# Attribute Overlap in the Modeling of Customer Product Portfolios

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Robert P. Leone and Rajendra Srivastava (1980) ,"Attribute Overlap in the Modeling of Customer Product Portfolios", in NA - Advances in Consumer Research Volume 07, eds. Jerry C. Olson, Ann Abor, MI : Association for Consumer Research, Pages: 762-766.

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http://acrwebsite.org/volumes/9773/volumes/v07/NA-07

This paper investigates the importance of considering attribute overlap when modeling consumer choices of several products out of a single product class. It extends the hypotheses and model employed by McAlister in order to accommodate a measure of attribute overlap. The model of simple satiation is compared statistically with the satiation-overlap model, and the two are found to provide the same fit and forecasting performances. However, the estimates found are quite different between the two models, and therefore, the value of considering attribute overlap is pointed out. In addition, due to the unstable nature of the model's parameters, the question of the correctness of the model's form is raised.

INTRODUCTION

In the past, the emphasis of researchers in studying customer choice behavior has been on single choice situations, with the goal of obtaining a rank ordering of preference. These models have neglected an important choice situation which exists in many product purchases. If in fact, a second or set of products in the same class is to be chosen, the "interaction" which occurs between attributes of the products must be considered. Since most of the prior models have been derived from the Fishbein attitude model (Fishbein and Ajzen 1975), this "portfolio effect" has been ignored by most researchers.

Coombs (1964) was possibly first to point out the importance of considering these multiple choice situations. Farquhar and Rao (1976) developed a model for deriving an individual's utility for a set of items within a specific product class. Their model employed the variance of an individual's attribute values across all possible products in a product class to compute an individual "balance model." Recently, McAlister (1978) has suggested a more parsimonious explanation of this type of behavior. The model assumes that individuals can become satiated with a particular attribute at a certain level, and once this point is reached, the individual would pursue a higher level of their next best alternative (important) attribute. In this study, two basic assumptions were made to operationalize the concept of "attribute satiation."

The first assumption concerns the shape of the attribute utility curve. In order to accommodate the satiation hypothesis, it is assumed the marginal utility of attribute K is a decreasing function. This assumption is a logical one and is well documented in the literature (Coombs 1964, Fechner 1971). However, it should be noted that there are various functions which would satisfy this assumption, yet each could have a variety of shapes and quite different managerial implications, The second assumption is that the attribute levels are cumulative over all items which are consumed. That is to say, receiving 5 units of an attribute from Brand 1 and 3 units of that same attribute from Brand 2 is equivalent to receiving 8 total units of that attribute. This assumption is not necessarily true for several types of products. Consider an individual's evaluation of the "news content" of two magazines (an attribute employed by McAlister). If an important news event occurs during a certain time period, it is likely that both magazines would cover the story. Additionally, many of the "news magazines" often contain the identical cover story. Since it would not be correct to assume that the levels stated by the individual for "news content" merely sum, the joint value must be adjusted for this overlap. Adjusting for this overlap would change several values in the McAlister model, [This would be true unless one assumed that the items were all "different" from one another and no overlap existed. It is possible that a situation like this might exist for some product class.] and naturally could lead to different results and conclusions concerning the location of the satiation point. Another important question, not addressed by McAlister, is that given an individual becomes satiated with an attribute, at what level does this occur and how does this value compare to the current level obtainable for that attribute given the products available.

The objective of this paper is to replicate the study performed by McAlister considering the overlap in the attributes being measured. In so doing, by comparing McAlister's model to the overlap model, the differences in the models can be observed in terms of the "fit" of the estimated model, the predictive "fit" and the differences in satiation (ideal) points generated. The experiment will be discussed after a brief presentation of the model to be used.

MODEL

The model is identical in form to McAlister's, so only a brief overview will be presented in order to point out the difference in parameter definition between the two models. For a full discussion, see McAlister (1978).

In mathematical terms, the assumption stated earlier concerning the cumulative nature of attributes can be represented by

where

K = number of attributes,

k = index number of attribute k = 1,2,...,K,

g = a specific group of items,

G = the total number of items in each group,

X_{gjk} = the value of attribute k for item j in group g, and

X_{g,k} = the sum of the values for attribute k in group g.

This was the exact form of McAlister's model, and this measure will be computed and compared to a measure which considers attribute overlap. The overlap model is operationalized by defining

where 0_{xgjk} represents the overlap of attribute k for all items j in group g, and therefore, X^{o}_{g,k} represents the sum across products discounting the overlap of an attribute between brands, when one exists. This model would then consider any overlap between the attribute ratings provided by an individual. For example, if one brand was perceived as having 5 units of an attribute and a second brand as having 3 units of the same attribute, but, the brands were seen as having an overlap of 1 unit, the score would be 7 [(5+3) - 1] instead of 8 as in the earlier example.

The function used to operationalize the diminishing marginal utility assumption will also be the same as employed by McAlister. Graphically, Figure 1 shows the shape of the curve required by this assumption.

FUNCTION REPRESENTING SATIATION MODEL'S DIMINISHING MARGINAL UTILITY ASSUMPTION

Here, x_{k} represents the present level of attribute k and x_{k} the ideal level or satiation point. This representation is realistic in many marketing situations, but as already mentioned, the shape could be quite different and still fulfill the assumption (e.g. S-shaped curve).

McAlister showed how this quadratic relationship can be conveniently parameterized using Srinivasan and Shocker's (1973) LINMAP algorithm. Given the requirements stated, and due to the constraints which result from the linear programming formulation, the following quadratic form is found.

This form meets the requirements of the first assumption. The additivity assumption, allowing for overlap, combined with this function, yields an expression for the total utility of group g, AS(g),

where W_{k} is the importance or salience of attribute k and X_{k} the corresponding ideal point. This is the term which-McAlister observed to be the exact form of LINMAP, with the only difference being the sum of attribute values over items has been substituted for the attribute values for a single item. Therefore, with the input of n(n-1)/2 paired comparisons from a rank ordering of n items, the parameters of the model can be estimated.

EXPERIMENTAL DESIGN

The McAlister experiment was replicated in order to test the difference between the satiation model and the model considering attribute overlap. Twenty-one female subjects were presented with 32 magazine subscription packages which they were asked to rank order from the one to which they would most likely - to least likely subscribe, given they must pay the annual subscription rates. The five magazines used in the study were __Newsweek__, __Time__, __U.S. News and World Report__, __Cosmopolitan__ and __Playboy__. These five magazines generated the 32 (2^{5}) different subscription packages. The attributes measured were news content, entertainment features, eroticism and cost. It can be seen that these magazines not only possess the attribute news content, but could have an overlap in stories. While it is possible the magazines might overlap on several of the attributes (e.g. Playboy and Cosmopolitan on eroticism), the only attribute overlap which was measured dealt with news content(as the purpose of this paper is to merely illustrate the effect of attribute overlap). The task would be much more complex if all attributes were considered, and it was felt this would be a logical first step in investigating the importance of considering attribute overlap.

The overlap measures were gathered through the illustrative use of Venn diagrams. Figure 2 shows how the overlap in news content between magazines (combinations of magazines taken two, three, four and five at a time from the full set of five magazines) was measured.

This process was pretested to check for the clarity of presentation and difficulty to complete and found to be straightforward.

The instrument used to measure attribute levels (for news content, entertainment features, eroticism and cost) was identical to McAlister's, with data being gathered on an 11 point scale. For example, the scale for measuring news content (for each magazine) was:

No news at all 0 1 2 3 4 5 6 7 8 9 10 11 Maximum amount of news conceivable in a magazine

Each respondent went through four total tasks. First, they rank ordered the various packages; second, they provided perceptual information on the attribute levels; third, they provided information on the overlap of the attribute news content for 26 (32 - 6) groups of magazines; and, fourth, they were asked to provide measures concerning their feelings (involvement, fatigue) about the task completed. All of the questionnaires prepared were different and the order of the tasks performed, as well as the order in which the attributes and packages were presented, were randomized to prevent any biasing effect due to ordering. Out of the 21 subjects, only 19 were used since two of the individuals did not complete the entire questionnaire as instructed.

RESULTS

After administering the instrument discussed in the last section, the first step in the analysis was to analyze the respondents' answers to the questions concerning their feelings about the task. The responses of the subjects about their involvement in the task and the strength of their priorities were not as high as was expected. Therefore, a subset of the respondents [Since this was not planned at the outset of the experiment, we were only able to identify 12 of the respondents to complete this task.] was given a randomly selected group of six magazine packages out of the 32 and asked to rank order the items. The average Spearman-rho statistic for the respondents questioned was .94. The main difference in the two ratings they had given for the magazine packages occurred due to a switching of adjacent items in the middle of the set of six. It was therefore concluded that the respondents had provided valid information during the original session of questioning.

MEASUREMENT OF OVERLAP IN NEWS CONTENT

The rank orderings of 26 packages, the original 32 less a randomly selected holdout sample of six packages, were used to parameterize the models. Therefore, for both models, a correlation measure could be computed for the fitted and actual rank orderings of the 26 items used to estimate the model, along with a measure of correlation for the predicted and actual rank ordering for the six items in the holdout sample. Kendall's rank order correlation coefficient (Kendall's Tau) was used to measure the agreement, and since this was the same measure employed by McAlister, some general comparisons can be made.

The values for each individual are reported in Appendix 1. In Table 1, the average Kendall's Tau measure is reported for the satiation model and the satiation-overlap model.

AVERAGE KENDALL'S TAU MEASURES FOR QUADRATIC MODELS

Table 1 shows there is little difference between the performances of the two models in terms of fitting and forecasting the individual's orderings of the packages. However, after inspecting the parameters estimated (ideal points and importance weights) it is clear the implications of the results are very different. Table 2 presents the ideal points generated by the two models. The differences in the parameter's sizes and signs are quite

noticeable. In addition, the satiation levels implied by both models are extremely large, implying that if one assumes this is the proper model, it does not seem to be of strong managerial use since the values are so far from the individual's current levels. For example, individual 15 has an ideal point of -7.54 under the simple satiation model, yet when overlap is considered, the ideal point becomes 24.50. This change in sign occurs in several other places (with much larger differences) and is difficult to explain. In addition to this problem of changes in signs, the magnitudes of several ideal points seem to be extremely large. For example, individual 18 has an ideal point of 209 units of news content under the McAlister model, yet that individual's largest perception of news content was for the package of __Time__, __Newsweek__, and __U.S. News and World Report__ with a value of only 8 units. It would seem that it would be very difficult to "satiate" this individual with news content, given the high value she would require. McAlister did not report any of these values in her paper, so a comparison across papers is not possible. This instability of values and the unreasonable magnitudes of the ideal points raises a strong question concerning the correctness of the model's form.

Along with the ideal point model (quadratic), a very popular model used by marketers has been the vector (linear) model. Therefore, assuming that the model was incorrectly specified, the data were used to parameterize a linear model and the fit and predictive ability were tested. The average Kendall's Tau's are reported in Table 3, and once again there is no significant difference in the performances of the two models, although, the model which adjusted for overlap has a slightly better predictive performance.

AVERAGE KENDALL'S TAU MEASURES FOR THE VECTOR MODEL

CONCLUSION

This paper has replicated the work done by McAlister on attribute satiation, and attempted to extend its validity by considering attribute overlap in the model's formulation. The results of the tests indicate that the model which considers overlap does not fit the data significantly better, nor does it predict with any greater ability when paired against the simple satiation model. However, the results, and therefore, the managerial implications of the two models are quite different. Since all individuals noted overlap in the attribute news content, it is clear that attribute overlap is an important variable to consider when constructing models of consumer product portfolios.

One problem noted by this paper concerns the instability of the model used in the analysis. This raises a serious question concerning the correctness of the model form (i.e. quadratic) used by McAlister and in this paper. Since the only measure used to validate either model was purely correlational, there is really no basis to accept either model as giving a true representation of consumer's behavior. It is evident that there is a need for the development and testing of a theory concerning individual product portfolio selection which can be exposed to more rigorous tests than merely looking at the fit and forecasting ability of the model.

KENDALL'S TAUS FOR SATIATION-OVERLAP AND SATIATION MODELS

REFERENCES

Coombs, Clyde H. and Aurunin, George S. (1977), "Single Peaked Functions and Theory of Preference," __Psychological Review__, 84, 216-30.

Farquhar, Peter H., and Rao, Vithala R. (1976), "A Balance Model for Evaluating Subsets of Multi-attributed Items," __Management Science__, 5.

Fechner, G. T. (1971), "Zur Experimentellen Aesthetik," __Abhandlungen der Konigliche Sachsische Gesellschaft der Wissenschaften Mathematisch-Psysiche Klasse__, 9, 555-635.

Fishbein, Martin, and Ajzen, Icek (1975), __Belief, Attitude, Intention and Behavior: An Introduction to Theory and Research__, Reading, Mass.: Addison-Wesley.

McAlister, Leigh (1978), __Choosing Multiple Items from a Product Class__, Stanford University Research Paper No. 447, Graduate School of Business, Stanford, California.

Shocker, Allan D. and Srinivasan V. (1976), __LINMAP Version II - Users' Manual__,__ __(Mimeographed).

Srinivasan, V. and Shocker, Allan D. (1973), "Linear Programming Techniques for Multidimensional Analysis of Preferences," __Psychometrika__, 38.

Srinivasan, V. (1977), "A Theoretical Comparison of the Predictive Power of the Multiple Regression and Equal Weighting Procedure," Stanford, California: Stanford University Graduate School of Business Research Paper No. 347.

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