Ideal Point Models of Preference

ABSTRACT - Ideal point models possess an intuitive appeal which has contributed to their widespread acceptance by those modeling consumer behavior. This paper presents several technical suggestions for improving the models, but argues that conceptual problems severely limit their use, particularly in new product applications.

Citation:

Joel Huber (1976) ,"Ideal Point Models of Preference", in NA - Advances in Consumer Research Volume 03, eds. Beverlee B. Anderson, Cincinnati, OH : Association for Consumer Research, Pages: 138-142.

Advances in Consumer Research Volume 3, 1976      Pages 138-142

IDEAL POINT MODELS OF PREFERENCE

Joel Huber, Purdue University

ABSTRACT -

Ideal point models possess an intuitive appeal which has contributed to their widespread acceptance by those modeling consumer behavior. This paper presents several technical suggestions for improving the models, but argues that conceptual problems severely limit their use, particularly in new product applications.

INTRODUCTION

In an ideal point model, preference for a stimulus is represented as being inversely related to its distance from an ideal point within a stimulus space. This space is defined by the attributes of the stimulus set so that differences in preferences can be seen as the result of differing responses to those attributes. In a marketing context, the ideal point model provides an appealing geometric metaphor which can be used for defining new products, repositioning old products, and determining 'benefit" segments who desire similar attributes in a product.

In spite of its intuitive appeal, the concept has questionable validity. Predictive studies by Jones and Pessemier (1974), dealing with text books; Braun and Srinivasan (1973), on grooming products; and Green and Jain (1972), on dessert items, have produced predictions better than chance, but not much. This paper considers two reasons for the disappointing performance of the model. First, there are technical problems with the routines used to estimate the ideal points. Second, and more problematic, conceptual problems with the model render it inapplicable in a large number of situations, and particularly for the definition of new products.

Of concern here is the application of ideal points to scales of preference rather than the probability of choice. Preference measures have a stability lacking in choice probabilities. For example, the addition of Pepsi Cola to a product set would not be expected to change significantly the preference score for Coke. By contrast, the probability of choosing Coke might change radically. There are two reasons for this change. First, the existence of any new product in the product set is likely to reduce the probability of purchase of all brands; and secondly, because of its similarity to Coke, the addition of Pepsi is likely to have greater effect on the choice of Coke than other less similar brands. Thus, choice probabilities depend not only on the distance of a stimulus from an ideal, but also on the positions of the other stimuli in the choice set. A preference model is a necessary stage in a choice model, but not sufficient in itself.

This paper considers several routines for determining product spaces and positioning individual ideal points within those spaces. Implementation problems are then discussed, followed by a consideration of the conceptual problems with the models. An attempt is made to show why ideal point models have not, and probably will mot, be very effective modeling preference.

METHODS OF IMPLEMENTATION

This section is limited to the feasibility of placing individual ideal points in a space that applies across individuals. There are many ways to derive such a space. Following modifications of Coomb's (1964) unfolding paradigm, a joint space of stimuli and subjects can be derived from preference orderings. Alternatively, ideal points can be fitted into a space that already exists. Such spaces can have either a psychological or physical basis. Psychological spaces can be derived from similarity data using techniques such as TORSCA (Young and Torgerson, 1967), or INDSCAL (Carroll and Chang, 1970). Similar geometric representations can be produced through either the use of discriminant (Johnson, 1971) or principle components analysis (Morgan & Purnell, 1969) of the subjects' ratings of the stimuli by attributes. McKeon (1962) has shown that these apparently different methods are, under reasonable assumptions, theoretically equivalent. Furthermore, empirical work (Jones and Pessemier, 1974; Huber, 1973; Moore, 1974) indicates high correspondence between the spaces produced by alternate methodologies.

Several techniques are available to position ideal points in a defined space such that the closer a stimulus is to an ideal point, the more it is preferred. PREFMAP (Carroll, 1972) fits a quadratic function to the preference surface, the mode of which is the ideal point. The program assumes that preference is inversely proportional to the square of the distance from the ideal point. There are four phases to the program. Phase I allows the space to be rotated and differentially stretched for each subject. Phase II allows a differential stretching of the axes but no rotation. Phase III assumes no differential stretching and results in circular isopreference curves about the ideal points. Finally, Phase IV assumes a vector model which can be interpreted as placing the ideal point an infinite distance from the space. (See Green and Carmone, 1970, for a fuller explanation of this perspective.) The four phases are, in fact, progressively restricted quadratic regression, until in its final phase the coefficients of the squared terms are assumed to be zero and the result is simply linear regression. The phases form a linear hierarchy of models and the routine provides F-tests to ascertain whether the additional parameters of the more complex phases produce greater improvement in fit than could be expected by chance.

PREFMAP has a nonmetric option which allows the preference data to undergo a monotone transformation such that the fit of the linear version will be maximized. Under this option, the statistical tests comparing the fit of the models become descriptive at best. Further, unless one's preference data is metrically very weak, there is evidence (Huber, 1975) that the use of the nonmetric option reduces predictive accuracy simply because less of the available information is used.

A second routine that can be used to position ideal points in a space is LINMAP (Shocker and Srinivasan, 1974). It derives from the same assumptions as Phase II of PREFMAP: preference is inversely proportional to the square of the distance from the ideal point in a space that allows differential stretching of the axes for each subject. Instead of using quadratic regression to determine the position of the ideal points, LINMAP uses linear programming. This approach allows constraints to be put on the weights of the axes so that they are not negative. A negative weight implies the existence of an anti-ideal so that the modal point of the quadratic surface is the least, rather than the most, preferred. The ability to avoid anti-ideals is advantageous in two senses: (1) ideal points with negative weights cannot be clustered with those with positive weights thus making segmentation by clustering ideal points more difficult, and (9) anti-ideal points result in the counter-intuitive result of increasing marginal returns--the more one moves away from an anti-ideal along a given dimension, the more important that dimension becomes relative to other dimensions.

A second potential advantage of LINMAP is that it estimates the position of an ideal point directly. By contrast, PREFMAP uses least squares quadratic regression to estimate the preference surface and then takes a ratio of estimated parameters to arrive at the location. of the ideal point. Since the estimated parameters are not independent, their ratio is not unbiased. Furthermore, the position of the ideal point may be very sensitive to small changes in the denominator. This, in fact, occurs when the ideal point is positioned outside of the range of the stimulus space and the preference structure approaches that of a vector model. In such cases, the position of the ideal point has been found to be quite sensitive to small changes in the input data. More work needs to be done dealing with this problem and in assessing the relative effectiveness of the routines. Empirically, a comparison of the predictive effectiveness of LINMAP against PREFMAP would be valuable; while analytically, the statistical properties of the solutions of either of these routines have still to be worked out.

PROBLEMS OF IMPLEMENTATION

The foregoing has discussed a number of ways to form a stimulus space and to position ideal points within the resultant space. This section discusses limitations in these algorithms and suggests procedures that will minimize these problems. In particular, it considers the assumption of symmetry of preference about the ideal point and the problem of multiple ideal points. These are what might be called "technical problems" in that it is possible to specify the form of a correct solution within the context of the ideal point framework. Later, conceptual problems with the ideal point model are considered. These are far less tractable. Virtually all schemes for positioning ideal points in attribute space assume that, with respect to any attribute, deviation from either side of the ideal point has equal affective consequences. Pessemier's (1972) model is a notable exception but has the difficulty of requiring from respondents explicit judgments of ideal levels on each attribute. The reason implicit positioning of ideal points has assumed symmetry stems from considerations of degrees of freedom and programming ease. Unidirectional distance in a space of any dimensionality can be simply stored as one number. Directionality, on the other hand, requires attribute-by-attribute information since positive and negative directions only make sense within the context of an individual attribute. In terms of degrees of freedom, any model that allows differential stretching on either side of an ideal point requires at least one degree of freedom for each dimension.

One of the main problems with the assumption of symmetry is that it is so often counter-intuitive. Consider the ideal temperature for hot tea: being twenty degrees under the ideal point is certainly not equivalent to being twenty degrees over. One may be unpleasant, but the other is painful. Similarly, displeasure with vermouth in a martini might be far more sensitive to too much vermouth than too little. If symmetry is assumed, and it does not occur, systematic biases can occur in the location of the ideal point so that it no longer represents the most preferred point in the preference distribution. The problem is analogous to the distinction between the mode and the mean in skewed probability distributions. In the hypothetical case given in Figure 1, the assumption of symmetry results in an ideal point with less vermouth than is most preferred. This divergence is most easily observed by the lack of fit if the symmetrical model is moved so that its mode coincides with the mode of the actual surface.

FIGURE 1

EFFECT OF SKEWED PREFERENCES ON SYMMETRICAL IDEAL POINT MODEL

There are several possible solutions to this problem. If few dimensions are used, the best way to determine an ideal point is simply to look for it. SYMAP (Shepard, 1970) is a computer routine that provides such a perspective in two-dimensional space. Designed to produce geophysical contour maps, the routine estimates the height (preference) as a function of the level and trend of the surrounding points. The effect is similar to two-dimensional smoothing adjusted for trend. The mode of the surface is easily determined by inspection.

In more than two dimensions, other techniques are required. One possibility is to modify a technique such as PREFMAP to weigh inputs to the quadratic regression in terms of their preference. This would mean that points closest to the ideal point would have the greatest weight in determining its position. The generated ideal point would then be closer to the mode of the response surface.

A second possibility involves first smoothing the preference data to reduce the effect of random fluctuations. The techniques of SYMAP generalized to many dimensions could be used for such a task. once the random noise has been minimized, the method of steepest ascent or ridge analysis (see Myers, 1971) could be used to find the modal preference values for an individual.

Such a procedure might also be effective in determining whether there are multiple peaks within the space. Consider the choice of color for an automobile. Relatively small differences in hue, saturation and brightness can be expected to evoke different connotations and therefore, different preferences. Fairly specific shades of different colors might form local preference peaks. For a manufacturer offering a limited number of colors, it is important to estimate the distribution of these peaks, both within and across individuals. Local search routines on a smoothed preference surface could determine such local minima for individuals and these could then be aggregated across individuals.

To summarize, this section has dealt with the possibilities of asymmetry of preference and multiple peaks within a stimulus space. The notion of symmetry is critical to the ideal point model if one conceptualizes the ideal point to be the mode or peak of the preference distribution. Under such circumstances, routines can be modified to provide better estimates of the mode of the distribution. In the same way, the existence of multiple preference peaks, within a stimulus space, was shown to result in biased estimates unless an attempt is made to account for the different peaks. The next section considers those conditions under which, even given the best possible implementation procedures, ideal point models cannot be expected to work.

CONCEPTUAL PROBLEMS

The first requirement for an ideal point model--or any model-- to represent preferences in space, is that preferences be causally related to the underlying space. It is easy to think of examples where this would not be the case. Tucker (1964) shows that strong brand preference can develop for essentially identical brands of bread. There, the physical space would not predict preferences, while any psychological space would predict preferences only to the extent that a halo effect distorts attribute ratings (see Beckwith and Lehmann, 1975). For example, a psychological space might not be expected to predict the preferences of a man who buys a brand of automobile simply because his family has always bought that brand. Habit and familiarity are functions of the subject's past experiences rather than functions of the stimuli themselves. Where such components largely determine preference, a joint space of stimuli and subjects should not be used to specify preference for new brands simply because causality runs from usage to judged attributes rather than the other way around. Thus, any attribute space can be expected to be ineffective predicting preferences to the extent that preferences are based on individual habits or differential learning. Since most frequently purchased consumer goods fit into this category, it is perhaps not surprising that attributes have done such a poor job predicting preferences, except as a result of the halo effect.

Even given that the space is relevant, ideal point models have further problems. This section attempts to show that ideal point models are basically incompatible with vector models. Here, a vector model is one where preference never reaches a peak with respect to a given attribute. For example, more comfort is always preferred to less comfort, other things being held equal. An ideal point model, by contrast, assumes that beyond a certain point, preference falls with more of the attribute.

It is well-known that an ideal point model is equivalent to a vector model if the ideal point is an infinite distance away from the product set (Green and Carmone, 1970). Although the distance measure becomes undefined in such situations, the directional cosines of the lines from the centroid of the stimuli to the ideal do not. What is less well-known is that in a mixed mode situation, where some of the dimensions are monotone while others are singled peaked with respect to preference, both the distances and the directional cosines are undefined. This is not a problem empirically. In the quadratic regression of a routine such as PREFMAP, the isopreference curves are simply parabolas, and the probability of the denominator going to zero with real data is very low; but it does mean that the appealing geometri6 metaphor of the ideal point breaks down in this case. Furthermore, this consideration limits ideal points in the analysis of a very critical marketing variable, price. To the extent that a lower price is always preferred, price is monotone with preference; thus, the ideal point is undefined where price is one of the attributes.

If the foregoing argument is accepted, it implies that ideal point models should not be used where price, or any other vector attribute, is determinant. Ideal points might be' appropriate in formulating the mix of ingredients in a candy bar or the ideal image for a cigarette since for such items price would not be expected to vary as a function of a brand's position in the product space. In such models, price is implicitly assumed to remain constant. Unfortunately, such products (e.g. candy bars and cigarettes) are likely to be those where learning and habit are important determinants of preference. To the extent that this problem occurs, neither the ideal point, nor any other spatial representation, is likely to be very useful.

THE PURCHASE IDEAL: IDEAL POINT WITHIN A VECTOR FRAMEWORK

Having spoken of the incompatibility of joint vector and ideal point models, consideration must be given to a model of Pessemier's (1972) that derives ideal points in a space defined primarily by attributes which are monotone with respect to preference. This interior ideal point is defined by asking subjects to specify the level of attributes of the product they would most likely purchase. Such a purchase ideal is often an internal point in the attribute space reflecting the trade-offs and constraints that exist in actual purchase decisions. While such a model will result in approximate solutions, it will be shown that the divergence of the model from reality severely limits its use on many product sets and, in particular, for the evaluation of new products.

The purchase ideal is analyzed by positing a simple economic model of the purchase decision. Et will then be possible to highlight those situations where the purchase ideal results in improper predictions. To make the discussion concrete, we shall consider choice among automobiles and assume that the dimensions of sportiness, luxury and price Summarize all that is relevant in the purchase decision. Preference for purchase can be represented as:

Preference = F₁ (Sportiness, Luxury, Price)   (1)

Here, it is assumed that the partial derivatives of F₁ are always positive, though decreasing, for sportiness and luxury, and negative for price. Two models of the purchase ideal will be explored which account for Equation 1. The first of these positions the ideal in a space of only sportiness and luxury, while the second also includes price as an independent dimension.

FIGURE 2

PURCHASE IDEAL GIVEN IMPLICIT PRICE DIMENSION

Figure 2 provides an illustrative purchase ideal for automobiles in a two-dimensional space of sportiness and luxury. The internal ideal occurs at the point where the diminishing returns from improved sportiness and luxury are simply not worth the additional cost. Since price is not explicitly part of the Figure 2, this interpretation requires that price makes up an implicit third dimension which diminishes the overall value of a purchase as one moves beyond the purchase ideal. This dimension is contained in the dimensions of luxury and sportiness so that a given level of these attributes implies, on average, a given price. Or,

Price = F₂ (Sportiness, Luxury).   (2)

Equation 2 enables one to predict purchase preference in Equation 1 without using price. Its difficulty lies in the fact that even small error in Equation 2 can be expected to result in poor predictions. Consider a group of automobiles with approximately equal sportiness and luxury. These will have different prices which will strongly affect their probability of purchase. Yet, by the model in two dimensions, they would all be equally preferred for purchase. The implications for management might be even more misleading. Since the model assumes that a given level of attributes imply a certain price, any new product that does not conform to that implicit relationship will result in incorrect prediction. Specifically, a product with new lower price will not be conceived as superior to the original product with the same attributes.

This problem of causal misdetermination is general to many multivariate models. It occurs whenever one is working with a reduced space where (highly correlated) attributes are grouped in one factor. The predictions for a given product are only accurate insofar as that product is typical of the interrelations in the product set.

The problem can be mitigated, but not solved, by allowing each dimension to be independent of the others. In the automobile example, this can be illustrated by allowing price to be an independent dimension. For expositional purposes, sportiness and luxury are aggregated into a new dimension, "quality", and the decision is pictured as in Figure 3.

FIGURE 3

PURCHASE IDEAL GIVEN EXPLICIT PRICE DIMENSION

The production possibility frontier expresses the greatest perceived quality for a given price. The ideal point is where this curve is just tangent to the indifference curve between quality and price. This point, while being the most preferred from the available set, is not the "ideal point" in the normal sense of the phrase. Points beyond the ideal are not less preferred, but merely perceived to be infeasible. Furthermore, given the economic model postulated, a routine such as PREFMAP would not determine an ideal at the point shown, but rather a vector indicating the direction preference increases most rapidly. While this vector is clearly the appropriate model of consumer behavior in this hypothetical situation, it should be pointed out that the high correlation between price and quality will lead to a degree of indeterminacy with respect to the preference vector. This can be avoided by providing subjects with hypothetical choices on both sides of the production possibility frontier so that the dimensions of price and quality are more nearly orthogonal.

To summarize, then, internal ideal points can result in a reduced space of vector attributes where correlated attributes with opposite affective consequences (such as price with sportiness and luxury) are combined. Such a purchase ideal can result in misleading managerial recommendations to the extent that the reduced space slightly misrepresents potential offerings. By making each dimension independent, the internal ideal is seen to be an artifact of the reduced space. In the full space, the vector model, constrained by the production possibility frontier, is the appropriate model which does not result in incorrect preference predictions for potential new products that are atypical of the present set.

SUMMARY AND CONCLUSIONS

The paper has dealt with four problems encountered with the representation of individual preference as an ideal point in a geometric stimulus space.

1. Lack of symmetry of preference about the ideal.

2. Multiple ideal points for an individual in a given space.

3. Preference due to subject-specific attributes such as habit or past behavior rather than product-specific attributes such as density, or masculinity of image.

4. Situations where some or all of the attributes are monotone with respect to preference.

The first two can be considered technical problems, at least it is possible to specify the kind of solution that is necessary to solve the problem. The second two are conceptual problems, and more serious. They place a restriction of the applicability of ideal point models. The first restriction involves the inapplicability of ideal point or any spatial model for those product sets where habit or past usage are important determinants of preference. Because of the halo effect, such models may fit existing products fairly well; but since causality runs in the wrong direction, predictions on new products will generally be wrong. The second restriction involves the incompatibility of the ideal point model with one or more vector attributes. This means that many important attributes, such as price or perceived quality, simply cannot fit into the ideal point paradigm.

With such restrictions, it is difficult to envision many situations where an ideal point model would be useful. One wonders whether, in the light of the last two problems, it is worth fixing the first two.

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