Multidimensional preference scaling 9783110800814, 9789027975928

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Multidimensional Preference Scaling

methods and models in the social sciences

6

MOUTON • THE HAGUE • PARIS

multidimensional preference scaling

by GORDON G. BECHTEL University

of

Florida

MOUTON • THE HAGUE • PARIS

© Mouton & Co 1976 ISBN 90-279-7592-2 Jacket-design by Jurriaan Schrofer Printed in the

Netherlands

V

To Maria, Beth, and Tim

Preface

In recent years there has been an increasing use of multidimensional, or multiattribute, methods in various academic fields, as well as in the public and private sectors. Substantively, these applications primarily involve similarity perception and preference analysis. Since the former area has dominated the latter with respect to numerosity of techniques and frequency of application, the present monograph, Multidimensional

Preference

Scaling,

represents an attempt toward providing a more balanced set of scaling methods. The need for more extensive preference scaling stems f r o m an awakening interest in the infrastructure of utility, which is perhaps the most important construct unifying the social sciences. Latent preference structures have been neglected in many social and marketing research efforts, which simply report response percentages f o r particular segments of the population, or, at most, convert these percentages to manifest preference scales. Important exceptions, however, are found in the alternative multidimensional models for preferential choice contributed by C. H. Coombs and L. R Tucker. Tucker's scalar product formulation resembles classical multidimensional utility theory in economics, while Coombs' distance model for multidimensional unfolding represents a very different formulation of utility in terms of circular isopreference contours. The present work brings both of these formulations under a unifying analysis-of-variance framework for cardinal utility scaling. The resulting metric approach to multidimensional preference scaling permits more extensive statistical inference than that provided by previous, predominantly nonmetric, treatments of this topic. Part I of the book, Foundations,

establishes the analysis-of-variance model

for (unidimensional) utility measurement f r o m pairwise preferential choices.

VIII

Multidimensional

preference

scaling

The alternative multidimensional models are then generated as separate nonlinear constraints (hypotheses) which are placed upon the linear utility model. The detailed analyses for each of these submodels, including illustrative data applications, follow in Part II, Scalar Product Models, and Part III, Distance Models, which form the core of the book. This development of multidimensional structures as explicit submodels of a given linear model is a prominent feature of the present approach to latent parameter estimation. The final Part IV, Generalizations,

extends the preceding developments to

the multidimensional scaling of nonpreferential choices in the areas of value and similarity.

The work is then concluded upon a practical note with a

presentation of incomplete pairwise and single stimulus methods for unidimensional utility scaling. These methods, which involve fewer items, provide utility measurements that also may be multidimensionally scaled by any of the procedures in Parts II and III. The course background necessary for a thorough reading of the present monograph should include calculus, linear algebra, and one or two statistics courses covering regression and the analysis of variance. Therefore, in an academic setting the book is probably most useful for core or supplemental reading in advanced graduate seminars in the social sciences and quantitatively oriented colleges of business and education. For example, it has recently served as a text in a seminar, Multidimensional Societal

Analysis,

Scaling for Market and

which is cross-listed in the departments of Marketing and

Psychology at the University of Florida. The book also may be useful in certain departments of applied statistics in which multidimensional scaling is a topic of interest. Outside of the classroom the monograph is intended for faculty reference usage, as well as for scaling practitioners in fields such as marketing, survey research, and industrial psychology. By reaching these latter audiences, as well as graduate students in training, it is hoped that the present work will stimulate further theoretical and applied research upon the latent structures underlying individual and group preferences. These latent structures were first addressed by C. H. Coombs, who, in his A Theory of Data seminar at the University of Michigan, originally stimulated the author's interest in preference scaling. This interest and research effort has been complemented and further developed through consultations with L. R Tucker at the Educational Testing Service and the Oregon Research Institute, as well as in discussions with P. H. Schónemann, J. O. Ramsay, T. G. G. Bezembinder, W. Chaplin, W.-C. Chang, E. Lovelace, and M. N. Layman. Of course, the results of these interactions, as reflected in parts of the

Preface

IX

present monograph, are entirely the responsibility of the author. The impetus for this work was provided by the supportive encouragement of H. A. J. F. Misset at the Netherlands Institute for Advanced Study and L. R. Goldberg and J. S. Wiggins at the Oregon Research Institute. The monograph was initiated in 1972 at the former institution under funding from the Dutch government and was completed at the latter under Grant MH 12972 from the National Institute of Mental Health, United States Public Health Se-.vice. The author would also like to express his appreciation to J. P. van dc Geer and A. D. de Groot for their aid in bringing the monograph to final publication, as well as to L. G. Cooper, J. P. van de Geer, and L. Delbeke for providing data sets for illustrative reanalysis. These data analyses have been carried out under the skillful programming of W. E. Jones at the University of Oregon Computing Center and that of M. N. Layman and W. Chaplin of the Oregon Research Institute. Computing assistance at the latter institution was obtained from the Health Sciences Computing Facility, University of California, Los Angeles, sponsored by Grant FR-3 from the National Institutes of Health. A final word of thanks for the typing of this technical manuscript is extended to Pilar van Breda-Burgueno of the Netherlands Institute for Advanced Study and to Linda Mushkatel and Christina Hardy of the Oregon Research Institute. Netherlands Institute for Advanced Study Wassenaar, August, 1975

G. G. BECHTEL

Contents

P A R T I:

FOUNDATIONS

Chapter 1. Introduction and Overview

3

Chapter 2. An Orthogonal Analysis of Variance for Preferential Choices

14

Chapter 3. Spatial Constraints upon the Utility Matrix

P A R T II:

SCALAR PRODUCT

33

MODELS

Chapter 4. The Scalar Product Model with Known Stimulus Coordinates

47

Chapter 5. The General Scalar Product Model

P A R T III:

60

DISTANCE MODELS

Chapter 6. The Powered Distance Model with Known Stimulus Coordinates Chapter 7. The General Powered Distance Model

P A R T IV:

GENERALIZATIONS

Chapter 8. Multidimensional

Scaling of Choice

129

Chapter 9. Other Observational Techniques for Obtaining a Utility Matrix

148

XII

Contents

Bibliography

158

Author Index

163

Subject Index

165

PARTI

Foundations

CHAPTER 1

Introduction and Overview

The term multidimensional scaling traditionally refers to a class of methods for spatially representing observed similarities among stimuli, and, with several exceptions (e.g., see Tucker & Messick, 1963; Carroll & Chang, 1970), only the stimuli are represented in the similarity model. A description of methods for processing similarity data may be found in a recent volume edited by Shepard, Romney, and Nerlove (1972), who also include metric and nonmetric techniques for analyzing preference data. Due to the importance of preference data in the social sciences, as well as to the link between preference and choice behavior in general, the present book is devoted to the further development of metric methods for multidimensional preference scaling. It is hoped that the justification for the metric approach to preference analysis will become apparent within the book itself and through any research which this work may stimulate.

1.1

The purpose and nature of multidimensional preference scaling

Unlike most of their similarity counterparts, multidimensional preference models parameterize respondents as well as stimuli, thereby addressing the motivational aspects of individuals (or groups) in conjunction with their perceived stimulus structures. The dual parameterization in these preference methods also sets them aside from the classic unidimensional preference models associated with the method of paired comparisons. Again, these latter models give only the stimuli representation, generating unidimensional stimulus scales by averaging over sets of respondents. The representation of respondents as well as stimuli in quantitative models

4

Multidimensional

preference

scaling

actually stems f r o m another classical area of unidimensional psychometrics, namely, test theory. This antecedent for multidimensional preference scaling has been formally acknowledged in Coombs' (1964) A Theory of Data, where stimulus models are distinguished from stimulus-respondent models. In Coombs' formulation unidimensional test models and multidimensional preference models are classified as joint (stimulus and respondent) parameterizations. These models are to be contrasted with unidimensional stimulus comparison models and multidimensional similarity models, the latter being stimulus parameterizations. Hence, the representation of individuals and stimuli in multidimensional preference scaling is in the test theoretic tradition of joint, or dual, parameterization, which emphasizes, rather than obliterates, information about individual and intergroup differences. Having thus distinguished preference scaling from similarity scaling, one hastens t o observe that both types of multidimensional scaling, like factor analysis, are more inductive than deductive in nature (cf. Cronbach, 1957). The primarily inductive character of all multidimensional scaling methods follows f r o m the purpose of these data analytic techniques, for they surely share many of the technical characteristics of deductive methods such as the analysis of variance. That is, a multidimensional scaling model is, like an analysis-of-variance model, a decomposition form for a particular data layout, and this fact will enable us in the following chapters to repeatedly exploit a linear, analysis-of-variance model as a point of departure for the multidimensional scaling of preference. However, whereas in scaling applications the linear model stops short by providing descriptive, unidimensional preference scales, a multidimensional model inductively ferrets out the several latent attributes postulated as mediating these unidimensional scales. For example, if each of several individuals expresses his preferences among all pairs in a set of stimuli, an appropriate linear model will describe each individual's preference scale for these stimuli. However, we shall have no idea as to what latent structure may underly these preferences until taking the further step of employing some multifaceted representational form. Since they are primarily inductive in nature, multidimensional scaling methods are probably most useful in the early stages of the social sciences when taxonomies are sought and exploration is in order. When addressing the question, 'What determines voters' choices among political candidates?', one is certainly confronting multifaceted latent structure which is shrouded by confusion and ignorance. Naturalistic field surveys pursuing questions such as this might well utilize multidimensional scaling as one possible approach to

Introduction

and overview

5

this underlying structure. Subsequently, if the model proves to be fruitful, and if more is learned about the number of latent dimensions and the labels of these dimensions, the research emphasis will inevitably shift from exploratory t o confirmatory data analysis. However, although multidimensional scaling must give way to other, more intricate methods of observation and experimental design, it will also play a role in the more exacting confirmatory phase as well. This role, which is especially appropriate for metric multidimensional scaling, results from the carry over of exploratory research. That is, once dimensionality and labeling have been tentatively established, attention turns toward the metric structure upon particular dimensions. For example, the relative spacing among political leaders upon a dimension labeled liberalism-conservatism may give rise to research questions of the following forms: is scale structure altered by changing social, political, or economic conditions? (hypothesis testing); is the scale label correct? (correlational validation against a similarly labeled criterion measure of the scaled leaders); is the scale relevant? (correlational prediction of important external criteria). As these examples demonstrate, the distinction between exploratory and confirmatory multidimensional scaling should be only loosely drawn. It is observed here as a matter of emphasis that exploratory data analysis will be more heavily dominated by multidimensional scaling than confirmatory data analysis, and that, due to its inductive nature, multidimensional scaling may well prove most useful in exploratory social science research. However, the usefulness of multidimensional scaling in confirmatory applications should not be obscured, and it should be noted that the metric preference models in the following chapters may be deployed in either way.

1.2

Types of unidimensional scales for multidimensional scaling

At our observational base we shall utilize a unidimensional ratio scale of preference strength which, when linearly transformed, gives rise to a unidimensional interval scale of utility for each respondent. If the derived set of interval utility scales 'fit' the preference observations (see Chapter 2), they will then be used to construct our multidimensional scale. If not, these utility scales serve to terminate the procedure, and hence this metric approach is vulnerable to empirical testing even prior to the spatial representation itself. One should also note that, although the initial linear transformation from the

6

Multidimensional

preference

scaling

preference strengths to the utilities introduces an arbitrary additive constant for each respondent's scale, we shall have no 'additive constant problem' (see Torgerson, 1958, Chapter 11), i.e., these constants will not be needed in spatially representing the unidimensional interval scales. Also, we shall avoid classifying the final spatial representation as to scale type (e.g., ordinal, interval, ratio, etc.) simply because a taxonomy of transformations delineating multidimensional scale types is unavailable. The unidimensional classifications for our observations and descriptive scales stem from the original taxonomy of Stevens (1951) and from its clarification by Suppes and Zinnes (1963). The purpose of our spatial decomposition of the utilities is to gain a more penetrating insight into the latent preference structure than that provided by the utilities alone. However, as we shall see, this second, spatial representation is obtained at the cost of greater fitting error. This error increment stems from the fact that our spatial decomposition of the descriptive linear parameters is itself a nonlinear model containing even fewer parameters. Therefore, this latter representation is, of necessity, attended by a poorer fit to the original preference data. This two-stage approach to multidimensional preference scaling is depicted in Figure 1.1, which also shows the kinds of scales used in our hierarchical schema. A similar two-stage schema may be found in the metric preference scaling of Schonemann and Wang (1972), who (nonlinearly) transform preference proportions into utility scales — the latter being 'unfolded' into a spatial representation. An earlier use of this general twostage approach in metric similarity scaling occurs, of course, in Torgerson's distinction between distance and spatial models (Torgerson, 1958, Chapter 11). That is, Torgerson (linearly) transforms triadic similarity data into distances, which are subsequently fitted by a set of spatial coordinates. Because we shall employ ratio and interval unidimensional scales to estimate the numerical coordinates of the spatial models, all of the analyses presented in this book are metric multidimensional methods. These metric methods involve, of course, much stronger assumptions than the original nonmetric procedures of Coombs (1952; 1964), who recovered ordinal dimensions from ordinal scale data (see also Bennett & Hays, 1960). Each of these latter dimensions rank orders the scaled objects but contains no numerical coordinates. The more recent nonmetric methods pioneered by Goode (see Coombs, 1964) and Shepard (1962, a, b), which construct numerical coordinates from ordinal observations, also represent a somewhat 'weaker' data analytic approach than the fully metric one presented here. (See also Lingoes, 1973, which is based upon the work of Guttman and Lingoes; and

Introduction

and overview

1

Kruskal, Young, and Seery, 1973, which is based upon the work of Kruskal, Young, Shepard, and Torgerson.)

1.3

The preference observations

The spatial coordinates of the stimuli (and/or respondents) we wish to scale will be estimated f r o m choice responses to the stimuli presented in pairs. The metric estimation procedures to be employed require the strength as well as the direction of each pairwise preference, and hence some choice-strength indicator constitutes the basic element of our observational base in Figure 1.1. This operationalized choice strength is unique up to multiplication by a positive constant, i.e., it constitutes a ratio scale upon which the natural origin is t h e point of indifference between two stimuli.

SPATIAL MODEL (MULTIDIMENSIONAL SCALING)

LINEAR UTILITY MODEL

(UNIDIMENSIONAL INTERVAL SCALING)

O B S E R V A T I O N S

p]

(UNIDIMENSIONAL RATIO SCALING) Figure 1.1

Two-layered penetration of the preference data

8

Multidimensional

preference

scaling

A typical numerical choice strength

indicates the degree to which

respondent (i.e., individual or group) i prefers stimulus; to stimulus k. Either Pijk ^

Pijk ^

or pjjfr = 0, indicating, respectively, the choice of j over k,

k over /', or an indifference between j and k on the part of i. The numerical indicator p ^ may be a transform of an objective behavioral index of pairwise preference, such as a response porportion or latency, or it may be a numerical rating reflecting i's subjective judgment of the preference strength for j over k. Thus, preference applications of the techniques to be presented may span a broad class of settings ranging f r o m the experimental laboratory to the field survey. Also, these applications may involve animals or infants, as well as verbally responsive humans who can express preferences in a wide variety of stimulus domains.

1.4

The reliability of the spatial coordinates

Individually rated degrees of preference will in most instances contain greater measurement error than expressions of group choice strength such as transformed proportions. However, whatever amounts of error pervade the various indicators of pairwise preference, it is felt here that more reliable spatial representations can be constructed by exploiting the metric properties of the preference data. This belief is based upon the overdetermination that the ratio-scale data exert upon the derived interval-scale utilities, in conjunction with the further overdetermination which these latter measurements place upon the final multidimensional scale (see Figure 1.1). In statistical terms this means that the estimates of the parameters of the fitted spatial model have far less sampling variation, i.e., much greater reliability, than do the observations themselves. However, we will be able to obtain analytic information about the reliability of our spatial solutions only in Chapter 4 of the present book. That is, only in t h e models of Chapter 4 are the spatial coordinates linear parameters whose estimates have explicit formulas for their sampling variation. Unfortunately, this kind of analytic information is not available for any of the other spatial models, which are all nonlinear. Hence the argument for the greater reliability of metric methods awaits empirical support f r o m 'sensitivity' studies of artificial data generated f r o m known spatial configurations. In these investigations one would hope to find relatively minor departures (from the known configuration) in the metric spaces recovered when these artificial data are increasingly error perturbed.

Introduction

and overview

9

A preoccupation with the reliability of the spatial coordinates stems from the fact that the stability of the solution structure is a necessary (but, of course, not sufficient) condition for its validity. Since this validity is the raison d'être of multidimensional scaling, we turn now to a brief consideration of the substantive appropriateness of spatially representing an underlying preference process.

1.5

The spatial representation of preference processes: validity

Although sometimes viewed as data reduction techniques, multidimensional preference and similarity scaling can be used most fruitfully to represent latent psychological or social processes postulated as generating the data. This representation requires a decomposition of the data by means of an explicit model with estimable parameters characterizing social or psychological objects (individuals, groups, stimuli, concepts, etc.). Since the parameters are the coordinates of these objects upon postulated latent dimensions, the representation provided by multidimensional scaling is geometric and spatial. Geometric models must be distinguished, of course, from other representational forms, such as graphs, trees, and hierarchical clusters, which are very different kinds of latent structures (e.g., see Harary, Norman, and Cartwright, 1965 and Johnson, 1967). Therefore, the use of multidimensional scaling in a given instance should bè regarded as a choice of a particular psychological or social theory about the data (Coombs, 1964). The appropriateness of this choice depends upon the properties of the data, but these properties have been studied less for preference data than for similarity data. For example, in situations of known dimensionality, Beals, Krantz, and Tversky (1968) have given axioms which constitute a set of necessary and sufficient conditions among similarity data for the Minkowski-r metric class of dimensional representations. The violation of those axioms which are empirically testable casts doubt upon the validity of any Minkowski-r metric model, including the Euclidean distance model, as a representation of the particular set of similarity data. In the case of preference data, however, we must still rely heavily upon the traditional concept of 'goodness of fit', even though arbitrary increases in the number of dimensions result in progressively better fits to real, errorful data in situations of unknown dimensionality. For this reason, when plotting a goodness-of-fit index (e.g., error sum of squares) against the number of dimensions, one prefers to find gradual

10

Multidimensional

preference

scaling

decrements in the index over the first few dimensions, then a sharp descent almost to zero, and a very mild decline thereafter. The number of dimensions accepted will correspond to that at which the sudden drop occurs. Moreover, it is well to supplement this criterion with the requirement that each of the accepted dimensions, perhaps after a transformation, e.g., a rotation, be substantively interpretable in terms of the social or psychological process being studied. These criteria — goodness of fit and interpretability — are validation devices which, like testable axioms, are internal to a given set of data. Finally, it should be noted that these internal validation devices may be accompanied by external procedures for establishing the validity of dimensional representations. Several external, confirmatory uses of multidimensional scaling have already been mentioned. For example, one might predict f r o m theory that a given condition will cause a particular alteration in the positions of the stimulus coordinates on some labeled dimension. If this dimension emerges f r o m the preferences observed before and after the condition is imposed, and if the predicted coordinate changes are also realized, then an instance of construct validation for that dimension may be recorded (Cronbach & Meehl, 1955). Alternatively, the dimension may be established empirically by another kind of external validation, e.g., through the correspondence between the stimulus coordinates and a similarly labeled criterion measure of the stimuli. Chapters 5 and 7 contain methods for transforming stimulus and respondent coordinates into maximal congruence with 'target' measurements upon external criterion dimensions. The validity of spatially representing preferences is, of course, the most serious problem of multidimensional preference scaling. Although questions concerning validity, like those of reliability, are yet unanswered in the various fields of application, the preceding discussion at least points up the availability of empirical procedures for addressing them. In the case of the more difficult questions of validity, it appears that for the moment, and probably in the long run, we will have to rely upon a pattern of evidence garnered over several methods of validating particular dimensions. The increasing, and hopefully careful, application of multidimensional scaling to real data should provide this validational pattern in various fields. Meanwhile, other studies involving artificial and real data can supply us with complementary information about the reliability of metric preference spaces.

Introduction 1.6

and overview

11

Alternative representations of the respondent: individual and intergroup differences

The various representations in both preference and similarity scaling all utilize a multidimensional stimulus space, i.e., a Cartesian space in which the stimuli reside as points. However, as emphasized at the beginning of the chapter, preference models are distinguished from most similarity models in that the former contain some representation of the respondents vis-à-vis this stimulus space. Hence, preference models also serve to quantify individual or intergroup differences. Since alternative treatments of the respondent in these models will guide our work, it is necessary to pursue this aspect of preference scaling in greater detail. Parameterizations of respondents in spatial preference models have taken two primary directions. On the one hand, the respondent has been represented as an ideal (stimulus) point residing in multidimensional stimulus space; on the other hand, the individual (or group) has been characterized in terms of weights assigned to the several dimensions of this space. The former distance type of preference model originated with Coombs and his associates in a procedure known as multidimensional unfolding (Coombs, 1964). The latter scalar product type of model was first used to analyze preferences by Tucker (1960) and Slater (1960). More recently Carroll (1972) has interrelated distance and scalar product models in his hierarchy of preference models, which share a space of stimulus points but vary with respect to their representations of the respondent. The present book is confined to detailed metric analyses of the simpler, classical distance and scalar product models. Although these models stand within a multitude of possible respondent representations, their highly distinct and useful structures would appear to justify further parametric development. This development will be carried out for the general form of each model, i.e., when the stimulus and respondent coordinates are unknown, as well as under the simplifying condition in which the stimulus coordinates are specified a priori. It is hoped that these analyses will stimulate important, and perhaps more realistic, variations both upon these simpler models themselves and upon the present methods for metric preference analysis.

12

Multidimensional

1.7

preference

scaling

Multidimensional similarity scaling as an adjunct to preference research

As we observed at the outset, similarity scaling is primarily a method for spatially representing perceived stimulus structures, or, to use Goodman's (1951) term, 'the structure of appearance.' However, the structures derived f r o m similarity observations may also be put to use in multidimensional preference scaling when the stimulus coordinates are to be specified in advance. That is, the subjective stimulus scale may be given by an adjunct similarity scaling of the stimuli employed in the preference analysis. Hence, similarity scaling can be extremely useful in those preference studies in which one desires to determine empirically the stimulus space f r o m data quite apart f r o m the preference observations themselves. The investigator may want to turn to this strategy with less reliable preference data from which as few parameters as possible, i.e., only the respondent parameters, should be estimated. In this situation the preference data would only be used to place the respondents into a stimulus structure already constructed f r o m separate similarity data; or, stating this strategy alternatively, one would use more data, i.e., similarity and preference observations, rather than just preference observations, to estimate the stimulus and respondent parameters more reliably. This estimation procedure is sequential because part of the data (the similarities) are used to estimate part of the parameters (the stimulus coordinates), and this subset of estimates is then used with the rest of the data (the preferences) to estimate the rest of the parameters (the respondent coordinates). One caution concerning this procedure should be mentioned, however. In certain preferential situations the stimulus space mediating the pairwise choices may be motivational rather than perceptual in nature. If these two kinds of stimulus structures differ in a given application, then the perceptual stimulus space constructed f r o m similarity data will not provide an appropriate framework for embedding the estimates of the respondent parameters derived f r o m the preference data. In this case the placement of the stimulus points should be regarded at best as an initial hypothesis which is subject to alteration, i.e., the ancillary multidimensional similarity scaling would not fix the stimulus space, but, rather, would serve as a rough approximation which the investigator would alter upon other rational and/or empirical grounds. For example, if a distance type of preference model is being invoked, one could still employ the similarity data to find a 'starting configuration' for the

Introduction

and overview

13

stimulus points. Subsequently, on the basis of the preference data, this stimulus configuration, along with a corresponding pattern for the respondent points, may be iteratively altered to a final solution. This solution, presumably, would be characterized by a somewhat different stimulus pattern than the initial one gleaned f r o m the similarity data. Iterative procedures for both the stimulus and respondent points in the distance model are treated in Chapter 7, while solutions for the respondent points, given a fixed stimulus space, are presented in Chapter 6. Finally, we note that the initial construction of a stimulus space f r o m adjunct similarity data may be carried out by any of the available techniques for multidimensional similarity scaling. Chapter 8 includes one of these techniques, which is particularly convenient since it is formally identical to that used in multidimensionally unfolding preferential choices. That is, given triadic similarity choices (Torgerson, 1958), and replacing the respondent with a standard stimulus, the unfolding procedure becomes a method for multidimensional similarity scaling (cf. Coombs, 1964). Of course, this procedure for the multidimensional unfolding of similarity is not solely an ancillary technique for preference research. It may also be used to represent perceived stimulus structures per se. Therefore, the method given in Chapter 8 takes its place among various alternatives for the multidimensional scaling of similarity.

CHAPTER 2

An Orthogonal Analysis of Variance for Preferential Choices

We shall see in t h e following chapters that different spatial preference models can be generated as alternative nonlinear breakdowns of a common linear model. Since this descriptive, linear decomposition involves relatively many parameters, it usually produces a close fit to the observations. Therefore, we may regard the analysis-of-variance model as only a half-way house to the understanding of underlying preference processes, i.e., we shall utilize this linear decomposition as a point of departure for further nonlinear breakdowns containing the parameters of the multidimensional models (cf. Gollob, 1968a, 1968b, 1968c; Tucker, 1968). The basic conceptualization, which is summarized in Figure 1.1, will be developed in the following chapters as a two-stage parameter estimation procedure. Since preferential choices are to be conceptualized as linear in the respondent's utility values for the stimuli, the first stage of our procedure consists of the linear transformation of preference strengths into descriptive utility values. These latter values are least squares estimates of the utility parameters contained in an orthogonal linear model, which is treated in the present chapter. Subsequently, at t h e second estimation stage (treated in Parts II and III) the properties of this orthogonal model provide the least squares estimates of the spatial coordinates of both the scalar product and distance submodels. It shall be shown that, by least squares fitting the coordinates to the utility values obtained in the first stage, one immediately obtains estimates of the eeordinates which are least squares fit to the preference observations. Thus, the two-stage procedure guarantees a close coupling between the spatial submodels and the original pairwise data. With the exception of the scalar product submodels in Chapter 4, this two-stage estimation procedure will be applied in fitting nonlinear spatial

A n orthogonal analysis of variance for preferential choices

15

models to pairwise preference data. The two-stage procedure is particularly convenient since it reduces a difficult nonlinear least squares problem, i.e., the fitting of spatial coordinates to pairwise preferences, to a simpler one, i.e., the fitting of spatial coordinates to utility values. Moreover, the use of an intervening linear utility model provides least squares estimation and hypothesis testing procedures for unscalability parameters, which are common to the utility model and its spatial submodels. These latter parameters may be analytically estimated and tested at the first stage (in the present chapter) prior to the estimation of the spatial coordinates (in Parts II and III).

Figure 2.1

The layout of pairwise preference strengths

16

Multidimensional

2.1

preference scaling

The layout of preference strengths

Our observational base in Figure 1.1 actually takes the form of the layout of pairwise choice responses illustrated in Figure 2.1. This experimental design includes m sets of responses to all pairs in a set of n stimuli. As described in Chapter 1, the strength as well as direction of choice is recorded in each cell, and thus a typical observation (in the darkened cell) indicates the degree to which respondent i prefers stimulus j to stimulus k. These w ^ ) strength-of-preference measures will be linked to the coordinates of our spatial models through the linear utility model in Section 2.2.

2.2

The linear utility model

The lineai decomposition of a preference strength p i j k is assumed to be Pijk

=

v

ij -

u

ik

+ Jjk

+

e

ijk

(1)

where, L = vik = fjk = ei)lc =

I'S utility f o r / , ¿'s utility for k, unscalability for the pair random error.

In (1) i's preference strength depends upon the difference between i's utilities f o r ; and k, as well as upon the yjk parameter, which we identify as systematic error present in pijk. This component represents unscalability in the observed preferences associated with the stimulus pair . Unscalability may stem from the falsity of the utility differencing model itself or from other sources of systematic error, such as response bias, in the pairwise choice strengths (e.g., toward the middle or the extremes). Since 7 r e f l e c t s invalidity in the initial linear model, it is to be contrasted with the unsystematic, or random, component e^k associated with the stimulus pair for respondent i. This latter component is identified with errors of measurement due to unreliability. 2.2.1

The design matrix

Table 2.1 gives the design matrix for (1), which is partitioned

into

An orthogonal analysis of variance for preferential choices

17

m x (m + 1) submatrices. The submatrixS is the (![) x n matrix in Table 2.2, while I is the (%) x ( j ) identity matrix in Table 2.3. The rows of each submatrix are labeled by the observations for respondent i, and the columns are labeled by the associated parameters. Table 2.1

The partitioned design matrix

1 2

m

m+ 1

1 2

SO OS

0 0

I I

m

0 0

S

I

Table 2.2

The pairs x singles scale matrix S

v

v

i2

.

.

vif

Pm

1

-1

.

.

0

. . .

Pijk

0

0

.

.

1

Pi,n~\,n

0

0

.

.

0

i1

.

.

.

v

• •

•

v

i,n— 1

"in

0

.

.

.

0

0

. . .

-1

.

.

.

0

0

. . .

0

.

.

1

-1

ik

Table 2.3 The pairs x pairs identity matrix I •

- y/k

. . .

y

n - 1 ,n

Pi 12

1

.

.

o

. . .

0

Pijk

0

.

.

l

. . .

0

Pi,n— 1 ,n

0

.

.

0

. . .

1

18

Multidimensional

2.2.2

preference scaling

Side conditions upon the model

The design matrix in Table 2.1 has rank \ (n — 1)(2m + n — 2), which is its number of columns mn + less its column deficiency n + m — 1. This deficiency may be seen by deleting the first n columns, along with the first column in each of the subsequent m— 1 submatrices. The remaining | (n — 1)(2m + n — 2) columns, which are linearly independent, generate each deleted column. Due to the rank deficiency of model (1), n + (m — 1) = m + (n — 1) linearly independent side conditions are needed to identify the parameters, and we choose the 'natural' ones given by 2 vtj - 0

0 = 1,... , m ) ,

(2.a)

Z7ik = 0 k '

(/' = ! , . . . , « ) ,

(2.b)

where Ikj = - t i k

and

In

=

Equations (2.a) constitute m linearly independent side conditions which set the origin for each of the m utility scales. The n equations (2.b), however, impose only n — 1 linearly independent side conditions due to the skew-symmetry of the y)k.

2.3

The separation of the error sum of squares and the least squares estimates

We shall obtain the least squares (LS) estimates and demonstrate the orthogonality of (1) and (2) by expressing the error sum of squares for this model as the weighted sum of several other, simpler sums of squares. For convenience, the pseudo-observations Pikj = -Pijk

and

P,jj =

0

( 3 )

are used to extend the layout in Figure 2.1 to a three-way block of m skewsymmetric matrices. In this block the linear forms 0,-y vjk

= pjj. = j's (estimated) utility for;', = pik. = i's (estimated) utility for k,

(4.a) (4.b)

An orthogonal

analysis of variance for preferential

choices

19

Jjk

= p jk ~ v.j + v.k = average (estimated) unscalability over i, (4.c)

¿¡jk

— (Pijk ~

+ 0, fc ) - 7jk = (estimated) random error,

(4.d)

may then be defined, noting that 2u,7 = 0

(¿=1, ...,»!),

(5.a)

S7/fc=0 k '