# Multidimensional Psychophysics: an Overview

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Vithala R. Rao and Gordon W. Wilcox (1974) ,"Multidimensional Psychophysics: an Overview", in NA - Advances in Consumer Research Volume 01, eds. Scott Ward and Peter Wright, Ann Abor, MI : Association for Consumer Research, Pages: 154-165.

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http://acrwebsite.org/volumes/5709/volumes/v01/NA-01

INTRODUCTION

Over the last several years, there has been an increasing application of multidimensional scaling methods (MDS) to research problems in marketing and consumer research. It is perhaps fair to say that contribution of these methods to the practice and theory of consumer behavior is impeded owing to the lack of resolution of two basic issues. These relate to the determination of relationships between measurable stimulus dimensions and subject judged measures of similarity (perception) and preference (evaluation) and the ability to design a new stimulus which will produce a prespecified measure of similarity or preference. Lack of progress in these two areas, we feel, is attributable to at least three factors: (a) relative unfamiliarity of consumer researchers with the traditional psychophysics and its potential when combined with the methods of multidimensional scaling; (b) need to digest the ramifications of the theory and foundations of measurement, which have been only recently developed; and (c) relative unavailability of the MDS algorithms for applied researchers.

This paper is an attempt to review some of the current thinking about psychophysics with emphasis on the multidimensional aspects of psychological scaling. Although this paper is not an attempt to present a comprehensive review of the substantive studies in the area, we will attempt to lay out the goals of multidimensional psychophysics, some research issues, and selected methods of analysis.

Our discussion falls somewhat naturally to a brief consideration of a small part of theories of measurement (Krantz, Luce, Suppes, and Tversky, 1971) deemed to be relevant to the types of numerical representation which may be available to summarize empirical knowledge in this area, and then to a more practical introduction to the scaling procedures currently available for finding numerical representations from several data types. We also suggest ways of finding relationships between objective stimulus measures and the scaling solutions. To begin with, however, we give a precis of the classical (unidimensional) psychophysics with which to contrast the recent developments in multidimensional psychophysics.

CLASSICAL PSYCHOPHYSICS

Fechner (1860) apparently thought that quantitative sensation magnitudes are evoked by intensive and extensive physical stimuli and that this is a self-evident fact of our perceptual experience. The problem he investigated was to scale the sensation magnitudes in terms of the physical scales of intensity (e.g., pressure, amplitude, molar concentration) and extent (e.g., length, duration). In principle two methods, later labeled "indirect" and "direct," seemed open to him. The first method, "indirect method" which he adopted, was to measure sensation in terms of the change in a physical variable necessary to produce equally often noticed sensation differences. These scaling methods require that the physical continuum be finely divisible so that the interval between two stimuli which are "just noticeably different," (that is, correctly judged as different 75% of the time) may be determined.

At the time of Fechner's investigations, Weber had already discovered that for many physical quantities (it appeared that) the necessary increment in the physical variable was proportional to the magnitude of that variable (Weber's Law). Fechner proposed that equally often noticed sensation differences are equal increments in sensation (Fechner's Principle) and hence can be used to establish a unit of measurement for sensation magnitude. When such units are added up by being placed end-to-end they constitute a scale of sensation. And if Weber's Law is assumed, the scale becomes a logarithmic function of physical intensity or extent. Statistical procedures are available to obtain the "best" scale from various response methods used in making the judgments (Torgerson, 1958, Bock and Jones, 1968). This is an example of an unidimensional "psychophysical function" and is known as the now famous Weber-Fechner Law.

The other method of psychophysical scaling considered by Fechner, and later called the "direct method," would be to obtain the differences in physical magnitudes necessary to produce equal ratios of sensation magnitude. However, he felt it was not possible to measure sensation magnitudes directly, and so rejected this possibility in favor of defining such magnitudes in terms of his aforementioned Principle.

Fechner's negative views (S.S. Stevens called them myopic) on the observability of sensation magnitudes were not shared by all psychophysicists: the assumption that a subjective report of equal ratios of intensity indicates equal ratios of sensation magnitude (a so-called Mapping Assumption) leads to the popular Power Law form of the psychophysical function (Luce, Bush, & Galanter, 1963, p. 291). It has recently been shown that a mapping assumption is not necessary as a justification for the major empirical results of scaling with direct methods, including the Power Law. A weaker set of assumptions requiring that "ratio" judgments preserve the order of a relation on pairs of sensations due to the psychophysical process is sufficient (Krantz, 1972).

Direct response methods in psychophysics such as magnitude estimation, "ratio" judgments, bisection, etc., are of particular interest to psychologists and behaviorists because of the success these methods have had in predicting the psychophysical functions in cross-modality matching [Cross-modality matching involves a direct comparison of "sensation magnitudes" between different stimulus types, e.g., a response of the form "this light is as bright as that noise is loud" or "I love you just as much as (I think) you love me." (See Stevens, 1972 for a summary of his views.)] (Stevens, Mach, & Stevens, 1960). These positive empirical results on cross-modality matching tend to increase our confidence that the subjective reports by observers reflect genuine relationships in their psychophysical processes.

In spite of the success of direct response methods for stimulus evaluation, most work involving the psychophysics of sensory continua has had a performance orientation which has depended primarily on indirect methods. The journals in Psychology abound with articles on discrimination of intensive stimuli under various conditions of degradation of the stimulus ("noise") and placing the observer in contexts which increase the difficulty of a simple task, such as having few signals over long periods of time (vigilance), and having many confusable stimuli among which a choice must be made (recognition). Much less effort has been given to complex stimuli described multidimensionally. We turn to this more general problem in psychophysics which involves the determination of multidimensional mappings of stimulus properties into attribute dimensions in psychological space.

PSYCHOPHYSICS AND SCALING

In a discussion of psychophysics we start with certain observable quantities, such as the physical properties of stimuli and the responses which observers use for the evaluation of those stimuli. We wish to use our representation of these overt variables together with data in an inferential process to discover a representation of internal variables or covert process which serve to organize and simplify our description of the relationship between the observable quantities. To this end, there would appear to be a meta-theoretical organization imposed upon the solutions to the problem of finding an internal representation for stimuli. A block diagram for this organization is presented in Figure 1. The inference to a psychological representation space for stimuli (,) from a physical representation space for stimuli(P) characterizes psychophysics, whereas the representation of stimuli in z is the main endeavor of psychological scaling.

In classical psychophysics separate representation spaces for responses and internal stimuli are not provided. More recently, particularly as a result of the development of theories of choice behavior (Luce, Bush, Galanter, Vol. I., 1963), there has been a greater emphasis on response biases and making corrections for these biases during the construction of a stimulus representation space.

Psychophysics and scaling can be viewed as inseparable aspects of the problem of obtaining a transformation from stimulus to response with the psychological representation space for stimuli serving as an intervening construct. Psychological scaling is also performed on responses to stimuli defined in terms of psychological attributes. To the extent that independent scaling has been performed on such psychological attributes (even at the nominal level such as brand names) they may be considered formally equivalent to having specified physical values for the stimuli. In practice, the attributes may either be totally psychological (e.g., sportiness for a car or crunchiness for a cereal) or be mixed [Of course, there is a possibility that an investigator would like to assess psychological attributes in parallel with the stimuli. We, however, will not address ourselves to this "conjoint" scaling problem, nor to the less serious question of a possible dependence between the psychological and physical attributes which are represented in P. The question of dependence among stimulus attributes is not unique to the mixed case.] including both physical and psychological ones. In any case, the judgments of the subjects will be utilized in determining the "psychological" dimensions on which responses may have been based. The derived psychological dimensions are related to the attributes on which the stimuli are described. These relationships are in fact the multidimensional psychophysical functions. However, it must be kept in mind that the invariance of the psychological representation space, and hence the psychophysical functions, is a major empirical concern in any practical application of these methods.

MULTIDIMENSIONAL PSYCHOPHYSICAL FUNCTIONS

From the foregoing discussion it should be-clear that the goal of multidimensionaL psychophysics is that of determining the specific transformation functions used by a subject (judge) or a group of subjects (judges) in giving responses to a set of stimuli described on a number of attributes.

Formally, the problem can be stated as follows: Let

n = number of stimuli.

q = number of attributes.

P = (P

_{1}, P_{2}, ..., P_{q}) = set off attributes (physical or psychological or mixed) on which stimuli are described.P

_{ju}= value of the j th stimulus on the u th attribute; j=1, 2, ..., n; u=1, 2, ..., q.r = number of "psychological" dimensions derived from the judgments of the subject (r < q).

E = (S

_{1}, S_{2}, ..., S_{r}) = vector of r variables representing the derived psychological dimensions.S

_{jt}= value of the j th stimulus on the t th psychological dimension: j=1, 2, ..., n; t=1, 2, ..., r.

Using this notation, the problem of multidimensional psychophysics can be stated as that of determining the set of functions, (ht; t=1, 2, ..., r) relating the E-variables to the P-variables. That is

E

_{t}= h_{t}(P_{1}, P_{2}, ..., P_{q}); t=1, 2, ..., r. (1)

The equations (1) are an immediate generalization to the univariate psychophysical functions. The functions ht need not be restricted to be linear or additive. Although all forms are possible in scaling, only simple polynomial functions monotonic in their arguments have been thoroughly investigated from the perspective of measurement theory--that is, as to the degree to which one can state the appropriateness of the representation beyond the methods of curve fitting. There is no general guide available to researchers as to which form of a psychophysical function will be most appropriate, although considerations of simplicity, uniqueness, and empirical content should all contribute to the choice.

Noting that the "psychological" dimensions need not be independent of one another, the problem of multidimensional psychophysics can be stated in most general terms as one of determining two sets of transformation functions Gs(i) and Gp(i) such that

G

_{s}^{(i)}(E) = G_{p}^{(i)}(P); i=1, 2, ..., r_{0}, (2)

where r0 is less than or equal to r.

The equations (2) perhaps reach a nearly intractable level of abstraction. They do represent the possibility of a maximal (functionally) independent set of higher order invariances between derived structures and physical attributes. We expect, however, that it is unlikely that meaningful invariances with this degree of generality will be found by sheer numerical methods in the absence of powerful conceptual representations. In our discussion to follow, we will, however, suggest methods for determining linear forms of invariance. With G-functions restricted to be linear. equation (2) becomes

G

_{s}E = G_{p}P, (3)

where Gs and Gp are matrices of suitable transformations. Notice that when Gs is an identity matrix, the equations expressed by (3) are less general than those of equation (1).

The problem of estimating the transformation functions in equations (1) and (3) can proceed (to the extent of available techniques) and , and P are known and when constraints are placed upon the transformations Gs and Gp. The determination of P is always highly context dependent and usually straightforward. Hence, the next section is primarily devoted to a brief review of the scaling methods available for obtaining ,. Review is also made of a variety of experimental methods for obtaining psychophysical judgments. Later, we pick up the problem of determining the psychophysical functions.

EXPERIMENTS FOR MULTIDIMENSIONAL PSYCHOPHYSICS

The researcher in any prototypical experiment on multidimensional psychophysics faces three major decisions. These relate to the construction or development of the stimulus set, nature of responses to be elicited from the subject and method of collecting such data, and methods of analysis of responses for determining the implicit stimulus structure. These decisions, no doubt, depend upon the objectives of the study. Our goal here is to briefly discuss the problems and alternatives available to the experimenter for each of the decision areas in general terms in order to present an overview of the process. We will not cover the problems associated with determination of sample size, physical design of the experiment, and controls and other methods used for reducing bias and errors in an experimental layout. Interested readers may consult standard books on classical psychophysics or better yet read Chapter 2 of Coombs (1964) (where further references are given) and Runkel and McGrath (1972). And for information regarding questions on experimental design see J.L. Myers (1972).

Development of a Stimulus Set

The first question to be considered in this area is whether the stimuli used are real (e.g., brand names of several automobiles) or hypothetical. Even when real stimuli are employed, it would be useful to include some test or replicated stimuli in order to get an idea of the error involved. In either case, the question of significance is the choice of the q specific attributes on which stimuli will be described. As noted before, the attributes can either be physical or psychological or some combination of the two. The method of Kelly's repertory grid [Briefly, this method involves presenting to the subject three of the existing stimuli and asking him to think of a way in which any two of the three are similar to each other and different from the third. After an attribute is elicited in this manner, all stimuli are rated by the subject on this attribute. The question is repeated for the same triple until no additional attributes are derived and the whole process is repeated for another triple of stimuli until an exhaustive set of attributes are identified and the stimuli rated on them. The resulting matrix of ratings is analyzed in order to select the most salient attributes.] (Kelly, 1955) is highly suitable for determining the salient attributes. Once the attributes are selected, the specific levels for each attribute have to be decided. In the case of real stimuli the experimenter has no option but to use the available measures on the physical attributes of the stimuli. However, in the case of hypothetical stimuli, this decision has to take into account the meaningful range and step sizes for each attribute (Stevens, 1972). Once the levels are fixed hypothetical stimuli can be generated from all combinations of attributes on the respective levels with constraints on size of stimulus set and the meaningfulness of the stimulus itself. Some forms of fractional factorial designs (Cochran and Cox, 1957 and R.H. Myers, 1972) can be employed when the total number of all attribute combinations, each representing a stimulus, is too large for the subject's task. For example, when there are four salient attributes each describable at three levels, there are 34 = 81 combinations in all. These numbers quickly become too large for a subject to make comparative judgments. The number can be reduced by selecting combinations according to a fractional factorial which confounds some higher order, hopefully, less important interactions. Another method of coping with the large stimulus set problem is to use a balanced incomplete block design (BIBD) or a partially balanced incomplete block design (PBIB).

Selection of Response Method

An important variable regarding responses is the specification of the situation or scenario under which the subject is asked to make judgments. We will assume that the decision on scenario(s) will be totally guided by the objectives of the experiment. This will be taken as given in the following discussion. While many kinds of responses for a prespecified scenario are possible, three basic types seem to be significant for our consideration. These are similarity judgments (overall or with respect to a prespecified attribute or construct); profile ratings along a set of prespecified dimensions; and preferential judgments. A set of behavioral measures (e.g., endorsement) are not explicitly mentioned since they can be analyzed either as profile data or preferential judgments. Also not included are responses relating the subject to the stimuli in the case of similarities and profiles. These could be obtained, if necessary, by augmenting the stimulus set with an explicit "ideal" stimulus. Several methods (e.g., magnitude estimation, sort, pick, rank order) exist for collecting these data. Interested readers should consult Torgerson (1958), Coombs (1964) and Shepard, et. al. (1972).

Stimulus Scaling

We will assume that data are collected from a sample of N subjects in the experiment. Utilizing the above scheme the experimenter could obtain one of several types of data which for sake of discussion we shall label (A) overall similarity judgments, (B) similarity judgments with respect to a prespecified attribute, (C) profile ratings on a set of prespecified attributes, or (D) preferential judgments.

Invariably, studies of multidimensional psychophysics are conducted with more than one subject. However, situations do arise when data are collected from only one subject or when data from many subjects are aggregated and assumed to represent data from a hypothetical "average" subject. In this situation the methods of deriving stimulus spaces discussed below do not necessarily apply. This special case can be handled by other methods according to the type of response. For data types A and B from a single subject and for data type C for more than one attribute, any one of the two-way scaling methods e.g., KYST (Kruskal et. al., 1973) can be employed to yield a psychological representation of stimuli. The methods of determining psychophysical transformations applicable to the equation (3) are pertinent to this situation. However, in the case of data type C when only one attribute is employed, it is not possible to obtain a multidimensional representation from a single subject's data. The same is the case with respect to the preference judgments. For these two kinds of data, it is appropriate to consider the responses as resulting from a undimensional "stimulus" space S ; i.e., r=l in the above general formulation of psychophysical transformations. The methods for determining the transformations in equation (1) discussed later do apply to this case.

The analysis methods for determining stimulus structure depend upon the kind of data. While a large variety of analytical methods do exist, (Shepard et. al. 1972, Green and Rao, 1972) some suggestions will be made in this paper. These are succinctly summarized in Table 1. A brief explanation of the proposed analysis will follow.

A: Overall Similarity Judgments

This consists of N x n x n matrix of similarity judgments. We propose analyzing this matrix using the INDSCAL model (Carroll and Chang, 1970) which estimates the coordinate values for the n stimuli in r dimensions. The number of dimensions can be decided by judgment using the criterion of the proportion of variance accounted for in the original data. This constitutes the psychological space, E, i.e., the n x r matrix of stimulus coordinates on the r dimensions.

SELECTED METHODS OF STIMULUS SCALING

B: Attribute Specific Similarity Judgments

When only one attribute is used, the data consist of N pairwise matrices (n x n) with the similarity judgments as entries. The three-way matrix can be analyzed using the INDSCAL model referred to above. In the event the data cannot be represented in one dimension, the researcher can select the most salient dimensions. The above method of obtaining stimulus space can be repeated for additional attribute-specific similarity judgments.

C: Profile Ratings Along Prespecified Attributes

Here, the researcher has a matrix of N x n x q ratings on the n stimuli on q dimensions from N subjects. The method of multiple discriminant analysis (Anderson, 1958, and Johnson, 1969) can be used to derive stimulus coordinates on the reduced dimensions. As an alternative, either the method of CANDECOMP or canonical decomposition of N-way tables (Carroll and Chang, 1970) or the three-mode factor analysis (Tucker, 1972) can be used for the purposes of deriving psychological space.

D: Preferential Judgments

This consists of N x n preferences. The matrix can be analyzed using the internal method of preference analysis, namely, MDPREF (Chang and Carroll, 1969) or a multidimensional unfolding method, namely, KYST (Kruskal, et. al., 1973) to yield a matrix of stimulus coordinates on the psychological dimensions.

METHODS FOR DETERMINING PSYCHOPHYSICAL TRANSFORMATION FUNCTIONS

The methods which appear to be appropriate for the determination of transformation functions from the physical stimulus representation space P to the space z of psychological stimulus dimensions fall into two general classes according as whether equation (3),

G

_{s}E= G_{p}P,

is deemed adequate, or whether one must turn to the more general case represented by equation (1)

E

_{t}= h_{t}(P_{1}, ..., P_{q})

In equation (3) setting the Gs matrix equal to an identity matrix of operators is consistent with the spirit of classical psychophysics and solutions to the equations in that form would be expected to have a better chance of being interpreted in a meaningful way. Be that as it may, we shall consider methods for equation (1) first.

Decomposable Functional Transformations

Some methods are available with which to attack equation (1) when certain simplifying assumptions are made. By analogy to Krantz, et. al. (1971) we shall say that Et is decomposable if there exist real valued functions g1, ..., gq monotonically related to their arguments, P1, P2, ..., Pq, respectively, and a real valued function ht of q variables that is one-to-one in each variable separately such that for all stimuli,

E

_{t}= h_{t}(g_{1}(P_{1}), ..., g_{q}(p_{q})), t=1, ..., r. (4)

This equation expresses the idea that it is possible to represent physical attributes in a way that reveals their independent contributions to the level of the psychological attribute.

Both the additive and polynomial combinations of the gu functions are feasible as an explicit representation for ht. When stimuli have been constructed from a factorial design (i.e., Pu variables or attributes represent the factors of a q-way table), there are a number of computer programs available which attempt to fit the stimulus scale values on Et to a prespecified form for ht. These include the CM-series of programs (Lingoes, 1973), MONANOVA (Kruskal, 1965) and POLYCON (Young, 1972 in Shepard, et. al., 1972). It should be realized, however, that since the gu functions from these programs represent only a nominal scaling of the physical attributes Pu, the extent to which one of these solutions represents the investigator's notion of what "psychophysical scaling" should be will probably depend upon the extent to which the gu's are not arbitrary. At least, a good fit to equation (4) using these programs for some ht would indicate independence.

A function Et is said to be monotonically decomposable if it is decomposable such that ht is strictly monotonic in each of its arguments, gu, u=1, ..., q. In contrast to the preceding case, monotonicity requires that solutions to equations (4) must explicitly depend upon the ordinal properties of the physical attributes. A condition of monotonicity might be taken as the minimal requirement for multidimensional psychophysics to fall within the spirit of classical psychophysics.

The methods of additive conjoint measurement are available for an arbitrary number q of independent variables Pu, and polynomial conjoint measurement techniques have been worked out for a set of simple polynomials with a maximum degree of 2 and maximum number of variables q=3. These methods are extensively described by Krantz, et. al. (1971, Chapters 6 and 7) along with many valuable insights applicable to the present discussion. Rather than attempt to duplicate that effort in this paper, we turn to pointing out the methods available to obtain solutions to equation (3).

Linear Functional Transformations

Special cases of equation (3) permit the use of a variety of algorithms for obtaining scaling solutions. These are catalogued in Table 2. > e entries are self-explanatory and will not be elaborated except to point out that case 3 is identical to the problem of finding additive conjoint representations which was mentioned in the section above

METHODS FOR ESTIMATING G_{s} AND G_{p} FUNCTIONS OF THE EQUATION, G_{s}E = G_{p}P

SOME RESEARCH ISSUES

We have outlined in this paper a rich conceptual framework of psychophysics and a number of methods by which one could construct multidimensional psychophysical functions. Within this framework, there fits not only the traditional areas of research in psychology and physiology but also the endeavors of applied researchers in the field of consumer behavior. Consumer researchers might find the methods useful for a variety of current research problems which would include attempts to predict the impact of changes in the characteristics of existing stimuli on consumer behavior and prediction of responses to the introduction of a new stimulus. In addition to prediction problem, there is the more general question of designing stimuli which will produce prespecified reactions.

The recurrent problems of product quality, package design and advertising layout may eventually benefit from these methods. The question of developing suitable legislation for consumer protection can also be conceptualized as design of optimal stimuli.

The full potentiality of these methods shall be realized only when the basic research question of stability of the multidimensional representations has been explored. Existence of such stability needs to be investigated for variations over time as adaptation and learning take place, for changes in response task and situations, for changes in the composition of stimulus sets, and across product categories.

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