DISCUSSION PAPER SERIES
June 2014

No.2014-07

The Consideration Set Based Submarket Identification: An
Application of the Hierarchical Factor Analysis Model
Sotaro Katsumata
Akihiro Nishimoto

The Consideration Set Based Submarket Identification: An
Application of the Hierarchical Factor Analysis Model
Sotaro Katsumata(Nagasaki University)
Akihiro Nishimoto(Kwansei Gakuin University)

Abstract
In this study, we focus on the consumers’ consideration set and submarkets of the market. In themodern matured market, there are too many competitive brands, and consumers cannot evaluate
all the available brands. They only see the subset of the brands to compare them and choose
between them. The subset of comparison and evaluation is called a consideration set. Brands are
required to be in this consideration set, to be chosen. Therefore, identification of the consideration
set is an important issue for brand managers. However, examining the consideration set isrelatively difficult as compared with actual choice, because we cannot observe the compositionof the set from behavioral data such as purchase records. To address this issue, we examine theconsideration set collected through a research survey. In our analysis, we propose a model toexamine the submarket structure. Our model incorporates factor analysis into a discrete choicemodel and assumes hierarchical structure. To use the discrete choice model, we can obtain theconsideration probability of each brand. Factor analysis model enables us to identify submarkets.
Since we assume the hierarchical expression, we can examine the relationship between consumers’
characteristics and the consideration probability of each submarket.
Keywords
Consideration Set; Submarket; Discrete Choice Model; Factor Analysis; Hierarchical Model11.

Introduction

Consumers’ choice is one of the most important issues in marketing and consumer
research. One of the goals of every firm is to be the preferred choice of consumers. If the firm
products were not chosen, the firm would not grow. However, the choice behavior of realconsumers undergoes a complex process. The complexity occurs because of the selection ofalternatives. Many previous studies have indicated the existence of the choice subset. For example,
Simon (1947) defined the behavioral model of a human based on the cognitive limitation.
According to Simon (1947), humans evaluate only a few alternatives rationally and choose a“satisfied” alternative in the subset even if there are more preferable alternatives out of the subset.
Further, Howard and Sheth (1969) mention the existence of the “evoked set” as a kind of choicesubset within their consumer behavioral model. Firm managers have to consider the existence ofthe choice subset and recognize that it is the first step in the process of consumers choosing theirbrand. The consideration set is especially regarded as one of the most important subsets. As many
previous studies such as Shocker et al. (1991), Roberts and Lattin (1991, 1997), and Elrod and
Keane (1995) have highlighted, one of the most important choice subsets is the consideration set.
Consumers carefully evaluate the brands within the consideration set. Therefore, being in theconsideration set of consumers is an important milestone for managers. Thus, in this study, we
focus on the consideration set.
Consumers tend to evaluate and compare rationally; the choice problem within theconsideration set is relatively simple and consistent. However, it is difficult to identify whichalternatives are in the subset. Moreover, the number of candidate brands becomes large in themodern market. One of the clues to identifying the alternatives in the subset is the submarket
structure. There are some submarkets and brands within the same submarket, which tend to becompared to each other by consumers.
In this study, we focus on the consideration set and the market structure using the statedconsideration data. We collected the consideration dataset as a multivariate binomial observationfrom research survey and we construct a model to explain it. We apply the factor analysis model
to express the submarket of the market. Additionally, we assume the hierarchical structure to
examine the relationship between each factor (or consumer characteristic) and consideration set
memberships.

2.

Previous Studies on Choice Set

In this section, we briefly survey the previous studies and models of the consideration
set. Let 𝑃𝑖 (𝑗) be a choice probability of brand 𝑗 by consumer 𝑖 and let his/her choice set be 𝐶𝑖 .
The general formulation is given by Manski (1977) and Shocker et al. (1991) as follows:
𝑃𝑖 (𝑗) = ∑ 𝑃𝑖 (𝑗|𝐶)𝑃𝑖 (𝐶|𝐺)𝐶∈𝐺(𝑗)(1)

where 𝐺 is the set of all the possible consideration sets and 𝐺(𝑗) is the set of all elements of
𝐺 that contain brand 𝑗. This expression suggests that the choice probability of brand 𝑗 consistsof two conditional probabilities. 𝑃𝑖 (𝐶|𝐺) is the probability that the choice set 𝐶 is formulated
by consumer 𝑖, and 𝑃𝑖 (𝑗|𝐶) is the probability that brand 𝑗 is chosen from among the given
choice set 𝐶. From this expression, we can estimate the choice probabilities for each customer 𝑖.
For example, if there is a set of available brands {𝑎, 𝑏, 𝑐}, all of the possible choice sets2

are {𝑎}, {𝑏}, {𝑐}, {𝑎, 𝑏}, {𝑎, 𝑐}, {𝑏, 𝑐}, and {𝑎, 𝑏, 𝑐} (except the empty set). The set of choice sets
𝐺 consists of these seven subsets. For brand 𝑎 to be chosen, the choice set needs to be{𝑎}, {𝑎, 𝑏}, {𝑎, 𝑐}, or {𝑎, 𝑏, 𝑐}. These four subsets are elements of 𝐺(𝑎). Based on this formulation,some empirical models are provided, such as those of Andrews and Srinivasan (1995), Siddarthet al. (1995), and Chiang et al. (1999). However, this formulation has a serious limitation. The
number of possible subsets becomes very large as the number of alternative brands increases.
When the number of alternatives is 𝐽, the number of choice subset is 2 𝐽 − 1. Therefore, we haveto estimate probabilities for 2 𝐽 − 1 subsets.
To address this problem, Nierop et al. (2010) propose a model that estimates theprobability of a choice subset considered by a consumer for each alternative. In this model, themembership probability of the choice subset is estimated using a binomial discrete choice model.
Therefore, this model estimates the membership probabilities from their latent variables. In thisprobabilistic choice model, if the number of available brands is 𝐽, the number of the probability
is 𝐽. However, this model has a possibility that the empty set occurs. Nierop et al. (2010) introduce
an assumption that the brand that has highest membership probability shall automatically be anelement of the choice set even if the latent variable of the membership probability does not exceed
a certain threshold.
In this study, we also assume the membership probability of the choice set based on
Nierop et al. (2010). Thus, we suppose that the observed choice probability of brand 𝑗 byconsumer 𝑖 is obtained from the product of two probabilities. The first term, 𝑃𝑖 (𝑗 ∈ 𝐶𝑖 ), is theprobability of 𝑗 being an element of the choice set. The second term, 𝑃𝑖 (𝑗|𝐶𝑖 ), is the probabilityof 𝑗, chosen from among the given choice set 𝐶𝑖 .
𝑃𝑖 (𝑗, 𝑗 ∈ 𝐶𝑖 ) = 𝑃𝑖 (𝑗|𝐶𝑖 ) × 𝑃𝑖 (𝑗 ∈ 𝐶𝑖 )(2)

As mentioned in the previous section, our research purpose is to examine theformulation of the choice set. Therefore, in this study, we only focus on the right term, 𝑃𝑖 (𝑗 ∈ 𝐶𝑖 ).
Of course, we adopt our model to the conditional choice model such as the incidence, choice, andquantity model proposed in Bucklin et al. (1998) and van Heerde and Neslin (2008). In thefollowing section, we propose a model based on the factor analysis model.

3.

The Probabilistic Model of the Consideration Set

3.1.

Consideration Probability
At first, we define an observed indicator variable 𝑧𝑖𝑗 , where brand 𝑗 is an element of𝐶𝑖 , which is the consideration set of consumer 𝑖.
1
𝑧𝑖𝑗 = {
0𝑖𝑓𝑒𝑙𝑠𝑒

𝑗 ∈ 𝐶𝑖(3)

The observation variable 𝑧𝑖𝑗 is a binary measure so that if consumer 𝑖 has brand 𝑗 in∗
his/her choice set, 𝑧𝑖𝑗 = 1; otherwise, 0. We assume the latent variable 𝑧𝑖𝑗
based on Albert andChib (1993) as follows:3

1
𝑧𝑖𝑗 = {
0

𝑖𝑓
𝑒𝑙𝑠𝑒

∗
𝑧𝑖𝑗
>0

(4)

To estimate the membership probability of each brand, we introduce the factor analysis
model. The factor model has good properties to examine the submarket structure andconsideration set composition (Roberts and Lattin 1991, 1997). For example, Elrod and Keane(1995) apply a discrete choice factor model to identify the submarket and draw a product map.
Although we can draw a scatter plot of the market if the dimension of the factor is two or three,
we do not aim to obtain an output figure in this study.
The factor model to explain the latent variable is as follows:𝑧𝑖∗ = 𝛼 + Λ𝑓𝑖 + ε𝑖 , ε𝑖 ~𝑁(0, 𝐼𝐽 )

(5)

′

∗
∗
where 𝑧𝑖∗ = (𝑧𝑖1, ⋯ , 𝑧𝑖𝐽
) , Λ is a 𝐽 × 𝑄 matrix parameter of factor loadings, and 𝑓𝑖 is a 𝑄 × 1

vector of factor scores. This is a 𝑄-dimensional factor model. Since the objective variable is notstandardized, we need to incorporate an intercept parameter 𝛼. In addition, the usual factor model
requires the diagonal unknown variance parameters. However, to satisfy the identification
condition, we assume an identity matrix for the variance-covariance matrix. From this model, we
can estimate the consumer 𝑖’s membership probability of brand 𝑗 from the factor loading Λ andthe factor score 𝑓𝑖 .
The dimension 𝑄 is interpreted as the number of the submarkets of the objective
competitive market. For example, if the value of 𝜆𝑗𝑝 , the (𝑗, 𝑝) element of the factor loadings
parameter Λ, is high, brand 𝑗 tends to be a member of submarket 𝑝. Note that we do not assume
the deterministic submarket membership. Our assumption is a fuzzy submarket membership suchas product mapping (Elrod and Keane 1995). This suggests that brands that have high values atthe 𝑝-th column of Λ are under competition within the submarket 𝑝.
As regards consumers, we can examine the submarket level consideration (preference)by 𝑓𝑖 . If the element 𝑓𝑖𝑝 is high, consumer 𝑖 tends to consider brands that have a high value at𝑝-th column of Λ. Since the factor score 𝑓𝑖 is obtained for each consumer, it is a source ofdifferentiation of the consideration probability of brand 𝑗. The latent variable of the membershipprobability of consumer 𝑖 for brand 𝑗 is the addition of the factor loadings and scores, as
follows:∗𝑧𝑖𝑗= 𝛼𝑗 + 𝜆𝑗1 𝑓𝑖1 + ⋯ + 𝜆𝑗𝑞 𝑓𝑖𝑞 + ⋯ + 𝜆𝑗𝑄 𝑓𝑖𝑄 + 𝜀𝑖𝑗(6)∗From the latent variable 𝑧𝑖𝑗, we can estimate the probability that consumer 𝑖 considers

brand 𝑗 in a purchase occasion, where 𝜙(𝑥|𝑚, 𝑣 2 ) is a density function of the univariate normal
distribution whose mean is 𝑚 and variable is 𝑣 2 .
∞𝑃𝑖 (𝑗 ∈ 𝐶𝑖 ) = ∫ 𝜙(𝑥|𝛼𝑗 + 𝜆𝑗′ 𝑓𝑖 , 1) 𝑑𝑥(7)0

In this study, we assume the prior structure for the model. Therefore, we relax the4assumption of the factor scores, which is usually assumed as 𝑓𝑖 ~𝑁𝑄 (0, 𝐼𝑄 ). In this study, we do
not assume the mean as 0, and set the following linear regression model:
𝑓𝑖 = Δ𝑤𝑖 + 𝜁𝑖 , 𝜁𝑖 ~𝑁𝑄 (0, 𝐼𝑄 )(8)

where 𝑤𝑖 is a consumer characteristics variable such as age or gender. In this model, these
consumer characteristics affect the factor scores as a prior. In this study, we incorporate theconsumer demographic variable and knowledge. The detailed setting is found in the next section.
Our factor analysis model, which assumes a discrete objective variable and hierarchicalstructure, has to adopt the Markov Chain Monte-Carlo (MCMC) method to estimate parameters.
The detailed procedure of the estimation is in Appendix A.1.
3.2.

Identification of the Factor Model
To obtain the estimates by the MCMC method, we have to satisfy the identification
condition of the factor analysis. For example, Geweke and Zhou (1996) propose a way to satisfy
the condition introducing the restriction in the factor loadings parameter. This restriction is also
adopted by Lopes and West (2004). Using another approach, Ansari and Jedidi (2000) restrict thecovariance matrix. In addition, this study incorporates a discrete objective variable. However, toestimate the restricted covariance matrix, we have to use the Metropolis-Hastings (M-H) method,
and the computational load is relatively high. Therefore, we use the restriction proposed by
Geweke and Zhou (1996).
In the model of Geweke and Zhou (1996), the factor loadings matrix is restricted to
identify the factor loadings and scores, which are able to rotate. In other words, for any 𝐽 × 𝐽
matrix 𝑅, where 𝑅𝑅 ′ = 𝐼, Δ∗ = 𝑃′ Δ and 𝑓 ∗ = 𝑃′ 𝑓𝑖 are also valid parameters. Therefore, toaddress the problem, we restrict Λ 𝑄 , which is a 𝑄 × 𝑄 sub-matrix of 𝐽 × 𝑄 factor loadings Λand Λ 𝑄 = Λ1:𝑄,1:𝑄 , as follows:
𝜆11
𝜆21
Λ𝑄 =
⋮
(𝜆𝑄1

0
𝜆22
𝜆𝑄2

⋯
⋱
⋯

0
⋮
0
𝜆𝑄𝑄 )

(9)

where the diagonal elements (𝜆11 , ⋯ , 𝜆𝑄𝑄 ) are all positive. We can obtain identified parameters
to introduce this restriction. However, it is difficult to interpret the estimated factors directly fromthe obtained samples. Therefore, we rotate posterior samples and discuss them. The detailed
procedures are available in section 3.4. The conditional posterior distributions are in the Appendix.
3.3.

Number of Factors
One of the biggest problems in the factor analysis is how to determine the dimension of
the factor. The most famous way is to determine from eigenvalues of the correlation matrix(Kaiser 1960; Guttman 1954). This method is called Kaiser Criterion, and suggests that the
number of the eigenvalue over 1 is an appropriate dimension of the model. In addition, we canapply the likelihood of the model to selection. Especially in Bayesian models, we can use themarginal likelihood or DIC (Deviance Information Criterion). In addition, Lopes and West (2004)
estimate the dimension of the model using the Reversible Jump algorithm developed by Green(1995).
5

In this study, we compare models using the Kaiser Criterion, marginal likelihood, andDIC. Note that eigenvalues are obtained from the correlation matrix of the observed variable.
3.4.

Rotation
As mentioned above, we can rotate the factor loadings and scores. To rotate the factors,
we can obtain useful managerial insights from the model. We also rotate the parameters using thefollowing procedure.
At first, we calculate the rotation matrix 𝑅 (where 𝑅 ′ 𝑅 = 𝐼). Let the ℎ-th sample ofthe posterior distribution of the factor loadings parameter be Λ(ℎ) , where ℎ = 1, ⋯ , 𝐻 . In
̅. Since we obtain the posterior mean value for each element,
addition, let the posterior mean be Λ(ℎ)
̅ = 𝐻 −1 ∑𝐻
the samples and posterior mean have the following relationship: 𝜆𝑗𝑞ℎ=1 𝜆𝑗𝑞 . Next, werotate the loadings for any criterion. In this study, we apply the varimax criterion. As a result, we
can obtain the rotation matrix 𝑅.
The rotation affects the factor loadings, factor scores, and hierarchical parameters.
(ℎ)

Therefore, we also rotate these samples. Let 𝜆𝑗 , 𝑓𝑖(ℎ), Δ(ℎ) , and Γ (ℎ) be the ℎ-th posterior

samples of each parameter. From these samples, the rotated samples are obtained by multiplying
the rotation matrix as follows:𝑅(ℎ)= 𝑅 ′ 𝜆𝑗𝑅(ℎ)= 𝑅 ′ 𝑓𝑖

𝜆𝑗
𝑓𝑖

(ℎ)
(ℎ)

(10)

Δ𝑅(ℎ) = 𝑅 ′ Δ(ℎ)Γ𝑅(ℎ) = 𝑅′Γ(ℎ)
Using these rotated samples, we can calculate posterior mean, posterior standard
deviation, HPD (highest posterior density) interval, and so on, to obtain useful insights.

4.

Objective Products

In this study, we collect consideration set data from a consumer research survey. In thissection, we propose the objective category, brands, and measures.
4.1.

Overview of Objective Markets
In this study, we focus on the beer markets and 21 major brands, as shown in Table 1.
The beer market is dominated by four manufacturers: Asahi, Kirin, Suntory, and Sapporo. The
sum of the market share of these four firms accounts for over 98% of the market. All the targeted
21 brands are made by these firms. In addition, there are four submarkets in this market. In general,brands categorized as “premium beer” are relatively expensive, and “beer” follows next. Brands
categorized as “low-malt” and “new genre” are relatively low-priced. Note that these four
categories do not exactly correspond to the categorization in Japan’s Liquor Tax Act. Strictly
speaking, premium beers are categorized as beer. However, this categorization is commonly usedby manufacturers and retailers. Although we can purchase 350ml cans, 500ml cans, and bottled
beer in convenience stores, we focus on the 350ml purchase occasion. Therefore, we show 350ml
can images in the questionnaire.
6

Table 1: Objective Brands and Submarkets
Manufacturer(Abbreviation)

Premium Beer (P)Kirin (K)Beer (B)

Low-Malt (L)New Genre (N)Lager Beer

Tanrei Green Label

Sumikiri

Ichiban Shibori

Nodogoshi Nama
Koi Aji

Asahi (A)

JukusenSuper Dry

Suntory (S)The Premium Malts

Malts

Style Free

Asahi Off
Clear Asahi
Kin Mugi
Jokki Nama

Sapporo (P)Yebisu Beer

Kuro Label

Hokkaido Namashibori

Kin No Off
Hokkaido Premium
Mugi To Hop

4.2.

Consideration Set
As mentioned in Brisoux and Laroche (1980) and Shocker et al. (1991), theconsideration set is the subset of the set of all available brands. The actual choice is based on thisset. In the exclusive choice model, the number of chosen brands is only one. Suppose that the
number of available alternatives is 𝑀 and the size of consumer 𝑖’s consideration set is 𝐶𝑖 , therange of the set is 1 ≤ 𝐶𝑖 ≤ 𝑀. Note that there are some different kinds of choice subsets. Some
previous studies propose a stepwise alternative selection model of consumer information
processing. The consideration set is defined as one of the choice subsets within the process.
We ask respondents whether the brand is considered during the purchase. It is possiblethat the consumer chooses all available brands or none of the brands. We explicitly require
respondents to choose “considered” brands. In the questionnaire, we show all 21 objective brands
with brand names and package images. We ask respondents to check all brands that are consideredduring the purchase: when you purchase a beer can, please check all brands that are consideredduring the purchase. We collect binary answers; {0: do not consider, 1: consider}.
4.3.

Consumer Attributes
As defined in the previous section, we try to explain factor scores by consumer
characteristics such as demographic or psychographic attributes. Our model enables us to findrelationships between each factor and consumer attributes.
In this study, we apply following five attributes: consumers’ age, sex, objective
knowledge (OK), subjective knowledge (SK), and familiarity (FM). The first two attributes aredemographic aspects, while the latter three attributes are sub constructs of consumer knowledge.
As Alba and Hutchinson (1987, 2000) pointed out, consumer knowledge of a product category
consists of two sub constructs—familiarity and expertise. Further, the expertise is also divided
into two parts—objective and subjective knowledge (Brucks 1985; Park and Lessig 1981).
Based on these studies, we define the effect of consumer knowledge as follows. The
objective and subjective knowledge directly affect the factor score, while we define that7

familiarity moderates these knowledge constructs.
𝑥𝑖 = {Intercept, Age, Sex, OK, SK, OK ∗ FM, SK ∗ FM}′(11)

where Age is the log of the observed age, and Sex is a binary value in which {0: male, 1: female}.
The measurement scales of the consumer knowledge constructs are listed in Appendix
A.2. The familiarity, objective knowledge, and subjective knowledge scales are based on
Moorman et al. (2004) and Brucks (1985).

5.

Results

5.1.

Data Collection
We collected data from October 11 to 15, 2013, through an internet survey. The 1,800
respondents in our sample were all over 20 years old. As a screening technique, we focus only on
the consumers who drink beer. There are 1,093 males and 707 females. The average age is 46.8
years. Within the sample, the average size of the consideration set is 5.15, the mode is 4, themaximum value is 21 (all brands), and the minimum value is 1. Comparing with the previous
study, Hauser and Wernerfelt (1990) report that the size of the consideration set in the beer
category is between 3 and 7. Our result is also within this range. Figure 2 shows the histogram ofthe size of the consideration set.

300
200
100
0

Frequency

400

500

Figure 1: Histogram of the Size of the Consideration Set

0

5

10

size of the consideration set

8

15

20

5.2.

Model Comparison: Number of Submarkets
In this subsection, we examine the number of submarkets in the market. In other words,
we determine the appropriate number of dimensions of the factor model. Table 2 shows the resultof indicators obtained from each model. In the beer market, there are 5 eigenvalues that exceed 1.
Therefore, from the Kaiser Criterion, the appropriate dimension is 5. In this study, we estimate 6
models, from the 1 factor model to the 6 factor model, and calculate the marginal likelihoods and
DICs. The detailed procedures to obtain marginal likelihood and DIC are found in Newton and
Raftery (1994) and Gelman et al. (2004), respectively. In addition, we also calculate the Bayes
factors from the marginal likelihoods.
As Table 2 shows, although we examine models until 6 factors, the marginal likelihood
and DIC of the 6 factor model is the highest. We additionally estimate the 7 factor model and findthat the marginal likelihood is −10317.5 and the DIC is −13260.7. This suggests that thefitness of the model will improve as the number of factors increase. However, our purpose is to
examine the submarkets. Although we need to interpret the results, it is difficult to interpret theresult if the number of factors is too large. Therefore, in this study, we choose the 5 factor model
based on the eigenvalue, and examine the results.

Table 2: Model Comparison Results
Log Marginal Log
Bayes# of factors
DIC
Eigenvalue
Likelihood
Factor1−16090.0−16849.2
4.01
2−14083.2
2006.7−15495.0
2.52
3−12985.0
1098.2−14858.3
1.52
4−12077.5
907.5−14341.5
1.49
5−11212.0
865.5−13789.7
1.10
6−10648.7
563.4−13287.7
0.95
Note: Log Bayes Factor of the 𝑘 factor model is 𝑚𝑘 − 𝑚𝑘−1 where, 𝑚𝑘 is the log marginal
likelihood of the 𝑘 factor model.

5.3.

Parameters
Table 3 shows the posterior mean of the factor loadings obtained from the 5 factor model.
All the results are based on the rotated samples shown in the previous section. In Table 3, bold
fonts indicate especially higher (lower) values among each factor. We find there are somesubmarkets in this market. For example, in the Factor 2 submarket, values of traditional andrelatively high priced beer brands such as “Lager beer (K/B)” and “Ichian Shibori (K/B)” arenegatively high. In contrast, the values of low calorie content and low price brands such as “Asahi
Off (A/N)” and “Kin no Off (P/N)” are positively high. This suggests that in the Factor 2
submarket, consumers who have a positively high factor score tend to prefer relatively new low
calorie content brands and avoid traditional brands. In contrast, the consideration probabilities ofthese traditional brands of consumers who have a negatively high score in this factor tend to behigher.
Similarly, values of low calorie content and low price brands such as “Asahi Off (A/N),”“Kin no Off (P/N),” and “Style Free (A/L)” are positively high in Factor 1. In Factor 3, the valuesof brands related to Hokkaido, “Hokkaido Premium,” and “Hokkaido Namashibori” are high. In
9Factor 4, the premium beer brands called “The Premium Malts (S/P)” and “Yebisu Beer” havenegatively higher scores. In Factor 5, brands that are classified as “nama (non pasteurized)” havehigher scores, such as “Nodogoshi Nama (K/N)” and “Jokki Nama (S/N).” These brands also
have the common property that they are both lower priced (new genre) brands.
From the factor loadings, we find that there are some submarkets. Based on this result,
we further examine the relationships between the factor (submarket) preference and consumer
attributes. In the next subsection, we examine the prior structure of the factor scores 𝑓𝑖 .

Table 3: Factor Loadings
Brands (Manufacturer/Category)Tanrei Green Label (A/P)
Asahi Off (A/N)
Kuro Label (P/B)
Kin no Off (P/N)
The Premium Malts (S/P)
Sumikiri (K/N)
Malts (S/B)
Style Free (A/L)
Hokkaido Premium (P/N)
Lager Beer (K/B)
Kin Mugi (S/N)
Ichiban Shibori (K/B)
Nodogoshi Nama (K/N)
Hokkaido Namashibori (P/L)
Koi Aji (K/N)
Yebisu Beer (P/P)
Mugi to Hop (P/N)
Super Dry (A/B)
Jokki Nama (S/N)
Jukusen (A/P)
Clear Asahi (A/N)

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

1.02
𝟏. 𝟓𝟒
−0.13
𝟏. 𝟓𝟑
−0.21
0.36
−0.03
𝟏. 𝟕𝟑
0.30
−0.11
0.16
0.10
0.33
0.33
0.88
−0.24
0.11
−0.06
0.17
0.26
0.36

−0.35
𝟎. 𝟑𝟗
−0.73
𝟎. 𝟑𝟔
−0.63
−0.12
−0.54
0.11
−0.06
−𝟏. 𝟔𝟓−0.01−𝟏. 𝟑𝟐
−0.47
−0.34
0.05
−0.69
0.02
−0.42
−0.11
−0.12
0.06

−0.04
−0.03
0.71
0.65
0.09
0.46
0.45
0.11
𝟏. 𝟗𝟑
0.11
0.21
0.14
−0.08
𝟏. 𝟖𝟑
0.48
0.49
0.69
−0.01
0.28
0.87
0.12

0.22
−0.10
−0.31
0.03
−𝟐. 𝟏𝟒
−0.10
−0.76
0.18
−0.25
−0.08
−0.22
−0.39
𝟎. 𝟑𝟔
−0.03
−0.09
−𝟎. 𝟖𝟖
−0.17
−0.11
0.04
−0.63
−0.01

0.19
0.32
−0.10
0.54
0.13
0.65
0.09
0.30
0.58
0.07
0.95
0.05
𝟏. 𝟐𝟕
0.54
0.36
−𝟎. 𝟏𝟕
0.66
0.11
𝟏. 𝟐𝟕
0.05
0.76

Note: For manufacturers, A: Asahi, K: Kirin, S: Suntory, and P: Sapporo. For categories: P:
Premium Beer, B: Beer, L: Low-malt beer, and N: New Genre. Bold fonts indicate especiallyhigher (lower) values among each factor.

5.4.

Consumer Characteristics and Factors
Our model assumes the prior structure on each consumer’s factor scores 𝑓𝑖 . In this study,
we assume that consumer characteristics are explained by their demographic traits (age and sex)
and knowledge (objective knowledge, subjective knowledge, and familiarity). Table 4 shows theresult of the parameter Δ. In this table, underlined fonts indicate that 0 lies outside the 10% HPD(highest probability density) intervals of the parameter. Further, bold fonts indicate that 0 lies
outside the 5% HPD intervals. The detailed procedure to obtain HPD intervals is found, forexample, in Chen et al. (2000).

10

Table 4 Result of Prior Parameter 𝚫
Factor 1
Factor 2
Factor 3
Factor 4
Factor 5
Δ
Intercept
0.79
𝟏. 𝟐𝟔
−0.07
0.28
𝟏. 𝟐𝟖
Sex
𝟎. 𝟐𝟔
𝟎. 𝟏𝟖
0.08
−0.09
0.10
Age
−𝟎. 𝟑𝟐−𝟎. 𝟑𝟎−0.16
0.04−𝟎. 𝟑𝟐
SK
0.61
0.30
𝟏. 𝟎𝟕
−0.30
−0.08
OK
0.20
−𝟎. 𝟔𝟖
𝟏. 𝟎𝟐−𝟏. 𝟎𝟐
−0.60
SK*FM
0.02
−0.59
−0.74
−0.64
−0.27
OK*FM
0.06
0.45
0.08
0.60
𝟏. 𝟔𝟎
Note: SK: Subjective Knowledge, OK: Objective Knowledge, FM: Familiarity. Underlined fonts
indicate that 0 lies outside the 10% HPD interval, and Bold fonts indicate that 0 lies outside the
5% HPD interval.

To assume the consumer attribution for the prior structure of the factor scores 𝑓𝑖 , we canfind useful implications of consumer behaviors for each submarket. For example, we find that inFactor 2, if 𝑓𝑖2 is positively higher, the consumer tends to consider low calorie contained brands,
while if 𝑓𝑖2 is negatively higher, the consumer prefers traditional well-tasted brands in Table 3.
Based on this, we can interpret the Table 4 that the result implies that younger female consumers
tend to consider low calorie content brands and older male customers consider traditional brands.
In addition, we find that consumers who have higher objective knowledge tend to prefertraditional brands.
Similarly, in Factor 1, we find that younger female consumers tend to consider low
calorie content new genre brands. In addition, we find from Factor 3 that consumers who havehigher objective knowledge and subjective knowledge tend to consider low-malt or new genre“Hokkaido” related brands. In Factor 4, premium beer category brands are considered byconsumers who have higher objective knowledge. Finally, in Factor 5, “Nama (non pasteurized)”brands are considered by consumers who are young and have high familiarity and subjective
knowledge. Since the construct of familiarity is measured by the frequency of use and purchase,and subjective knowledge is measured by the confidence of consumers’ own knowledge, thisresult implies that consumers who drink beer and have more confidence in their knowledge thanordinary people tend to prefer these “Nama” brands.
As a summary, our model enables managers not only to identify the submarket structure
but also to show the relationship between each submarket and consumer attribution such as
demographic aspects and knowledge.
5.5.

Consumer Attribution and Consideration Probability
Since our model is based on the probabilistic choice model, we can estimate theconsideration probability using the factor loadings Λ and prior variable of the factor scores.
Suppose that a consumer whose attribution is 𝑤
̃, the consideration probability of the consumer isobtained from Φ(ΛΔ𝑤
̃), where Φ(𝑥) is the standard normal distribution function evaluated on𝑥.
The estimated considered probability is shown in Figure 3. The upper figure shows thedifference between male and female consumers. In this figure, other attributes are as follows: 40
years old, SK=OK=FM=0.5. This suggests that female consumers tend to prefer low calorie
brands and male customers prefer beer category (B) brands. The lower figure of Figure 3 shows11that younger customers tend to consider new genre (N) and low-malt (L) category brands, whileolder customers prefer beer category (B) brands.

Figure 2: Difference in the Consideration Probability (Upper: Sex, Lower: Age)
0.9

Male

Female

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0.9

Age 20Age 40Age 60

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Note: The upper figure shows the probabilities of 40 years old consumers. The lower figure shows
the probabilities of male consumers.

12

6.

Conclusion

In this study, we focus on the competitive market, which has many brands, and identify
the submarket structure using the discrete choice factor analysis model. This study makes twocontributions, as follows.
First, our study identifies the consideration set based submarket structure. Since we canscarcely observe the consideration set of consumers, the number of studies related to theconsideration set formulation and its component are relatively few compared with actual choice(purchase) studies. Our findings shed light on the field.
Second, we apply the hierarchical factor analysis model to examine the market structure.
Our model allows the binomial objective variable and enables us to estimate probabilities. This isfeasible to examine the choice problem and we succeed in applying the model to the consideration
probability. Further, we assume the prior linear combination mean on the factor scores 𝑓𝑖 . This
enables us to identify the relationships between submarkets and consumer characteristics. Further,to rotate the posterior samples, we can easily interpret the result and obtain useful implications
about the market.
As issues for future research, we propose two points. First, we have to incorporate themodel as part of the conditional choice models proposed in Bucklin et al. (1998) and van Heerde
and Neslin (2008). In this study, we only focus on the consideration set composition. However,our goal is to estimate the choice probability of each brand. Second, we have to find a criterion todecide the appropriate number of factors. In this study, the marginal likelihood and DIC did notwork well. Although these indicators tend to prefer large dimensional models, these are difficult
to interpret for managers and analysts. We have to find a criterion to solve the trade-off.

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14

A. Appendix
A.1. Posterior Distributions∗
For the latent variable 𝑧𝑖𝑗, we can generate samples from the following truncatednormal distribution:𝑧𝑖𝑗 | ⋅ ~ {𝑇𝑁(0,∞) (𝛼𝑗 + 𝜆𝑗′ 𝑓𝑖 ,1)𝑇𝑁(−∞,0) (𝛼𝑗 + 𝜆𝑗′ 𝑓𝑖 ,1)(A.1)

The sample of the intercept parameter 𝛼𝑗 is obtained from the following univariate
normal distribution:
𝛼𝑗 | ⋅∼ 𝑁(𝑚𝑗 , 𝑠12 )

(A.2)
−1

∗
′
−2
where 𝑠12 = (𝑠0−2 + 𝑁)−1 and 𝑚𝑗 = 𝑣𝑗2 (∑𝑁
𝑖=1(𝑧𝑖𝑗 − 𝜆𝑗 𝑓𝑖 ) + 𝑠0 𝑎0 ) .

The factor loadings Λ are obtained for each row 𝜆𝑗 . However, we have to consider the
first 𝑄 rows. Therefore, in this section, we will explain the first to 𝑄-th row and 𝑄-th to 𝐽-th
row separately.
At first, in the first to 𝑄-th row, 𝜆𝑗 is constrained (1 ≤ 𝑗 ≤ 𝑄); that from the (𝑗 + 1)th element to the 𝑄-th element is 0. Additionally, the 𝑗-th element is constrained to be positive.
Considering these conditions, the posterior distribution is following a constrained multivariate
normal regression.
𝜆𝑗,1:𝑗 | ⋅∼ 𝑁𝑗 (𝜇, Ψ)𝟏(𝜆𝑗𝑗 > 0)
−1

(A.3)

−1

−1
−1
where Ψ = (𝑉0𝑗
+ 𝐸𝑗′ 𝐸𝑗 ) , 𝜇 = Ψ(𝑉0𝑗
𝑔0𝑗 + 𝐸𝑗′ 𝑌𝑗 ) , 𝑉0𝑗 = 𝑣02 𝐼𝑗 , 𝑔0𝑗 = 𝑔0 𝟏𝑗 （𝟏𝑗 is a 𝑗∗dimensional 1 vector）, 𝐸𝑗 = 𝐹⋅,1:𝑗 , and 𝑌𝑗 = 𝑍⋅,1:𝑗. We divide this posterior distribution into twoparts, from the first to (𝑗 − 1)-th element 𝜆𝑗,1:(𝑗−1) and 𝑗-th element 𝜆𝑗𝑗 . Hereafter, let 𝑗 ∗ =1: (𝑗 − 1).
𝜆𝑗,𝑗 ∗ |𝜆𝑗𝑗 ,⋅ ∼ 𝑁𝑗 (𝜇∗ , Ψ ∗ )(A.4)

𝜆𝑗𝑗 |𝜆𝑗,𝑗 ∗ ,⋅ ∼ 𝑇𝑁(0,∞) (𝜇 † , Ψ † )where𝜇∗ = 𝜇𝑗 ∗ + Ψ𝑗∗ ,𝑗 Ψ𝑗𝑗−1 (𝜆𝑗𝑗 − 𝜇𝑗 ), Ψ ∗ = Ψ𝑗∗ ,𝑗∗ − Ψ𝑗∗ ,𝑗 Ψ𝑗𝑗−1 Ψ𝑗,𝑗∗−1†∗ ,𝑗 ∗ (𝜆𝑗 ∗ − 𝜇𝑗 ∗ ), Ψ = Ψ𝑗𝑗 − Ψ𝑗,𝑗 ∗ Ψ𝑗 ∗ ,𝑗 ∗ Ψ𝑗 ∗ ,𝑗 .
Ψ𝑗,𝑗 ∗ Ψ𝑗−1

15

,

and

𝜇 † = 𝜇𝑗 +

While for ( 𝑄 + 1 )-th to 𝐽 -th row of the factor loading, there are no constraints.
Therefore, we can generate samples from the following multivariate normal distribution.
𝜆𝑗 | ⋅∼ 𝑁𝐽 (𝜇, Ψ)
−1

(A.5)

−1

where Ψ = (𝑉0𝐽 + 𝐹′𝐹) , 𝜇 = Ψ(𝑉0𝐽 𝑔0𝐽 + 𝐹 ′ 𝑍 ∗ ) , 𝑉0𝐽 = 𝑣02 𝐼𝐽 , and 𝑔0𝑗 = 𝑔0 𝟏𝐽 .
Posterior samples of factor scores are obtained from the following 𝑄 -dimensional
multivariate normal distribution.
−1−1

𝑓𝑖 | ⋅∼ 𝑁𝑄 ((𝐼𝑄 + Λ′ Λ) (Δ𝑤𝑖 + Λ′ 𝑧𝑖∗ ), (𝐼𝑄 + Λ′ Λ) )(A.6)

The sample of parameter 𝛿𝑝 , which affects the 𝑝-th column of the factor score 𝑓𝑝 =′(𝑓1𝑝 , ⋯ , 𝑓𝑁𝑝 ) , is generated from the following 𝐾-dimensional multivariate normal distribution.
−1

𝛿𝑝 | ⋅∼ 𝑁𝐾 ((Σ0−1 + 𝑊 ′ 𝑊)−1 (Σ0−1 𝑑0 + 𝑊 ′ 𝑓𝑝 ) , (Σ0−1 + 𝑊 ′ 𝑊)−1 )where 𝑊 = (𝑤1 , ⋯ , 𝑤𝑁 )′ .

16

(A.7)

A.2. Measures
Table A.1 Measurement Scales
Familiarity (2 items, 7-point scale)
Reliability: 0.893
Directions: I would like to ask you about canned beers (or alcohol that tastes like beer). Pleaseanswer the following questions.
Items
Anchors1) How often do you purchase canned beers?
1) less than one can a month2) How often do you drink canned beers?
2) one can a month3) few (2 to 3) cans a month4) one can a week5) few (2 to 3) cans a week6) one can a day7) more than two cans a day
Subjective Knowledge (3 items, 7-point scale)
Reliability: 0.958
Directions: Please answer the following questions.
Items
Anchors1) Rate your knowledge of beer product information 1) much lesscompared to the average consumer.
2) less2) Rate your confidence in using beer product information 3) a little lesscompared to the average consumer.
4) average3) I feel confident about my ability to comprehend beer 5) a little more
information on product labels.
6) more7) much more

Objective Knowledge (21 items, binary scale)
Reliability: 0.804
Directions: For the following canned beer brands, match the category and manufacturer to thebrand.
Items
Anchors
1) Jukusen (1)1) beer brand manufactured by Asahi
2) The Premium Malts (5)2) low malt or new genre brand manufactured by Asahi
3) Yebisu (5)3) beer brand manufactured by Kirin
4) Super Dry (1)4) low malt or new genre brand manufactured by Kirin
5) Ichiban Shibori (3)5) beer brand manufactured by Suntory6) Ragger Beer (3)6) low malt or new genre brand manufactured by
7) Malts (5)Suntory8) Kuro Label (7)7) beer brand manufactured by Sapporo
9) Style Free (2)8) low malt or new genre brand manufactured by
10) Tanrei Green Label (4)Sapporo11) Hokkaido Namashibori (8)1712) Asahi Off (2)13) Clear Asahi (2)14) Koiaji (4)15) Sumikiri (4)16) Nodogoshi Nama (4)17) Kin Mugi (6)18) Jokki Nama (6)19) Kin no Off (8)20) Mugi to Hop (8)21) Hookkaido Premium (8)(Correct answers are in brackets)18