# A Note on the Fishbein Attitude Model As an Expected Utility Model

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Joel Huber and Robert P. Leone (1979) ,"A Note on the Fishbein Attitude Model As an Expected Utility Model", in NA - Advances in Consumer Research Volume 06, eds. William L. Wilkie, Ann Abor, MI : Association for Consumer Research, Pages: 245-247.

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http://acrwebsite.org/volumes/9562/volumes/v06/NA-06

Fishbein's attitude models and similar expectancy-value formulations, code the association measure as bipolar. Although there is good empirical justification for this coding, it results in the theoretically embarrassing prospect of negative probabilities. If, however, the Fishbein model is seen as a truncated version of the full expected utility model, then the bipolar measures of association are required by, rather than in contradiction of, the axioms of probability.

INTRODUCTION

One of the more appealing aspects of the Fishbein Attitude model (Fishbein and Ajzen, 1975) is its parallel with an expected utility formulation. That is, in the Fishbein model, attitude is a function of the evaluation of an attribute multiplied' by its degree of association with the object. This is analogous to an expected utility model which multiplies the utility or value of each event times the probability of occurrence. This point has been made by Etter (1975, p. 482) who concluded that the Fishbein model "...can be regarded as equivalent to a multi-attribute decision theory model where the decision is made under conditions of risk and utility is additive." Unfortunately the equivalence is marred by Fishbein's bipolar scaling of association. Ahtola (1975, p. 52) has commented that "the operationalization of B_{i} (the measure of association) should be more in accordance with the axioms of probabilities." Other researchers have expressed similar concern with respect to this coding (Lutz, 1976; Bettman, Capon and Lutz, 1975).

The purpose of this note is to argue that the Fishbein scaling of the association measure as bipolar does not contradict the axioms of probability but follows from them. The problem lies in the fact that the Fishbein model does not include the negative outcome and so represents a truncated version of the full expected utility model. Given the full model and certain reasonable assumptions, it can be shown that the axioms of probability __require__ that the measures of association in the truncated model be rescaled from the range 0,1 to a range that is symmetrical about zero. Thus the negative values are a mathematical necessity brought about by the use of an incomplete expected utility model.

Figure 1 shows the needed inputs to a full expected utility model for a hypothetical automobile. Here it is assumed that there are just two attributes, reliability and comfort, although the results generalize easily to any number of dimensions. Assume further that the attributes are binary, i.e., a car is either reliable ("x") or it is not ("x") and is either comfortable ("y") or not ("y") and that an automobile may take any of the four joint states reflecting combinations of these attributes. In an expected utility model the value of a particular car is the sum of the products of the joint values times their probabilities, or

UTILITIES AND PROBABILITIES REQUIRED FOR FULL MODEL: TRUNCATED MODEL USES MARGINAL TOTALS

This is the full expected utility model. It can be contrasted with the simpler Fishbein model which for two attributes is:

Ao = a_{x}b_{x} + a_{y}b_{y} (2)

where a_{x} is the marginal goodness or badness of x, and b_{x} is the degree of association between the object and the attribute x. As the model is operationalized, respondents are not asked questions about the value of both reliability __and__ unreliability but are asked to respond to only one of the two poles of the attribute. The measure of association, b_{x}, is coded to be symmetric about zero so that high levels of association result in positive values and low levels result in negative values.

Equation (2) is certainly easier to estimate and understand than the full model. Moreover, it will be shown that given reasonable assumptions, it produces identical results to the full model if the measures of association are coded to be symmetrical about zero. Although shown for only two attributes, this result holds for any number of attributes and thus applies to the general Fishbein model.

To prove this equivalence certain assumptions are necessary about the relation of the joint to the marginal values in Figure 1. These are provided below:

If these assumptions and their comparable forms for the different arguments are inserted into Equation (1) the result is:

A* = u_{x} (2P_{x}-1)/k + u_{y} (2P_{y}-1(/k (7)

Or

A* = a_{x}b_{x} + a_{y}b_{y} = A_{o} (8)

If

a_{x} = u_{x}, a_{y} = u_{y} (9)

And

b_{x} = (2P_{x}-1)/k, b_{y} = (2P_{y}-1)/k . (10)

The above analysis indicates that the two models produce proportional results (up to a multiplicative constant, k), if the Fishbein measure of affect equals the utility score, and if the Fishbein measures of association are derived from a symmetric rescaling of the marginal probabilities. The form of this rescaling is illustrated in Figure 2.

TRANSFORMATION OF PROBABILITIES REQUIRED TO MAKE TRUNCATED MODEL EQUIVALENT TO FULL MODEL

Notice that this recoding does not imply that __probabilities__ actually range from -1/k to 1/k. On the contrary, it asserts that the probabilities must be modified to satisfy the axioms of probability given a truncated model. This result is consistent with Fishbein's assertion that a probability of less than 50% of achieving a desired attribute results in a lessening of the total attitude score. In the full model this lessening occurs when P_{x} is less than 50% because the negation of a desired attribute (-u_{x}) times its occurrence (1-P_{x}) is larger than the value of the positive occurrence (u_{x}) times its occurrence (P_{x})- In the truncated model the coding of the association measure to be symmetrical about zero is necessary to produce this same effect.

Notice further that this result, while framed in the context of the Fishbein Model, applies to all expected utility models. That is, if a researcher uses bipolar attributes and includes only one pole in the evaluation of an object, then the probabilities should be recoded to be symmetric about zero to account for the pole that was left out. This result, however, depends critically on the validity of the assumptions which are discussed in the next section.

THE VALIDITY OF THE ASSUMPTIONS

The first two assumptions require that the subjective probabilities follow the axioms of probability. In a series of experiments Wyer (1974) provided evidence for the logical consistency of subjective probability judgments. Probabilities were measured on an 11-point scale ranging from extremely improbable (0) to extremely likely (10). While the experiments did not directly test (3) and (4), they tested more complex syllogisms that are true only if these assumptions hold. Thus these experiments provide support for the argument that subjective probabilities can be seen as being derived from mutually exclusive and exhaustive events.

It should be emphasized that these assumptions about probabilities are fairly weak. Notice that they do not require that the probabilities be independent across attributes. For example, a car that is perceived as comfortable may be more likely to be perceived as reliable, but the use of the truncated model is not damaged by these correlated attributes. This is a somewhat surprising result, but quite fortunate since most attributes are perceptually correlated.

By contrast, the use of the truncated model does assume the independence of utilities across attributes. Thus, assumption (5) requires that the value of two components together be proportional to the sum of their values separately. It is possible to think of cases where such additivity would not hold. For example, the perceived attractiveness of the grill section of a car might interact with the perceived attractiveness of the hood, so that if one aspect is unattractive the value of the other aspect is attenuated (e.g. the Edsel). Generally, however, the utility of components can be considered additive with little loss in accuracy. For example, in the current case, the __value__ of reliability would not be expected to change appreciably depending on the level of comfort. Furthermore, decompositional models have typically not found the deviation from additivity to be significant even in cases where such deviance was expected (Huber, 1975). Paradoxically, it is the very robustness of the additive model that has lead some to question whether it provides a distorted description of the decision process (Anderson, 1971). However, from a predictive standpoint there can be little doubt that an additive model accounts for a vast majority of the variance. Thus the truncated model can be expected to be predictively quite accurate even if it fails to account for expected nonadditivity of the component values.

Of the assumptions, the bipolarity of utilities is the most problematic. It asserts that the value of attaining an attribute has the opposite sign but same value as not attaining it. It is similar to the familiar constraint in additive models that the sum of a factor's values equals zero. Since in the present case each attribute is assumed to have just two values for its sum to equal zero one must be the negation of the other.

However, it is an empirical question whether, in fact, assumption (6) is justified. Bettman, Capon and Lutz (1975), and Lutz (1976) have provided some support for the assumption in that probabilities that are coded in a bipolar fashion fit data better than those that are coded in a unipolar fashion. A more complete test requires one to estimate the full model and then see which assumptions best lead to approximation of the full model by a truncated version. It is likely that certain attributes will require different rescaling. For example, the attribute "perfect repair record" might have a very high partworth value while the attribute "not perfect repair record" could not be expected to have a correspondingly high negative value. By contrast, the value for "reliable" might reasonably be considered the negation of "not reliable." Thus it is likely that the truncated model will only work for certain kinds of attributes. For other attributes the full model or even more complex models such as Ahtola's (1975) are needed.

SUMMARY

The purpose of this note has not been to test the assumptions of the model, although in the light of the inconclusiveness of testing the predictive accuracy of competing attitude models, it might be desirable for attitude researchers to focus on and test the assumptions instead of the overall predictiveness of their models. Rather, its purpose has been to argue that given certain reasonable assumptions, the Fishbein model yields results equivalent to the full expected utility model only if the probabilities are rescaled to be symmetrical about zero. Thus the Fishbein scaling of the association measure is not in conflict with either the axioms of probability or the expected utility model but actually follows from them.

REFERENCES

Oli T. Ahtola, "The Vector Model of Preferences: An Alternative to the Fishbein Model," __Journal of Marketing Research, __12(February, 1975), 52-9.

Norman Anderson, "Integration Theory and Attitude Change," __Psychological Review__, 78(May, 1971), 171-206.

J. Bettman, N. Capon and R. J. Lutz, "Cognitive Algebra in Multi-Attribute Attitude Models," __Journal of Marketing Research__, 12(May, 1975), 151-64.

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

Joel Huber, "Predicting Preferences on Experimental Bundles of Attributes: A Comparison of Models," __Journal of Marketing Research__, 12(August, 1975), 290-7.

Richard J. Lutz, "Conceptual and Operational Issues in the Extended Fishbein Model," __Advances in Consumer Research__, v. III, Beverlee Anderson, ed., Association for Consumer Research, 1976.

Robert S. Wyer, __Cognitive Organization and Change: An Information Processing Approach__ (Potomac, Maryland: --Lawrence Erlbaum Associates, 1974).

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