# A Structural Analysis of Buying Style Variables

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Jamshid C. Hosseini, Rajshekhar G. Javalgi, and Robert R. Harmon (1988) ,"A Structural Analysis of Buying Style Variables", in NA - Advances in Consumer Research Volume 15, eds. Micheal J. Houston, Provo, UT : Association for Consumer Research, Pages: 476-480.

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http://acrwebsite.org/volumes/6849/volumes/v15/NA-15

This paper, employing the information theoretic approach, reanalyzes the buying style variables for a sample of working women, initially analyzed by Clogg and Munch (1984) using latent structure analysis. The information theoretic method is shown to be useful in obtaining structural models within an a priori selected criterion variable.

INTRODUCTION

In marketing research, perhaps the most common method of exploration of the relationships between two (or more) qualitative variables involves the construction and analysis of cross-tabulations. Here, the cell entries of an n-way tabulation represent the joint frequencies (or joint probabilities3 in which each combination of the variables' categories appears.

The present paper investigates the structural models and the analysis of association involving multidimensional contingency tables composed of qualitative data. A variety of alternative techniques, including log-linear, logit/probit, and latent structure analysis (LSA), are available for examining the extent of association in a discrete multivariate contingency table (Green et al. 1977, Perreault and Barksdale 1980, DeSarbo and Hildebrand 1980, Clogg and Munch 1984. Green et al. (1977) point out that " ... other approaches to the modeling categorical data can be. mentioned." One of these methods, among others, is based on information theory.

The present study employs an information theoretic approach to analyze multidimensional contingency tables. Though no statistical technique can replace theory, the present paper demonstrates that the information theoretic framework can provide researchers with a powerful tool for gaining valuable insight into qualitative/ categorical data. To demonstrate the potential usefulness of this approach in marketing research, the primary focus of the study presented next.

PURPOSE OF THE STUDY

This paper reanalyzes the buying style variables (e.g., style-consciousness, conformity, brand-loyalty, etc.) for working women, initially analyzed by Clogg and Munch 1984 who used the probability based latent structure analysis (LSA) technique. This approach begins with a multidimensional contingency table constructed from the variables under investigation. The aim of LSA is to identify a set of underlying (unobservable) variables which best characterize the structure of relationships among these variables.

Latent structure modeling is quite flexible and provides a means of testing (via goodness-of-fit tests) whether latent factors explain the observed association among variables, the substantive meaning of the latent variables, and how to assign individuals to the classes of the latent factor itself. In their article, Clogg and Munch (1984) compared two groups of working women, those who perceived their job as a "career" and those whose work was "just a job," in terms of the latent structural model parameters. Using this approach, they noted no significant difference between buying style attitudes of the "career" vs. "just a job" women once the brand loyalty variable was excluded.

Instead of searching for latent structures within the data, the observable variables are analyzed here, by employing the information theoretic framework. More clearly, the information theoretic approach, in conjunction with the recently developed procedures for structure analysis is utilized to analyze observable variables (i.e., buying style variables) in order to: (1) assess interrelationships between such variables, (2) identify and separate those variables which are orthogonal (or nearly orthogonal) to the rest of the variables, and (3) identify a theoretically sound variable, a priori, and then obtain structural models within different classes of this variable.

It should be mentioned that this paper is not intended to directly compare the present approach with latent structure analysis and/or log-linear models. However, it is demonstrated that the information theoretic approach offers some advantages and provides some insight into the analysis of multivariable contingency tables which have often been analyzed by such procedures as log-linear and latent structure analysis.

METHODOLOGY

Uncertainty (variability) in a discrete probability distribution can be measured using Shannon's entropy (Shannon and Weaver 1949). This is a "unique, unambiguous" measure based on the intuitive notion that a broad distribution represents more uncertainty than does a narrow, sharply peaked distribution (Herniter 1973). For a given system with N possible states, entropy is given by:

where P(X_{i}) is the probability of the system being in the state Xi (i=1,2,...,N) and the base of the logarithm is arbitrary (when assumed at two, the measurement unit of entropy is known as Binary digit, BIT).

A system with two variables, X and Y, can be viewed as having total (joint) variability--or"behavioral freedom" (Broekstra 1981)CH(X,Y). If this quantity has its maximum possible value, Hm Ix(X,Y) = H(X) + H(Y), independence exists between X and Y. On the other hand, if the two variables are mutually associated, their joint behavioral freedom is reduced as they impose behavioral constraint upon one another. The degree of association between X and Y can be expressed as T(X:Y), the difference between maximal and actual joint variability,

T(X:Y) = H

_{max}(X,Y) - H(X,Y)= H(X) + H(Y) - H(X,Y) (2)

T(X:Y) is known as "transmission." For the two variable case, transmission can be interpreted as the amount of variability removed from one variable once the other variable is known (i.e., amount of covariability between X and Y). This measure is completely decomposable which can be partitioned into variability between and variability within sets of variables, allowing for an ANOVA like analysis of variables (Garner and McGill 1954).

Furthermore, the uncertainty of variable in a multivariate system can be decomposed into the portion explained by other variables (i.e., transmissions and conditional transmissions between that variable and the remaining variables), and the proportion not explained (Ashby 1964, Krippendorff 1986). For instance in a four-variable system WXYZ, the following breakdown of the uncertainty of variable W may result:

H(W) = T(W:X) + T

_{X}(W:Y) + T_{XY}(W:Z) + H_{XYZ}(W) (3)

This equation can be used to prioritize the degree of explanation of variable W by variables X, Y, and Z, by sequentially selecting the highest contributors. For instance, in choosing T(W:i'l i=X, Y, or Z, i should be selected such that it renders the highest bivariate transmission; in selecting Ti(W:j), j=X, Y, or Z, j=i, j should result in the highest conditional transmission, conditioned on i; etc.

Transmission is used in this paper to first develop a bivariate transmission table between the buying style data for two groups of women. Through this process, bivariate interrelationships between variables will become apparent. This table is then used to eliminate variables which are unrelated to all other variables from further consideration.

Next, a theoretically sound variable is selected as the criterion variable (in this case, "brand loyalty" was selected), and Equation (3) is utilized in order to prioritize the contributions of the other (descriptor) variables to the determination of this (criterion) variable. This is in turn used to further isolate irrelevant variables.

Finally, after selecting a group of interrelated variable (and excluding insignificant variables), Krippendorff's (1986) spectral analysis of relations is utilized to develop structural models for each class of the criterion variable. We also revised Krippendorff's method by utilizing the transmission method used by Broekstra (1981) to assess the degree of mutual association of the system captured by each model.

Spectral analysis develops a tree of structure types from the most complex, which is the multivariate system without alterations, to the simplest structure which is a disjoint set of all the variables by systematically removing n-way, (n-l)-way, etc. interactions and replacing each with their embedded lower levels of ordinality. The reader unfamiliar with spectral analysis should review Krippendorff (1986) and Broekstra (1981) for a thorough exposition of this technique and its related issues.

Spectral analysis will yield the "best" structural model within different classes of the criterion variableC The best structural model is defined as the model with the lowest possible ordinalities within its components which accounts for a statistically significant portion of the information contained in the top level structure. This can then be extended to include the strength and direction of the interrelationships in each class.

ANALYSIS

The data which was adopted from Clogg and Munch (1984), contained information regarding the buying style attitudes (and behavior) of white, married, full-time employed females in the middle three of the five "social class" categories. This data was extracted from a 1979 study of Media and Markets conducted by the Simmons Market Research Bureau.

The women in the sample were divided into two groups (G); those who consider their job as a "career", and those who consider their occupation as "just a job." The purpose of this partitioning was to assess the differences between the two groups based on their buying style attitudes. We shall demonstrate later that at least in this sample there is no appreciable difference between the two groups based on the buying style variables considered, and then, concentrate on the buying style variables for the whole group.

The selected buying style items were "style-conscious" (S), "persuadable" (P), "conformist" (C), "experimenters" (E), - and "brand loyalty" (B). The original data were collected as five-category Likert type sales, but for the demonstration purposes, they were all dichotomized to agree-a-lot/disagree-a-lot categories. Then, a multidimensional contingency table of these six variables was developed and analyzed to identify different latent structures. It is this contingency table that we adopted in this paper. This data is presented in Table 1.

Obviously, in cases where variables are collapsed on their categories (i.e., recoded to lower number of classes), particularly, dichotomized variables, a myriad of questions and skepticism are invoked. Also, in assessing multivariate contingency tables, problems associated with sparse cells arises. These issues, however, are not addressed here. Interested reader is referred to Feick (1984) and Dillon et al (1981) for excellent discussions of such issues.

The bottom triangle in Table 2 presents the bivariate transmission table, while the top triangle shows their corresponding likelihood ratio chi-square (L2). It is evident that variable G, group, has very little association with the rest of the variables. The only statistically significant association for this variable is with S, style-conscious, variable which has a likelihood ratio chi-square of 5.2 on one degree of freedom. However, due to lack of any apparent association with the remaining variables, this variable is considered nearly independent of the others and is dropped from further consideration.

Next, we collapsed the contingency table over G, the group variable. Among the remaining variables, B, brand loyalty, was selected as the criterion variable. The reason for this selection was the fact that, in practice, many marketing strategies begin by determining who are the loyal consumers of the product and what it takes to retain them, and on the other hand, who are the brand switchers or the competitors' brand loyals and what will persuade them to switch over to and remain with the company's brand.

Then, Equation (3) was applied to the five-variable cross-tabulation with B, brand loyalty, variable treated as the criterion variable. The following breakdown of uncertainty in B, brand loyalty, and prioritization of the other, descriptor, variables was obtained:

H(B) = T(B:S) + T

_{S}(B:C) + T_{SC}(B:E)(0.894) = (0.019) + (0.015) + (0.012)

L

^{2}= 33.1 26.1 20.9+ T

_{SCE}(B:P) + H_{SPCE}(B) (4)+ (0.009) + (0.839)

15.7

It is seen that all of the remaining variables are statistically significant (with df = 1 and p - 0 for all). Therefore, no other variables are dropped from further consideration. In general though, variables with no significant association with the criterion variable would be eliminated in the order of their prioritization (i.e., the last in the priority list would be dropped first).

CROSS-CLASSIFICATION TABLE FOR THE SIX-VARIABLE SYSTEM

Finally, the structure analysis technique was utilized to assess the interrelationships between each class of the criterion variable B, brand loyalty, (i.e., brand loyal/brand disloyal). Tables 3 and 4 show the best representative structures obtained from all permutations of the variables for brand loyal and brand disloyal, respectively. In these two tables, column 1, Structure, lists the structure types, column 2, indicates the amount of transmission captured by structure type, column 3 and 4 contain the chi-square value and the degrees of freedom for each structure type, respectively, and column 5 lists the absolute difference between chi-squares of each structure type and the top level structure.

BIVARIATE TRANSMISSION TABLE OF THE SIX-VARIABLE SYSTEM

STRUCTURE ANALYSIS OF THE BRAND LOYAL GROUP

STRUCTURE ANALYSIS OF THE BRAND DISLOYAL GROUP

To clarify what we mean by the "best representative structure," consider the model PCS/PSE/CSE in table 3 (i.e., the third structure type from the top). All permutations of this particular structure type were SPC/SCE/PCE, SPE/SEC/PEC, SCP/SPE/CPE, PCS/PSE/CSE. The absolute difference in chi-square values between PCSE, the top level structure, and each one of these four structures was calculated as 1.35, 1.64, 1.03, and 0.61, respectively. Therefore, PCS/PSE/CSE was selected as the structure which is the best representative among the four, as it approximates the top structure most closely.

Clearly, in Table 3 the best structure type for the brand loyal class (B=B1) is the structure ESP/SC/PC (p=0 and diff.=0.25). Similarly, the best structure type in Table 4, for the brand disloyal group (B=B2) is the structure type PE/PC/EC/CS (p=0, and diff.=O.11) These two structure types are depicted in Figures (1a) and (1b), respectively.

The structure type in Figure (1a), ESP/SC/PC, can be interpreted in the following manner Among brand loyal individuals .in the sample, experimenter quality (E), style-consciousness (S), and persuadability (P) have three-way direct interactions,--style consciousness (S) and conformity (C), and persuadability (P) and conformity (C) have two-way direct interactions, while experimenter quality (E) and conformity (C) have only indirect interactions.

Similarly, the structure type in Figure (l.b), PE/PC/EC/CS, can be interpreted in the following way: Among brand disloyal individuals in the sample, conformity (C) has direct two-way interactions with Persuadability (P), experimenter quality (E), and style consciousness (S), while Style consciousness (S) has only indirect interactions with Persuadability (P) and Experimenter quality.

GRAPHICAL REPRESENTATION OF THE BEST STRUCTURAL MODELS

CONCLUSIONS

The foregoing analysis demonstrated that the information theoretic approach is capable of analyzing multidimensional contingency tables and addressing fundamental issues such as:

a) Assessing the interrelationships among variables-- e.g., by constructing bivariate (or higher order) transmission tables and identifying the most (and the least) interrelated variables. This is useful in identifying and excluding variables which are least associated with other variables.

b) Prioritizing the extent of contribution of each (descriptor) variable to the reduction of uncertainty about an a priori selected criterion variable. This would enable the researcher to select the most significant descriptor variables and then, assess the criterion variable based on this reduced set of variables.

c) Conducting structural analysis for each class of the criterion variable and identifying the best structural model within each class. Structural models in general, are useful for the purposes of simplification and substantive interpretation of relationships among variables.

These analyses are performed straight forwardly, with readily available substantive interpretations. Clogg and Munch (1984) primarily focused on the latent class identification problem. Here, the focus is on the observable variables. It is natural to attempt to draw conclusions based of observable variables first. Only when the observable variables do not provide satisfactory results, latent variables should be sought.

In the present treatment, it is demonstrated that in order to assess the buying style variables which determine brand loyalty of a consumer, reasonably simple structures (which are statistically significant) for each class of this variable can be identified. Information theory provides the main vehicle for this structure identification problem.

REFERENCES

Ashby, W.R. (1964) "Constraint Analysis of Many Dimensional Relations," General Systems Yearbook, 9, 99-105.

Broekstra, G. (1981) "C-Analysis of C-Structures: Representation and Evaluation of Reconstruction Hypotheses by Information Measures," International Journal of Systems, 7, 33-61.

Clogg, C.C., and J.M. Munch (1984) "Using Simultaneous Latent Structure Models to Analyze Group Differences: Exploratory Analysis of Buying Style Items," Journal of Business Research, 12, 319-336.

DeSarbo, W.S., and D.K. Hildebrand (1980) "A Marketer's Guide to Log-linear Models for Qualitative Data Analysis," Journal of Marketing, 44 (Summer), 40-51.

Dillon, W.R., M. Goldstein, and L. Lement (1981) "Analyzing Qualitative Predictors with Too Few Data: As Alternative Approach to Handling Sparse-Cell Values," Journal of Marketing Research, XVIII, (February), 63-72.

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Green, P.E., F.J. Carmone, and D.P. Wachspress (1977) "On the Analysis of Qualitative Data in Marketing Research," Journal of Marketing Research, 14 (February), 52-59.

Herniter, J.D. (1973) "An Entropy Model of Brand Purchase Behavior," Journal of Marketing Research, X (November), 361-375.

Krippendorff, K. (1986), "Information Theory: Structural Models for Qualitative Data ," Sage University Paper Series on Quantitative Applications in the Social Sciences, 07-062, Beverly Hills and London: Sage Publications.

Perreault, W.D., Jr., and H.C. Barksdale, Jr. (1980), "A Model Free Approach for Analysis of Complex Contingency Data in Survey Research," Journal of Marketing Research, 17 (November) 503-515.

Shannon, C.E. and W. Weaver (1949) The Mathematical Theory of Communication, University of Illinois Press. Urbana, Chicago, London.

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