# Mathematical Considerations in the Use of Linear Attitude Models

##### Citation:

Frederick W. Winter (1972) ,"Mathematical Considerations in the Use of Linear Attitude Models", in SV - Proceedings of the Third Annual Conference of the Association for Consumer Research, eds. M. Venkatesan, Chicago, IL : Association for Consumer Research, Pages: 790-795.

Proceedings of the Third Annual Conference of the Association for Consumer Research, 1972      Pages 790-795

MATHEMATICAL CONSIDERATIONS IN THE USE OF LINEAR ATTITUDE MODELS

Frederick W. Winter, University of Illinois

[Frederick W. Winter is Assistant Professor of Business Administration. University of Illinois at Urbana-Champaign.]

Expectancy value models have made a significant impact in the analysis of consumer behavior; the intuitive appeal of the model has led to a broad range of empirical research findings many of which suggest unresolved issues. The basic expectancy value model has been formulated in a variety of ways (Rosenberg, 1956; Fishbein, 1963) and relates the individual's tendency to act as a function of two components:

1.  the strength of expectancy that a given act will result in a set of consequences

2.  the value of the consequences to an individual

The strength of expectancies are generally multiplied by the value of the consequences and summed over all the salient attributes. It is not the purpose of this paper to review the history of expectancy value models as overall summaries of the research efforts and issues are readily available (Wilkie and Pessemier, 1972; Cohen, 1972). What will be questioned is the operational framework in which consumers' perceptions are translated into the individual's attitudinal measure.

THE BASIC CONCEPT

In mathematical language, the researcher is asking the individual for his perceptions along some scale, x, and the functional relationship between x and utility or value, g(x). In simple terms one might first ask, "how economical is the Volkswagen?", and "what is your utility for the level of economy offered by the Volkswagen?". Undoubtedly more than one attribute is salient to the individual, and, thus, xi represents the perception along the ith dimension and the "utility' function gi(xi) expresses the relation between value and xi.

In the majority of research applications xi is expressed better as a continuous variable as opposed to a dichotomous variable, consequently considering Volkswagen economy over many outcome or consequence states is probably more valid than the two-state representation of "economical" or "uneconomical." As with most perceptual processes, it is unlikely that an individual's perceptions along one attribute can be summarized with a point estimate. Uncertainty is one example where a probability distribution is needed to adequately describe the perceptual framework.

EXPANSION OF THE BASIC CONCEPT

Figure 1 can serve as an example of a consumer's perception of Volkswagen economy. Researchers have, in the past, asked for one of two types of responses:

1)  "on a 1 to 7 scale, how economical is the Volkswagen?"

2)  "how improbable (1) - probable (7) is it that volkswagen is economical?"

The first question presumably asks for a modal or mean estimate from the individual's perceptual distribution; this is designated as xi,m in Figure 1. To respond to the second question the individual must first designate a boundary point that divides the economy continuum into the uneconomical-economical dichotomy. This boundary point is designated as xi,b, and the question asks for the probability that Volkswagen is greater than xi,b [i.e., EQUATION]. Whether or not consumers' responses correspond to their distribution is a question to be answered by future research efforts.

The second component of the expectancy value model is a functional form of Xi j to relate consequences to utility gi(xi ) A monotonic relation between utility and perception is shown in Figure 2. Clearly it is possible to imagine a variety of circumstances where the relation would not be monotonic (e.g., temperature or sweetness of coffee, power in an automobile, etc.)

Utilizing the information provided by question 1, the estimate of VW utility if a modal response is assumed is:

(1a)   EQUATION

where n = total number of salient attributes.

A mean response results in:

(1b)   EQUATION

Question 2 responses yield:

(2)   UVW = (Expected Utility of xij where j > b) (probability xi > xi,b)

An alternative measure might be to consider an expected utility model that considers probability multiplied by utilities over all possible attribute values. A model of this nature is currently undergoing testing (Ahtola, 1971). Using current notation: n X

(3)    EQUATION

Although it is impossible to specify (1), (2), or (3) as being more desirable at this time, it is obvious that different perceptual measures will result in different attitude measures. Equation (3) does appear to be the more appealing measure since (1) and (2) eliminate some of the information. It can be seen that (1a) and (3) will yield comparable results only when:

gi(xi,j) = constant, for all j

or, f(xi,j) = 0 when j = m            (total certainty)

= 1 when j = m

or some less than obvious sets of f(xi,j), gi(xi,j)

The same conditions result in identical (1b) and (3) measures with the additional possibility when the utility function is linear:

if gi(xi,j) = a + b (xi,j)

(3)    EQUATION

Equations (2) and (3) yield similar measures when:

EQUATION

or, gi(xi,j) = 0 for all j < b

or, some less than obvious sets of f(xi,j), gi(xi,j)

Equation (3) seems to correspond best to the spirit of the expectancy-value concept by considering probabilities and utility over all possible ranges of xi,j values; only in selected cases will (1a), (1b), and (2) result in identical measures to (3). It will be important in future research efforts to not only empirically compare the alternative models, but to also determine why different results occurred. A natural starting point is to determine how consumers respond to alternative questions and how these responses relate to the perceptual distribution of Figure 1.

The lack of correspondence between (1a), (1b), (2), and (3) is particularly evident in the study of attitude change. Figure 3 illustrates what might be pre-post measures of the perceptual space. In this example, mean or modal [(1a) and (1b)] perception measures have not changed. Boundary probability measures [(2)] reflect change that is due to uncertainty reduction alone. Equation (3), probability measures over all possible consequences, utilize the entire information that mean responses haven't changed while uncertainty measures have.

FIGURE 1

PERCEPTUAL REPRESENTATION OF VW ECONOMY

FIGURE 2

UTILITY FUNCTION OF xi,J

FIGURE 3

PRE-POST MEASURES OF THE PERCEPTUAL SPACE

SUMMARY

There is no one correct way to measure perceptions and utility-values in expectancy value models. It is important, however, to be aware that different measures provide different information about potential consumer behavior. Hopefully, this view of the measures will stimulate empirical research to consider not only which measures may be applicable but also to understand why the measures may differ.

REFERENCES

Ahtola, Olli T. "An Investigation of Cognitive Structure Within Expectancy-Value Response Model," dissertation proposal presented at College of Commerce and Business Administration, University of Illinois, 1971.

Fishbein, Martin. "An Investigation of the Relationships Between Beliefs About an Object and the Attitude Toward that Object," Human Relations, 1963, 16, 233-40.

Rosenberg, Milton, J. "Cognitive Structure and Attitudinal Affect," Journal of Abnormal and Social Psychology, 1956, 53, 367-72.

Wilkie, William L. and Edgar A. Pessemier. "Issues in Marketing's Use of Multi-Attribute Attitude Models," Paper No. 365, Institute for Research in the Behavioral, Economic, and Management Sciences, Krannert School of Industrial Administrations Purdue University (August, 1972).

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##### Authors

Frederick W. Winter, University of Illinois

##### Volume

SV - Proceedings of the Third Annual Conference of the Association for Consumer Research | 1972

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