 6: Experimental probabilities for brands 1 and 4 (x – price index, y - probability)
In order to examine the demand for the outside good, the logit model estimation should be performed. One of the
aims of this procedure is to compare the fixed effect and random coefficients logit models. The main difference
between these two models is related to individual consumer taste for brands. In fixed effect or simple logit
model, individual taste is accounted for in the logistic regression procedure. In contrast to this model, the random
coefficients logit model is accomplished with consideration of all choice experiments, without any consideration
of individual tastes.
As a result of logit model estimation, price and brand coefficients can be obtained. Hence, the probability of
given brand to be first ranked at given price can be calculated as follows:

 

i
b p
jp
i p
e
e
P
 
, (15)
where  p , j are price and taste coefficients for the current brand. Here we have used the fact that the
particular choice of consumer depends on the brand type and price.
After the logistic regression, one can obtain the list of estimated coefficients. The coefficients are
usually presented in two ways. The first option is to estimate the coefficients in log-odds units. This means that
 
 

   


 


Nprices
p
p p
Nbrands
j
jbj p
p
p
1 1 1
(16)
These estimates describe the relationship between the independent variables (brand and price) and the dependent
variable rank, where the dependent variable is on the logit scale. These estimates tell us the amount of increase in
the log odds of rank that would be predicted by a 1-unit increase in that predictor, holding all other predictors
constant. Because these coefficients are in log-odds units, they can be difficult to interpret, so they are often
converted into odds ratios. This conversion is performed via a simple exponentiation. The results of the
estimation procedure for the logit and probit regressions with obtained odds ratios are illustrated in Table 1.

Vol.4, No.9, 2014
Table 1: Estimated odds ratios for two logit models and probit model
60
Coefficientss
Odds ratio (fixed -
effect)
Odds ratio (simple logit)
Odds ratio (ordered
probit)
1  25.675 0.069 0.353
2  19.672 0.094 0.569
3  18.909 0.100 1.324
4  16.992 0.113 0.343
5  14.173 0.124 0.609
6  13.102 0.132 0.545
7  13.570 0.127 0.612
8  12.508 0.127 0.784
9  12.324 0.130 0.834
1  3.044 0.402 0.960
2  1.839 0.670 1.071
3  0.868 1.197 1.146
4  2.972 0.423 1.098
5  1.729 0.726 1.140
6  1.600 0.637 1.140
From Table 1, one can observe that odds obtained via a fixed-effect logit indicates higher probabilities than
simple logit and probit models, suggesting that the fixed effect model best fits the solutions.
After fitting data to the model, an odds ratio cross-tab can be generated in order to illustrate the odds product to
be ranked first for particular scenarios, based on data. The crosstab is shown on Table 2.
Table 2: Calculated odds ratios for particular scenarios (fixed-effect logit model)
brand
price
1 2 3 4 5 6 7
1 78.01 47.06 22.14 76.49 44.31 41.10 25.68
2 59.77 36.05 16.96 58.60 33.95 31.49 19.67
3 57.45 34.66 16.31 56.33 32.63 30.27 18.90
4 51.63 31.14 14.65 50.62 29.32 27.20 16.99
5 43.06 25.97 12.22 42.22 24.46 22.69 14.17
6 39.81 24.01 11.30 39.03 22.61 20.97 13.10
7 41.23 24.87 11.70 40.43 23.42 21.72 13.57
8 38.00 22.92 10.78 37.26 21.588 20.02 12.50
9 37.44 22.58 10.63 36.71 21.27 19.73 12.32

Vol.4, No.9, 2014
One can observe from Table 2 that the highest odds ratios correspond to the brands 1 and 4. This means that
according to the estimated model, the probabilities of these 2 brands being ranked first are the largest. It is
important to emphasize that this result is correlated with the experimental probabilities obtained in this section.
( s Vni Vnj ) s




( s Vni Vnj ) ( s Vni Vnj )
 


    


   
P t e dt t e dt
61
4. CONCLUSIONS
The equations for logit model probabilities were derived and analyzed. The logit regression procedure was
applied in order to estimate the demand for different goods in the market. The results have shown that the fixed
effect logit model is more suitable than alternative models for analysis of real data. This is because this type of
model accounts for individual tastes of consumers. The obtained odds ratios for the fixed-effect logit model
show the correspondence with experimentally obtained probabilities. Obtained results can be used for price
management in real markets, in order to increase the demand of a given product.
APPENDIX
DERIVATION OF LOGIT PROBABILITIES: This appendix shows how the logit probabilities (12) can be
derived from (11).



 
 


    
 

 


s
s e
j i
e
ni P e e e ds
, (1)
where ni s  . One can rewrite the multiplication of the exponents as a summation in the one resulting
exponential term



   


 

 

  

 


     
s
s
j
s
s
j i
e
ni P e e ds e e ds
exp (2)
One can make a substitution t  exp(s),exp(s)ds  dt . After such substitution (1) is easily evaluated
   
 
 

 



t e
ni nj
exp( )
   


 

 

 


 

 



  

 


 

j i
V
ni
nj
j
V V
j
V V
j
V V
j
V V
ni
V
e
ni nj
ni nj ni nj
exp( )
|
exp
exp ( ) exp
0
0
0
. (3)
Bibliography
Adeogun, O. A., Ajana, A. M., Ayinla, O. A., Yarhere, M. T., & Adeogun, M. O. (2008). Application of logit
model in adoption decision: A study of hybrid clarias in Lagos State, Nigeria. American-Eurasian Journal
of Agriculture and Environmental Sciences, 4(4), 468-472.
Domencich, T. A., & McFadden, D. (1975). Urban Travel Demand - A Behavioral Analysis (No. Monograph).
Oxford: North-Holland Publishing Company, 1975.

Vol.4, No.9, 2014
Fisher, R. A., & Tippett, L. H. C. (1928, April). Limiting forms of the frequency distribution of the largest or
smallest member of a sample. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol.
24, No. 02, pp. 180-190). Cambridge University Press.
Huang, D., Rojas, C. (2010). Eliminating the Outside Good. Bias in Logit Models of Demand with Aggregate
Data. Amherst: University of Massachusetts at Amherst.
McFadden, D. (1974). The measurement of urban travel demand. Journal of Public Economics, 3(4), 303-328.
Johnson, J., Bruce, A., & Yu, J. (2010). The ordinal efficiency of betting markets: an exploded logit
approach. Applied Economics, 42(29), 3703-3709.
Kumar, S., & Kant, S. (2007). Exploded logit modeling of stakeholders' preferences for multiple forest
values. Forest Policy and Economics, 9(5), 516-526.
Malhotra, N. K. (1984). The use of linear logit models in marketing research. Journal of Marketing Research,
62
20-31.
Myung, I. J. (2002). Maximum likelihood estimation. Manuscript submitted for publication.
Persson, K., & Rydén, J. (2007). Exponentiated Gumbel distribution for estimation of return levels of significant
wave height. Department of Mathematics, Uppsala University.
Train, K. (2003). Discrete choice methods with simulation. Cambridge: Cambridge University Press.

The IISTE is a pioneer in the Open-Access hosting service and academic event
management. The aim of the firm is Accelerating Global Knowledge Sharing.
More information about the firm can be found on the homepage:
http://www.iiste.org
CALL FOR JOURNAL PAPERS
There are more than 30 peer-reviewed academic journals hosted under the hosting
platform.
Prospective authors of journals can find the submission instruction on the
following page: http://www.iiste.org/journals/ All the journals articles are available
online to the readers all over the world without financial, legal, or technical barriers
other than those inseparable from gaining access to the internet itself. Paper version
of the journals is also available upon request of readers and authors.
MORE RESOURCES
Book publication information: http://www.iiste.org/book/
IISTE Knowledge Sharing Partners
EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open
Archives Harvester, Bielefeld Academic Search Engine, Elektronische
Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial
Library , NewJour, Google Scholar

The application of discrete choice models in marketing

More Related Content

What's hot (16)

Similar to The application of discrete choice models in marketing (20)

More from Alexander Decker (20)

The application of discrete choice models in marketing