# Can the Multi-Attribute Attitude Model Be Utilized to Predict Probabilities of Brand Choice?

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David J. Reibstein (1977) ,"Can the Multi-Attribute Attitude Model Be Utilized to Predict Probabilities of Brand Choice?", in NA - Advances in Consumer Research Volume 04, eds. William D. Perreault, Jr., Atlanta, GA : Association for Consumer Research, Pages: 111-116.

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http://acrwebsite.org/volumes/9339/volumes/v04/NA-04

A method is developed and demonstrated which combines the multi-attribute attitude model with Luce's choice axiom, thereby providing a prediction of probabilities of brand choice on an individual basis. This approach thus establishes an important connection between systems of seemly stochastic consumer choice processes and the multi-attribute attitude model. An experiment involving repeated brand choices is conducted which enables the testing of the multi-attribute attitude model as a predictor of actual relative frequencies of choice.

INTRODUCTION

An issue which has been raised for several years, but has more recently received a revived emphasis in marketing journals, is the concept that individual brand preferences should be represented in a stochastic manner. To many consumer behaviorists this concept appears to be a direct affront on the work they had been pursuing for the past several years. When it is contended that, "It will never be possible to provide good predictions of individual consumer choice behavior for separate choice occasions," as Bass did (1974, p.9), this should not be interpreted to mean that all efforts to explain behavior should be abandoned, nor should Bass' statement be ignored. The interpretation of Bass' comments by this author is that research efforts on the consumer choice process should be re-directed. It is more reasonable to attempt to predict probabilities of brand choice than it is to predict the specific choice itself. This stochastic representation of "preferences" can also be found in the Markov chain literature (although there the emphasis was on predicting the probabilities of brand choice for an individual within a population of homogeneous individuals), the Hendry model (1966), Herniter's Entropy model (1973), and others.

One of the reasons behavioral research should be redirected is simply that the aspirations to predict individual choice behavior have not been successful. In many cases this has been attributed to either the fact that the models employed had excluded some essential variable(s) or that the measurement on these variables has been insufficient. Hence, we have witnessed the expansion of these models from strictly relying on demographics (Katona and Muellar, 1965; Myers, Stanton, and Haug, 1971; Akers, 1968; Frank, Massy, and Boyd, 1968) to the incorporation of measures of personality (Evans, 1959; Massy, Frank, and Lodahl, 1968; Dichter, 1964) to the inclusion of attitudinal variables (Wicker, 1969; Festinger, 1964), to the eventual creation of an all-inclusive model of consumer behavior (Howard and Sheth, 1969; Nicosia, 1966; Engel, Kollat, and Blackwell, 1973). In each instance, either the model has been too difficult to operationalize, as is the case with the models of Nicosia (1966) and Engel, Kollat, and Blackwell (1973) or when empirically tested have proven to be incapable of predicting individual choice behavior. For example, Farley and Ring (1970) operationalized all of the variables in the Howard and Sheth model (1969) and with actual choice behavior as the dependent variable, they found an R^{2} value of .088. This result was statistically significant, that is, better than a naive random model, yet still left a major portion of behavior unexplained. Once again the explanations offered for the relatively low level of explained variance were that there were substantial measurement errors and that the model did not accurately reflect the actual decision process (Lehmann, et al, 1971).

The lack of high R^{2} values discovered in these behavioral models should not be overly discouraging. The implication is not that the findings from these studies are unmeaningful, as pointed out by Bass, et al (1968), nor that efforts to show the relationship between different choice behavior should be curtailed. It may truly be possible that through time and continued effort, the appropriate model and accurate measurement techniques may be discovered. However, given that so many factors influence behavior, many of which are unpredictable in nature, then even if such a model could be developed it most likely would be too cumbersome and complex to be managerially useful.

It is contended by some that it will never be possible to completely predict individual choice behavior, as it is stochastic in nature (Klingsporn, 1973, p. 445; Bass, 1974, p. 1). One of the reasons for the inclusion of a disturbance term in econometric models, according to Johnston is, "... there is a basic and unpredictable element of randomness in human responses" (1972, p. 11). This implies that the low R values reported in studies of individual choice behavior may not be the result, at least not in total, of the lack of variables or poor measurement, as suggested by many authors, but a fundamental component of choice behavior. As such, minimization of error may not be possible beyond a certain point and, in fact, R^{2} values may be a poor criteria of a model's appropriateness.

Further, many empirical studies have discovered the existence of extensive brand switching in many product categories. Such observations have been made even when other potential influencing factors were held constant (McConnell, 1968a, b; Bass, Pessemier, and Lehman, 1972).

Whether behavior is intrinsically stochastic remains uncertain. However, what is clear is that deterministic models of individual choice behavior have been incapable of predicting individual behavior, that empirical data reflect considerable brand switching behavior, and that even if it were possible to develop a model which could explain individual behavior the data requirement would most likely make it difficult to implement. Therefore, it is reasonable to conclude that behavior should be represented stochastically.

Although it may be advisable and useful to represent preferences in terms of probabilities of brand choice on an individual basis, it continues to be worthwhile to attempt to understand consumer behavior. With this knowledge it would then be possible to design marketing strategies which could, generate a greater appeal to the consumer. The question then becomes, __is it possible to link the consumer behavior literature with the contention that behavior should be represented stochastically__?

MULTI-ATTRIBUTE ATTITUDE MODEL

Since early applications to marketing by Talarzyk (1969) and Lehmann (1969), the multi-attribute attitude model has received a large amount of attention in the marketing literature. The general notion in the model is that an individual's attitude about a brand is based on his perceptions of components of the brand. A person's judgment about a brand of toothpaste is contingent on his evaluation of the brand's taste, decay preventiveness, whitening ability, popularity, etc., and the importance of each of those components in his choice process. The multi-attribute attitude model is an aggregative model of these weighted components. The model in its most general form is

"_{xk} = [S_{j}(V_{jk}|B_{xjk}-I_{jk}|)^{r}]^{1/r} (1)

where,

"_{xk} = the attitude score towards brand x by individual k,

V_{jk} = the importance weight of attribute j for individual k,

B_{xjk} = the belief about the level of attribute j contained in brand x by individual k,

I_{jk} = the ideal level of attribute j for individual k,

r = the Minkowski measure.

A large portion of the research has been on testing various forms of the multi-attribute attitude model (Wilkie and Pessemier, 1973). In the majority of cases, the test of the alternative model forms has been a measure of the correlation between the model's attitude score and stated preference rankings. It has generally been accepted that the model (in all of its forms) produces reasonable scale values for each of the brands. The difference observed in the various model forms has been in its level of predictability of the stated preference rankings.

In spite of the potential shortcomings of the multi-attribute attitude model it still has proven to provide reasonable scale values for brands in a product class, reasonable in the sense that the model produces values which are significantly monotonically correlated with stated preference rankings. As such the model holds a prominent position in the behavioral marketing literature.

LUCE'S CHOICE AXIOM

The multi-attribute model produces positive scale values for a set of brands which could be used as input for the Luce (1959) choice model to derive probabilities of brand choice on an individual basis. The derivation of the model is quite simple. The choice axiom is

P(x;t) = P(x;R)P(R;T) (2)

where x is a brand contained in choice set R and R is a subset of set T (xeT, ReT), and P(x;T) is the probability of picking brand x from set T and S_{xeT} P(x;T) = 1.

This axiom states, for example, that the probability of choosing Coca Cola (x) from a set of soft drinks (T) is equivalent to the probability of choosing Coca Cola from a set of cola-flavored soft drinks (R, a subset of T) times the probability of selecting a cola-flavored soft drink. The axiom holds for all brands contained in both sets R and T. Thus,

P(y;T) = P(y;R)P(R;T), (3)

where y is another brand contained in sets R and T.

Dividing (2) by (3) yields,

which shows that the ratio of the probability of choosing brand x from a set with the probability of choosing brand y from a set is independent of the choice set R. Or, as is referred to as the constant ratio rule (Luce, 1959),

where

P(x>y) is the probability of an individual choosing brand x over brand y.

The constant ratio rule indicates that the ratio of the probability of choosing brand x over brand y with the probability of choosing brand y over brand x is equivalent to the ratio of the probability of choosing brand x from set T with the probability of choosing brand y from set T. This ratio is the same from all subsets of T which contain both brands x and y.

This shows that the pairwise choice probabilities contain all of the information necessary to determine the probability of choosing one brand from a given choice set.

The relationships could also be extended to a definite relationship among pairs of three or more brands, such that the probability of choosing one brand over another could be determined if all other pairwise probabilities are known. The results from such a model are similar to those derived from Thurstone's Case V model (Coombs, Dawes, and Tversky, 1970).

To determine the probability of choice for a particular brand from a choice set, let z be an arbitrary element of set T and define the scale value for x, S(x), equivalent to the ratio in (5),

Combining (5) and (9),

and from (7)

Therefore, the probability that a brand x will be chosen from choice T is

In other words, the probability of an individual selecting brand x from a choice set can be expressed as a function of the scale values of the brands in the choice set (Coombs, et al, 1970).

A limitation of the model in (12) is that the scale values for all brands are nonnegative, otherwise, the model would produce negative probabilities for the brands associated with negative scale values, and possibly probabilities greater than one for others, since S_{xeT} P(x;T) = 1.

The reasonableness of the choice probabilities depends on the reasonableness of the scale values used in the model although how the scale values are derived is not specified. There are several alternative methods of deriving scale values, such as, paired comparisons, a dollar metric approach, the multi-attribute attitude model, etc. Since the multi-attribute attitude model provides positive scale values for each of the brands in the choice set, these scale values have generally been accepted as relatively accurate representations of preference, and the model connects these scale values with information which could prove managerially insightful, it will be used to derive probabilities of choice in this study. However, the model provides the largest scale values for the least preferred brand, or most distant from the ideal. Thus, if the scales are reversed in desired order, it is necessary to alter the scale values while still maintaining the same ratio between brands. That is, if the scale values for brands x and y are such that S(x) < S(y), yet the intended interpretation is that brand x has a higher probability of choice than brand y, it is necessary to transform the data while not altering the ratio between the scale values of the brands. For example, let the corresponding scale values for brands x, y, and z be {2,4,6} respectively, and the interpretation of the scale value is that lower values imply a higher probability of choice, then it is desirable to maintain the ratio between brands such that,

The direct application of Luce's choice axiom,

provides

P(x;T) = 1/6, P(y;T) = 1/3, and P(z;T) = 1/2 (15)

which is opposite from the desired interpretation of the scale values.

Thus, it is necessary to reformulate the model as,

Since S S(x) is a constant, (16) can be written as,

Returning to the illustration given above,

and thus, the desired ratios are obtained. It is also important to notice that the necessary condition

exists.

Applying Luce's Choice Axiom and the scale reversal to the multi-attribute model, the scale value of brand x for individual k is,

and the probability of individual k choosing brand x given the choice set T is,

Or more simply,

Thus, it would be possible to create a vector of predicted probabilities of brand choice for each individual based on a set of attitudinal information. With an empirical base of repeated brand choices on an individual basis it would be possible to create a vector of relative frequencies of brand choice, a preference vector. To test the validity of the multi-attribute attitude model as a predictor of probabilities of brand choice these two vectors could be compared.

THE DATA

A simulated supermarket was constructed at Purdue University in the fall of 1974 with 138 students and secretary participants. They were paid to attend daily for six weeks where they would pay for a soft drink of their choice from a set of the six most popular brands. From their thirty brand selections it was possible to derive for each individual, a vector of relative frequencies which should approximate an actual preference vector. This is the vector we are interested in predicting.

There were three experimental groups - two which had varying prices and a third which had stable prices and equal between all brands. On one occasion the participants completed a questionnaire for the multi-attribute attitude model and on three visits they were asked to specify a preference ranking. (Details of the experiment can be found in [Reibstein, 1975]).

Based on the multi-attribute attitude model and the application of equation (21) it is possible to create a predicted preference vector which could be compared to the actual preference vector exhibited through individual choice behavior in the experiment.

RESULTS

The market shares for each of the brands were relatively stable from one day to the next, and for the third group, similar to the market shares each brand owned in the actual marketplace. As found in other empirical studies, the level of brand switching or choosing a different brand at trial t from that chosen on trial t-l, was considerable, ranging from 38.4% to 43.3% for the three experimental groups. The strict deterministic explanation for such behavior is that preferences must likewise be varying. However, given a product category with which people are familiar, and holding other extraneous factors stable, such as stockouts and in-store promotions, one would expect preferences to remain relatively stable. This contention can be examined by looking at the responses on the three different questionnaires to the inquiry, "What is your favorite brand of soft drinks?" A strictly random response would produce the same brand on all three trials with a probability of .028 or 1/36. The results indicate that 84% of the subjects provided the same response at each of the time intervals (each questionnaire was separated from the other by three weeks). Fourteen percent had the same response on two of the three questionnaires and only three individuals (2%) had different responses on each occasion.

The statistical significance level of this result is hard to determine since it depends on what the standard of comparison is, which is difficult to establish. However, the level of consistent responses indicates a strong stability of preferences. The few inconsistent cases could be attributed to a few individuals being indifferent between more than one brand. Hence, choice behavior is varying even though stated preferences appear to be stable. This may further substantiate the need to represent preferences in terms of probabilities of choice rather than mere specification of the preferred brand or a preference ranking.

Predicted Preference Vector

If the predicted preference vector is a good approximation of the actual relative frequency vector then there should he a high positive correlation between the two, and the better the predictive model the higher the correlation should be. Correlations could be obtained in several different manners: by individual across brands, by brand across individuals, and across individuals and brands. Since interest in this study concentrates on the prediction of individuals' preference vectors, and not the prediction of probabilities of choice for any one brand, the correlations by individuals and across brands is of primary interest.

This correlation measure is derived based on the number of brands minus one, since the sum of the brand probabilities equals one with n-1 probabilities of choice the nth can be derived as simply,

When correlations are obtained for proportions it is useful to use the following transformation on the data:

P'(x;T) = 2 arcsin P(x;T) (23)

so as to stabilize the variances (Winer, 1971, p. 400; Torgerson, 1958, p. 203). The results derived after the transformation are only slightly different from the correlations derived on the raw data.

The correlation measure averaged across individuals was clearly nonsignificant for all three groups. There were some individuals for which the model provided high correlations, but this number constituted less than six percent of the subjects.

The results tend to indicate that the model is not related to the actual choice behavior exhibited in the experiment for the overwhelming majority of the subjects. How can this be possible given the popularity of similar models in marketing research and the high validity levels exhibited in previous marketing studies? The answer may be that all previous studies employing similar forms of the multi-attribute attitude model were validated against stated preference ranking. That is, either a Spearman Rho rank correlation was computed between the model derived affect scores from equation (1), or variations thereof, and stated preference ranking, or the stated preference rankings were utilized as the dependent variable in a multiple regression analysis.

The implication usually drawn is that stated preference rankings give a strong indication of choice behavior. Prevalent theories are that either the most preferred brand would always be chosen, or, aware of brand choice variation, the theory is that the brand ranked first would be chosen most frequently, the brand ranked second would be the second most frequently chosen brand, and so forth, with the brand ranked last being chosen least. Thus, if the multi-attribute attitude model could predict stated preference ranking it should also be possible under the former theory to predict brand choice, and under the latter theory to approximate the relative levels of brand choice. However, what was discovered in this study, as mentioned earlier in this section, is that __there is not an isomorphic relationship between stated preference rankings and relative levels of brand choice__.

When the traditional analysis was performed on the multi-attribute attitude model, the results obtained were similar to those achieved in previous applications of the model (Wilkie and Pessemier, 1973). The average individual Spearman Rho rank correlation was .523, within the range of what has been discovered in other studies. Alternative model forms were experimented with - deletion of the importance weights and the exclusion of the ideal point. As has been discovered elsewhere, the importance weight had only a slight positive influence with an improvement in the Spearman Rho of .024. Since the ideal points for all attributes were not evaluated at an extreme end of the scale, the exclusion of the ideal point form was not entirely appropriate. The relationship between the attitude models and stated preference rankings was significantly better than a random model, and if the study of the attitude models were concluded at this stage it would be evaluated as a successful predictive model.

A comparison of the stated preference rankings with the actual frequency of choice was most informative. If the theories about the relative frequencies simply being a reflection of the stated preference rankings were true, then the rank correlation between the model and relative frequency of brand choice would be -1.00 (or 1.00 if the scale were reversed such that the brand preferred most was assigned the highest value). However, although an individual analysis reveals a significant Spearman Rho rank coefficient (a < .05) averaged across individuals of -.788, which was significantly less than zero for 74% of the subjects, this value is significantly greater than -1.000.

A possible explanation for this deviation from perfect correlation is that it is possible for the first and second most preferred brands to be very similar, with one preferred only slightly over the other. If the number one brand were not present in the choice set, then the second brand would most likely become the most frequently chosen brand. However, with the most preferred brand always included in the choice set, this slightly less preferred brand may not be chosen at all since it does not offer much of an alternative. Consequently, the brand may be ranked highly in a stated preference ordering, but the actual level of brand choice may be relatively small.

When metric data are desired, such as probabilities of brand choice, rather than mere rankings, there is apparently an even greater loss of information in the multi-attribute attitude model. That is, even if the multi-attribute attitude model could predict the relative ranking of brand __choice__ rather than just preference rankings, that does not mean that the model could discriminate between levels of choice. For example, suppose brands A, B, and C had probabilities of being chosen of .01, .02, and .97, respectively. A model which could predict brand C as the brand with the highest probability of brand choice and brand B as second highest, may not be able to distinguish between the relative levels of brand choice. Apparently this was the case with the multi-attribute attitude model. A problem encountered when rank correlations are utilized is that if the predicted ranking is in the general order of the actual ranking the test statistic will be significant.

CONCLUSION

In summarizing, an attempt was made in this study to bridge the two schools of thought - choice behavior as a deterministic process or as a stochastic process. As shown herein, it is possible to structure our behavioral models such that the output is the prediction of individual probabilities of brand choice coupled with the understanding of why these probabilities are of their given magnitudes and with some insight as to how they might be changed. However, it appears that although the multi-attribute attitude model can be used to significantly predict an individual's stated preference rankings, these stated preference rankings do not totally capture the rank orderings of brand choice.

That is, the inference should be drawn that attitudes and preferences are not equivalent to choice behavior, even when most intervening variables are experimentally excluded from the choice process, as in this study. Further, the prediction of rank order is different from the prediction in a metric scale of relative frequency of choice. These findings should caution against drawing too hasty a link between attitudes and behavior. The connection, if it exists, is subtle and should be explored further before conclusions are drawn.

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