METRIC MULTIDIMENSIONAL SCALmG AND commumcmom: . mEeRv Ana mmmammou - Thesis for the Degree om.- A. ‘ ' MICHEGM STAYE UMVERSIW ‘ . KEMBLASNESEROTA' - 1974_‘. 5"?" _.".M‘ ‘ :L ."- menu‘s.“ .“L,¢.. i" 5 .-.¢u— r—A n.‘ m“'M'-¥€"ku “ n :! 1:”1” \"gg '7 a. 1"" v —( . 'V.‘ , .9 .‘I' 4 sun". 2‘. 'D '0. “ “v 43‘ '; "‘ We: 4:- -‘. w tn ‘ _ ‘ o -: I ~ - ' ~ ~-c.--. . ‘ .' ~ .. Li.“ “a. a." 6" "I _,,' 4; ”J J" ‘V W w y w v v I , BINDING B's; ‘ h ~ , H HMS & 8095' ~ Manx amnm mc. LIBRARY BIN DEBS "MGM! ABSTRACT METRIC MULTIDIMENSIONAL SCALING AND COMMUNICATION: THEORY AND IMPLEMENTATION By Kim Blaine Serota This investigation attempts to examine the historical and methodol- ogical roots of multidimensional scaling in a measurement-theoretical framework and propose areas and methods for adoption in communication studies. Specifically, this work discusses scaling rigor, dimensional- ity, and isomorphism as criteria fOr the comparison of measurement tech- niques. The various contributions to the evolution of multidimensional scaling are examined with regard to these criteria and current problems identified. The thesis introduces metric multidimensional scaling as a response to these problems and argues for its application to longitudin- al and aggregation situations. Discussion of the relative conceptual measurement applications of ordinal and ratio scaling is presented. Four levels of comparison are generated from this discussion in conjunction with an examination of unidimensionality and multidimensionality. These levels are contrasted on the basis of isomorphism between data and numbering systems. Using the scaling discussion as a framework for comparison, the his- torical developments of mathematical transformation, factor analysis, and multidimensional scaling are traced. Multidimensional scaling is shown to draw on the mathematics of astronomy and the theory of psycho- physics, relying heavily on the contributions of Pearson, Garnett, and Hotelling and the seminal work of Richardson, Gulliksen, and Torgerson. Kim Blaine Serota Torgerson's model is contrasted with the later nonmetric work of Shepard and others. From these theoretical and historical foundations, metric multidim— ensional scaling is discussed in depth. The problems of judgement unre- liability, violations of linearity, and unknown dimensionality are shown to be overcome by this methodological approach. Further, the use of this technique to measure longitudinal, process variables and to examine con- ceptual relationships as a function of communicative interaction is devel- oped. The mathematical and theoretical components of the metric approach are detailed utilizing examples drawn from on-going political communica- tion research. Implications for further research are discussed. The Galileo computer package which supplies the necessary software to implement and facilitate the use of metric multidimensional scaling is appended. 'II/77 METRIC MULTIDIMENSIONAL SCALING AND COMMUNICATION: THEORY AND IMPLEMENTATION By Kim Blaine Serota A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS Department of Communication 197” Accepted by the faculty of the Department of Communication, College of Communication Arts, Michigan State University, in partial fulfillment of the requirements for the Master of Arts degree. Director omsis O I“ l , Chairman Guidance Committee: M QM //' W ‘7 ACKNOWLEDGEMENTS This thesis would not have been written without the concern and con- tributions of the people around me. To them, I am indebted. I would like to acknowledge and thank Edward Pink for his continuing feedback and invaluable guidance, and Erwin Bettinghaus, who for many years has encouraged me to continue and succeed when I might have given up. Others among my colleagues deserve credit fOr sharing their under- standing and their knowledge. Most prominent among them have been Virginia McDermott and George Barnett. I would particularly like to recognize two individuals who have made great contributions, and often sacrifices, to help me achieve my goals. Karen, by poking and prodding, and by loving and caring, has seen me through to the completion of this work, and in doing so has contributed as much as I. Joseph Woelfel has been both friend and sage, and has set the tone for this thesis and all of my future work by teaching me that science is more than repeating other people's mistakes. ii TABLE OF CONTENTS List of Tables List of Figures Chapter 1: Introduction Scaling Theory: A Conceptualization Ordinal Scaling Ordinal Scaling with a Natural Origin Interval Scaling Ratio Scaling Toward Isomorphism: Multidimensional Scaling Unidimensionality Versus Multidimensionality Objectives of the Thesis Chapter 2: Historical Development of Multidimensional Scaling Pre-Psychophysical Influences on Multidimensional Scaling Early Factor Analysis Development of Multidimensional Scaling ChaPter3: Galileo: A Procedure for Metric Multidimensional Scaling A Method of Ratio Judgements Aggregation Transformation to the Spatial Manifold iii vi 23 24 21+ 29 141+ 52 53 (J1 ‘D Longitudinal Data and Rotation Conclusion Bibliography Appendix A Appendix B iv 66 7O 73 85 87 Table Table 'Table Tadxle Tadile Tadile A-l. A-2. A-3. LIST OF TABLES Coefficients for the six tests represented in Figure 7. Coefficients for the six tests represented in Figure 8. Coordinate values for six political concepts in multidimensional space (June, 1972). Coordinate values for six political concepts in multidimensional space (August, 1972). Coordinate values for six political concepts in multidimensional space (November, 1972). Coordinate values for six political concepts in multidimensional space (June, 1973). 33 38 85 85 86 86 u.’ -.I vs, L-A- v.1 Figure l. IPigure 2. Figmue 3. Figpire u. Fignire S. Fignxre 6. Pignnee 7. Figure 9. LIST OF FIGURES Examples of transformations for each of the four types of scales satisfying the Stevens scheme. If the ab- scissa x is the linear continuum of possible observa- tions, the values on the ordinate y will fulfill the requirements of transfbrmation to the scale indicated. (Adopted from Torgerson, 1958). Changes in sweetness represented on two scales. The unrotated two-dimensional solution (n=l2) of the Wish data using INDSCAL on dissimilarities data. In this figure the coding: O, 1/2, 1, 2, H refers to the number of teaspoons of sugar Specified and fbr temper- ature: IC = ice cold, C = cold, LW = lukewarm, H = hot, and SH = steaming hot. (Adopted from Carroll, 1972). A multidimensional configuration at two points in time (magnitudes changed, correlations remain constant). Chronological development of pre-factor analytic con- tributions to the development of multidimensional scaling. Line of least squares best-fit. The "two—factor theory" applied to six tests. The values of the overlapping g factor are the correla- tions of each test with that factor. The 3 factor is represented by the residuals (S2 = 1.0 - g2). The bi—factor theory applied to six tests. The values of the overlapping g factor are the correlation of each test with that factor. The values of the overlapping group factors are the correlation of a specific test with a common factor for a subset of the tests with the general factor removed. The 3 factor is represented by the residuals. Projections of two variables on an independent coordin- ate system. Correlation between the two is represented by cosast. vi 14 18 22 3O 31 33 38 no Figure 10. Two-dimensional representation of six political concepts scaled by the Galileo procedure. 65 Figure ll. Representation of movement of six political concepts scaled at four points in time. 69 vii Chapter l Introduction Powerful and compelling new techniques of data analysis have been and.are being developed in the varied branches of social science; while 'taking many forms, they are intended to meet a common goal: to better (explain, predict, and control the exigencies of human behavior. In tine field of communication study it is the goal of the methodologist 1x) find techniques that provide both accuracy and parsimony for explor- ifug the dynamic, transactional processes that comprise human interac- tion). Among the most exciting, and potentially most valuable, of these is tflie technique of metric multidimensional scaling. It is the purpose of this thesis, to (a) examine, in general, the histuorical development of multidimensional techniques, both theoretical- LY eund mathematically, and (b) introduce Galileo, a metric multidimen— siorual_scaling algorithm, and show its advantages over more well-known teeihniques, particularly for longitudinal analysis of communication PPOCesses . Efiiégigpg Theory: A Conceptualization It is the object of measurement to classify and compare observa- ticnls in a meaningful way such that the representational measure and itsi'transformations are indicative of those observations and their Chihmges in reality. Our reality consists of constructs (such as, people, attitudes, relationships, and beliefs). These constructs are referred to by Torgerson (1958) as systems, the "things" which make up our conceptual universe. However, it is not the system which we measure, rather the properties of a system comprise the observable aspects of characteristics of that system which are present in the empirical universe. The act of measurement is one of assigning a numerical set to cor— respond with the properties of a system. The rigor of this process is expressed in the attempt to attain an isomorphism between properties and the systems which they describe. As many texts indicate (e.g., Carnap, 1959; Coombs, 196u; McNemar, 1969; Blalock, 1972) we can describe levels of scaling which express this degree of isomorphism, as it is achieved by a particular measuring technique or device, as a function of ordinality, linearity, and origin; the closer a scale conforms to these criteria, the greater the likelihood it will achieve a one-to—one corres- pondence between the properties of a system and the numbers used to represent those properties. Perhaps the most well—known expression of the relative isomorphism of scaling levels can be seen in the organization of transformation groups by Stevens (1951). These serve to reduce the arbitrariness of selecting a numerical system to represent a property by the quality of transformation which may be imposed upon that numerical system: Ordinal scaling. If objects can be ordered only on the basis of the relative position or magnitude of some property, they then lack the distinctiveness desirable for achieving sophisticated mathematical trans- formation. Since the numbers are assigned such that they are order- preserving, the ordinal scale is said to determine relationships to with- in a monotonic-increasing transformation (Figure 1a). Such "scales" are the minimum expression of relationship between two or more variables (excluding the notion of nominal scales which are not truly measures of properties but rather categorical representations for the classification of the systems themselves). Ordinal Scaling with a natural origin. If, in addition to the monotonic transformation described above, the scale has a unique point of origin, the ordered relationship of two or more systems can be more accurately specified. This allows us to indicate from the scale value of zero that there is an absence of any amount of the property being observed. Thus, according to Stevens' scheme, this type of scale can only generate those transformations which leave the origin unchanged (Figure lb). Interval scaling. If the scale lacks an absolute point of origin but the numerical differences reflect equal intervals between finite amounts of a property, then the relationship between two or more systems on the basis of that property can be specified exactly. However, this does not yet describe a ratio scale since the absolute magnitude cannot be expressed. As is indicated by Stevens, this does yield a scale that is not affected by a transformation of the form: y = ax + b, where a is any positive real number and b is any real number. This simply describes the basic form of a linear transformation. As an example, the difference, or interval, of two units on the low end of a Scale will be equal to the interval of two units on the high end of the Scale, and the slope of a line defined by these units will remain con- stant (Figure 1c). Ratio scaling. A ratio scale is any scale which meets the criteria for an interval scale, and in addition, satisfies the linear transfor— mation such that b=0, or in other words, has a natural origin (Figure 1d). Such a scale has a unique quality such that any set of properties may be expressed as a ratio of the magnitude of one to another, and the absence of any magnitude of a property is represented by a value of zero. Y Y (a) Ordinal scale. (b) Ordinal scale with natural origin. J~< ‘< -7/c ox _/ (c) Interval scale. (d) Ratio scale. Figure 1. Examples of transformations for each of the four types of scales satisfying the Stevens scheme. If the abscissa x is the linear continuum of possible observations, the values on the ordinate y will fulfill the requirements of transformation to the scale indicated. (Adopted from Torgerson, 1958). The most important issue of scaling is thus one of achieving an isomorphic correspondence between the actual properties of systems and the numerical structure used to represent those properties. It is ob- vious that the more rigid the criteria for transforming a scale the better that scale is able to represent the state of a system with re- gard to any single property. Two major consequences of seeking isomor- phism exist. First, the closer measurement of the properties of a sys- tem comes to achieving one-to-one correspondence with the "reality" of that system the more adequately we can describe the structure of the system. Second, the selection of numerical representations that are more highly defined (as are those used to underlie higher level scales)* will allow us to perform more mathematically rigorous operations, upon the number set, which are equivalent to operations which might be per- formed upon the properties of the system under study. Toward Isomorphism: Multidimensional Scaling_ Historically, the majority of measuring devices have sought simplic- ity, at the expense of isomorphism, in the form of unitary scales. During *It should be noted that with highly defined (linear) scales the transformation process of subtraction will always yield a ratio scale of change scores (Woelfel, personal discussion). When performed upon inter- val and ratio scales this is consistent with our desire to generate scales capable of measuring processional changes in attitudes as functions of communicative behavior. It is also notable that this transformation to ratio scaling is readily observable in, and probably derived from, the physics of motion which uses the principle of relativity to deal with constant changes occuring in the universe (Einstein, 1923; Hempel, 1952). Further, this method of deriving ratio scales is not without foundation in the behavioral sciences; the various unfolding techniques use dissimi- larities as a second generation transformation to arrive at better quan— tifiable measures (Coombs, 196”; Coombs and Kao, 195a, 1955). the development of psychophysical measurement, it became apparent that judgement responses could not always be arrayed on a single undimen— sional scale (Thurstone, 1927; Richardson, 1938) and that the measure- ment excess must be attributed either to error or multiple influences. Klingberg (19u1) demonstrated that by interpreting the results as a multidimensional configuration rather than as a unitary judgement factor error was reduced considerably and that distinct bases (dimensions) for judgements could be identified. From these early presentations of multidimensional scaling the no— tion of representing judgements of the relationships between stimulus— objects as distances made it possible to conceive of conceptual struc- ture as analogous to Euclidean real space. In this format, dimensions, angles, and distances could be used to express data relationships more directly and more accurately than by repreSenting the single largest component of each interrelationship solely. A scale of this nature would, additionally, be capable of measuring very accurately changes in relationships which do not appear to occur on the single unitary factor. Currently, the majority of behavior measurement techniques subsumed under the rubric "multidimensional scaling" are static designs, struc— turally oriented. They are neither parsimonious with the intent of measuring "distance," failing to overcome difficulties in meeting the assumptions of measuring physical distance, nor process—oriented, fail- ing to provide a sound scale against which to measure change. It is Precisely the static nature which many of these design have that Roger Shepard describes, in the introduction to his major work on multidimen— sional scaling (Shepard, Romney, Nerlove, 1972:l) as rationale for these methods of analysis: The unifying purpose that these techniques share, despite their diversity, is the double one (a) of somehow getting hold of whatever pattern or structure may otherwise lie hidden in a matrix of empirical data and (b) of representing that structure in a form that is much more accessible to the human eyen-namely, as a geometrical model or picture. The objects under study (whether these be stimuli, persons, or nations) are represented by points in the spatial model in such a way that the significant features of the data about these objects are revealed in the geometrical relations among the points. To find a methodological technique satisfactory to the process de« mands of communication research, quantitatively more accurate than the present techniques of communication research, and sufficiently elegant to stand alone or easily enmesh with existing techniques requires that the technique meet, or attempt to meet, the following criteria: (a) it should be of the highest possible level of scaling, utilizing wholly ratio scales with natural origins, (b) it should be able to measurechanges in the relation- ship of the variables being scaled, with precision, (c) it should be able to clearly and simply represent those relationships to the researcher while maintainn ing a format readily transformable for less obvious means of analysis, (d) it should operate on the bases of theoretical and mathematical assumptions which do not force the loss of information through transformation, and (e) it should achieve isomorphism between the properties being measured and the characteristics of the system used to describe those properties such that transforma. tions and Operations performed upon the descriptive system are equivalent to transformations and operations which might be performed on the properties of the real system being represented. These are neither simple nor easily attainable criteria. The non-metric or "Shepard-Kruskal" approach (Kruskal, l96ub), while not meeting these requirements exactly, does come closer than any existing technique in standard usage among communication researchers. Examin- ing the foundations of the non-metric approach however yields a highly practical scaling model fbr communication which does satisfy the cri- teria: metric multidimensional scaling. Metric multidimensional scaling is a technique for the construction of spatial representations of inter- relationships from ratio judgement data. Judgements of dissimilarity (distance) between concepts are arrayed to depict the structure of all possible concepts, simultaneously, in a configuration analogous to Eucli— dean real space. This allows us to examine and describe structure, repre— sent change, and operate on the model in ways parallel to operations in reality without distortion of our original data measurements. The primary importance of multidimensional scaling (MDS) to commu— nication research is that it can provide an analytic tool for measuring and interpreting processes and change oriented hypotheses. It has been suggested (Berlo, 1969; Smith, 1966; Dance, 1970; Mortensen, 1972; Miller and Steinberg, 1974) that a major component of the concept of communica- tion is that it is a dynamic, on-going process. As such, it is necessary to seek ways to examine communication as a process rather than as a series of discrete events. Metric multidimensional scaling, by reliance on ratio scaling and the ability to generate a latent structure that is analogous to Euclidean continuous space, allows us to manipulate processes and observe change with a high degree of accuracy. It will be useful to the understanding of the relationship of the process variables found in communication to examine the differences that accrue from the application of unidimensional and multidimensional scales conjointly with the differences between the use of ordinal level and interval (ratio) level scaling. Unidimensionality Versus Multidimensionality_ The purpose of using unidimensional scaling is to measure single attributes or prOperties of a system. The unidimensional scale achieves correspondence by establishing a continuum of points on which the magni- tude of the attribute is represented by a point in the continuum (Russell, 1938). The primary test for unidimensional continua is the order pro- position of transitivity. Meeting Huntington's postulates is, thus, the first necessary condition for identifying the existence of a single un— derlying dimension in a set of measurements. Those postulates are (Stevens, 1951:1M): 1. If a # b, then either a < b or b < a. 2. If a < b, then a # b. 3. If a < b, and b < c, then a < c. However, a second condition must be met to sufficiently satisfy the condition of unidimensionality. Given a set of points, P, the selection of any three of those points, Pj’ P , and P]! where it is k known that Pk lies between Pj and P1’ must yield the equation: djk + dkl = djl' If any three points of the continuum fail to satisfy this equation, (Thurstone, 1927) the dimensionality is of a higher order (or is 10 imaginary) and we would posit two or more properties influencing the measurement of the system.* The limitation of a unidimensional scale is, by definition, its ability to represent only one property or attribute. This limitation is manifested in two ways: (a) if our purpose is to measure a uni- dimensional attribute and our results fail to satisfy the distance equation, then we must choose between the possible exPlanation that measurement error exists or that multiple properties were measured; (b) if our purpose is to measure all pr0perties of a system and identify those which are salient to interpretation, then we must construct numerous unidimensional scales, all of which face the first limitation, and any of which may be inconsistent with the others such that they are incomparable directly. The primary advantage of constructing a metric multidimensional scale is that it can overcome these problems by directly incorporating *The concept of distance is introduced at this point as a criterion for examining the rigor of scaling types. To satisfy this criterion we should limit our discussion to levels of scaling which have a known dis- tance function (e.g., interval and ratio scales). Since much of social and psychological measurement is based upon ordinal "scales" and since it is our purpose to examine the limitations of this, so-called, method of scaling, it is of some utility to note that what we previously (p. 2) referred to here as ordinal scales do not exist, as such. While the Sroperty of order may exist for a set of magnitudes, ordering itself oes not constitute measurement because it lacks the ability to be re- presented spatially; it lacks a distance component. More correctly we should refer to ordinal scales as ordinal relationships. We should dis- tinguish, separately, ordinal scales which are infiactuality interval scales which possess the quality of order while assuming that distances exist; this distance is, however, unknown and no attempt has been made to measure it. For a more lengthy philosophical discussion of this principle and an underlying rationale, the reader may examine Descartes (1685), Kant (1755), Newton (1686), Mach (1893), and others. ll them into the scale design. Since it is intended to measure many at- tributes at once, the first problem of measuring unitary properties becomes the purpose of multidimensional scaling. The second problem does not arise because, by constructing a single scale, attributes are treated as directly comparable. For communication variables, or other process-oriented variables, the problem of dimensionality is actually a problem of complexity; it is obvious that the ability to compare many attributes simultaneously increases our predictive or explanatory capabilities greatly. A rem search design can be simplified considerably if changes in many vari- ables can be measured with a single multidimensional instrument rather than with many lesser scales. A greater problem for communication research, which metric multi— dimensional scaling will be shown to overcome, is the effect of utilizing ordinal, distanceless, measures as opposed to interval/ratio measures. Because the ordinal scale represents only the order of properties and not their magnitude it can only reveal transpositions in order. It is easily seen that much change can occur without disturbing the order of a set of stimulus-objects; without being able to measure that change, how— ever, comparison and prediction are impossible. Torgerson (1958:31) underscores this view of measurement: The interval and ratio scales are by far the most use« ful measurement scales employed in science. As a matter of fact, the term measurement is often restricted to these kinds of scales, both in the ordinary use of the term and in the more advanced discussions of the topic. (See, for example, Rd at :1 \M 12 Carnap, 1959, p. 9; Hempel, 1952, p. 58; and Campbell in Ferguson, 1990, p. 3H7.) It might further be noted, that, in discussion of the nature of measurement, the distinc- tion between fUndamental and derived measurement is also commonly made only in terms of interval and ratio scales. An example, drawn from Myron Wish‘s "Tea-Tasting" experiment (Carroll, 1972), should clarify this discussion. We may imagine several different cups of tea, as stimuli to be judged on the basis of first one and then many properties, and scaled using different levels of sophistication. In the first case, we can imagine four cups of tea which vary in sweetness according to the amount of sugar present. If we present the subjects with pairs of teacups and ask them to judge which cup is sweeter we will arrive at an order list from least to most sweet such as: 1. cup b 2. cup c 3. cup a 11». cup do If we know that the stimuli are cup a = 2 teaspoons of sugar, cup b = 0 teaspoon, cup c = 1 teaspoon, and cup d = 3 teaspoons, then the result shown above would be considered correct within a monotonic transformation. However, we would not be able to recognize, in subsequent presentations of the stimuli, whether or not changes in the sugar levels had actually produced changes in judgements unless the order of the list was trans- posed. Again, imagine that we present these four stimulus-objects. However, 13 this time we use an interval or ratio scaling technique such as dis- similarity judgements (Woelfel, 1972) or an ordered metric (Coombs, 196u:80-u) so that values reported will be‘proportiOnal on a scale of equal intervals with a real distance component. This information might be represented:* cup cup cup cup b c a d I l --------- I ------- ---------------- O u 2u 3u where u represents a known but arbitrary unit of measurement. A longitudinal aspect of the experiment becomes available to us when the component of distance is added. As we have shown, unless the order of the stimulus-objects becomes transposed when changes in the system are introduced, the change can not be measured by an ordinal technique. However, with our ability to observe 35323 differences in distance along the scale from times, t to t2, longitudinal measurement 1 becomes a meaningful activity allowing us to add a class of process- oriented variables to our experiments. To represent this, suppose that we add one teaspoon of sugar to cup c, one half teaspoon to cup a, and no sugar to either cup b or cup d. Our scale (assuming the judgement values were proportional to actual sweetness levels, an empirical question) would now appear like this: cup cup cup cup a d C o—U‘ *In addition to the representation of distance we are now able to visualize the order relationship expressed in our earlier "measurement." 1H If the original distance from cup b to cup c had been 10 scale units, the judged change in sweetness would yield different results for the two types of scales discussed. Obviously, a significant event with respect to the overall structure of the stimulus-objects and the rela- tionship of the changed stimuli has occurred; however, only the inter- val scale takes into account enough information to explain that event, or even report that the event had occurred (figure 2). Interval Scale Ordinal Scale 2 :2_ ASweetness :_ f3 ASweetness cup a 20 25 +5 3 3 No change cup b 0 0 0 l 1 No change cup c 10 20 +10 2 2 No change cup d 30 30 0 H u No change Figure 2. Changes in sweetness represented on two scales, This example may now be expanded to the multidimensional case. We ihave sought to represent the property of sweetness as a measure of the difference between cups of tea, yet this difference might be portrayed as well by measuring the properties of temperature, strength, color, or age. Logically, if all of these and, potentially, other attributes can be used as measures of the difference between stimuli then some combina- tion of measures should increase the accuracy of description. One way in which we might achieve greater accuracy would be to list all the possible attributes and create a measuring instrument for each; numerous problems are incurred with this approach however, not the least of which is the fact that such a procedure requires that all attributes 15 along which the teas may differ must be known to the investigator in advance. A second method would be to measure the stimulus—objects by a direct technique, such as having the subjects judge dissimilarities without regard to any specified attribute. In this way, it is neces- sary for the judgement to be made on the basis of those attributes which are salient to the judge. This is the thrust of a multidimensional ap- proach to scaling attributes. The procedure for generating a multidimensional scale (which will be discussed in more detail in a later chapter) entails the derivation of a judgement of relationship for all possible pairs of stimulus- objects and the transformation of the judgement matrix into a matrix of loadings, or projections, on orthogonal axes of a real Euclidean space.* From these projections we may identify the axes (or dimensions), examine structure and, if the judgements are metric, use the scale to observe change over time.** *It may also entail projections in imaginary space. Hypotheses about the imaginary component will not be dealt with in this work. For a mathematical treatment of this problem, see Wilkinson (1961). **0ther models exist for the interpretation of multidimensional data, such as the "city-block" (Householder and Landahl, 19u5; Atteneave, 1950) and hierarchical cluster analysis (Johnson, 1967), however both rely on measures of distance and the "real-world" conception of Euclidean space for their particular representations. As suCh, the Euclidean model supplies both a simpler and more fundamental system for the examination of concept relationships. l6 Returning to the tea experiment, we may examine measurement differences for the multidimensional case (Carroll, 1972): Twenty-five hypothetical cups of tea were described in terms of pairs of description words or phrases. The first set of descriptions referred to the temperature of the cup of tea. Five temperature related words of phrases--ice cold, cold, lukewarm, hot, steaming hot--were used. The second set of descriptions specified the amount of sugar: no sugar, 1/2 teaspoon, 1 teaspoon, 2 teaspoons, u teaspoons. Subjects were shown a standard size styrofoam cup, in which they were to imagine tea of "moderate strength" with no cream or lemon. All 25 possible combinations of the two sets of descriptions were used to define the basic (verbally described) stimuli. The 300 possible pairs of stimuli were generated in a random order to form the basic questionnaire. There were a total of 12 subjects, di- vided randomly into two sets of subjects which respon- ded to the items in opposite orders. Each of the 12 subjects was asked to give . . . a rating of dissimilarity (called "degree of difference") of the pair on a scale from 0 (for an indistinguishable pair) to 9 (for an extremely dissimilar pair). Op AI 5% Us. D.“ Ar 4 v.‘ 17 The ordinal data from this procedure were then analyzed according to one of the major multidimensional algorithms:* INDSCAL (individual differences scaling) was applied to the dissimilarities, producing the group stimulus space shown in [Figure 3]. . . . Note that the basic lattice structure embodied in the factorial design used to gen— erate the stimuli is quite clearly in evidence (though a bit distorted) in the stimulus space, and, furthermore, that the "sides" of the lattice are very nicely parallel to the two coordinate axes. These axes are exactly as they came out of the analysis; no rotation whatsoever has been done. From this example, we can distinguish two levels of scaling rigor, ordinal and interval, for the multidimensional measurement approach. Ordinal, or nonmetric, multidimensional scales are generated by transformations based on an "unknown distance function" (Shepard, 1962a). That is, by using a fixed monotonic function based on identifiable relationships in ordinal data such as those generated by dominance measures (Coombs, 196n; Carroll, 1972) or profile measures * INDSCAL, developed by Carroll and Chang (1970), is a multidimen- sional scaling program for deriving the product matrix by the Eckart- Young rank reduction algorithm (Eckart and Young, 1936). Additionally, it generated weighting factors and a canonical decomposition of data to produce a space which accounts for individual differences in report- ing raw judgements. INDSCAL is limited by the normalization to unity of both scalar products and solution matrices which eliminates the original distances reported. 18 4,C ®'@ 4,LII ® , f ”J sq" @ (x 2,L a S ”H II. ’ \ j I c ”‘3 .‘“ s‘ a ®.@ 3 ® 63 ‘ ® ® a TEMPERATURE Figure 3. The unrotated two-dimensional solution (n=l2) of the Wish data using INDSCAL on dissimilarities data. In this figure the coding: 0, 1/2, 1, 2, u refers to the number of teaspoons of sugar specified and for temperature: IC = ice cold, C = cold, LW = lukewarm, H = hot, and SH = steaming hot. (Adopted from Carroll, 1972.) 19 (Shepard and Carroll, 1966) rather than actual distance estimates, a configuration approximating the latent structure of the data can be fOund. Metric multidimensional scales are generated directly from unidimensional judgements made with interval or ratio scales. Meas- ures for this type of scale are usually in the form of proxemities data (Torgerson, 1951, 1952; Woelfel, 1972); however, other forms such as frequency scores and individual differences may be found;* The metric procedure essentially asks the subject to make judge- ments about how each stimulus-object differs from all other stimulus- objects. Since the attributes on which the stimuli are to be judged are not specified, the individual is able to use those most important for distinguishing each pair. This results in a dissimilarities matrix *Some controversy exists as to which component of the technique, the original data or the multidimensional computational algorithms, constitutes the basis for discriminating metric and nonmetric scales. Traditionally, the computation and transformation processes have been used to judge the quality of the scale (Shepard, 1962a, 1962b; Kruskal, 196%, Guttman, 1968); those requiring an iterative procedure to adjust discrepancies are considered nonmetric, those using a direct derivation of the latent structure of a numerical set (such that differences bet- ween numbers in the set are maintained within a linear transformation) are considered metric. This assumes the ability to treat the data by rules of order and transformation (Hempel, 1952) which apply to more sophisticated numbering systems than we have actually utilized with our original measurement procedure. On this basis it would seem more natu- ral to distinguish multidimensional scaling'levels on the basis of data gathered. Obviously, the computational criterion is then also met if the appropriate transformation is applied to that data. A less rigid transformation would yield a nonmetric solution for interval type data, due to the standardization necessary to compute a nonmetric transforma- tion, while the use of a metric algorithm for ordinal data would only occur under strong theoretical scrutiny and with rigid restriction on the approximation of interval numbers. 20 in which all of the information about the data interrelationships is present but much of it (i.e. the attributed used to make judgements) is in a latent form. Generating the multidimensional scale, then, has two functions: (a) it attempts to derive the most parsimonious de- scription (i.e., the smallest set of numbers to represent all of the information) and (b) it allows us to make explicit some of the informa- tion that is latent in the original dissimilarities matrix. From the tea experiment it can be seen that a similar result for ordinal and interval data will be produced. Had the multidimensional transformations been performed upon a dissimilarities matrix derived from an interval scaling procedure, the result would have been a con- figuration based upon reported distances between the concepts scaled rather than a configuration based upon distances estimated from corre- lational strength. Figure 3 is a representation of the data in two dimensions; the configuration closely resembles an array of the stimulus—objects on the basis of actual physical properties. It is expected that the metric judgement procedure described would have further increased parsimony between the stimulus set and the final coordinate values. In either case, the quality of the data representation is improved to reflect the overall structure of the relationship rather than the order and/or distance of any single dimension of the relationship. However, the multidimensional ordinal configuration, like undimensional ordering is not comparable over time. The structure is an artifact of a monotonic transformation which provides only one of many possible interpretations. 21 The problem of representing change over time multidimensionally is that it requires both angle and distance. Since most nonmetric procedures rely on standardization of the data, they remove the dis- tances reported and substitute correlations. Correlation between any two concept values is rij; further, rij = cosaij, where a is the angle between two vectors representing the concepts i and j. Thus, the changes occurring over time may result in higher or lower correlations which will be represented as changes in angle (direction); however, changes in magnitude will not be represented at all. Figure n represents the same data set at two different points in time for which the magnitudes are changed. In both cases the angles remain constant and the correlation matrix, R1 is: 1.0 1.0 0.0 —1.0 0.0 . 1.0 0.0 —1.0 0.0 R1 = - 1.0 0.0 —1.0 - . 1.0 0.0 434.0 . . . LL 22 Figure u. A multidimensional configuration at two points in time (magnitudes changed, correlations remain constant). An interval configuration has both angle and magnitude which may be compared directly to determine longitudinal differences. It is then possible to introduce measures of change (such as velocity, accel- aration, and displacement) and posit change agents in mathematical terms (force, inertia, and momenta). It is the conceptual analogy to motion that we will refer to as the behavioral characteristics of process. 23 Objectives of the Thesis Torgerson seeks to remind us that science and measurement can be conceived simply (l958:2): Science can be thought of as consisting of theory on the one hand and data (empirical evidence) on the other. The interplay between the two makes science a going con- cern. The theoretical side consists of constructs and their relations to one another. The empirical side consists of the basic observable data. Connecting the two are rules of correspondence which serve the purpose of defining or partially defining certain theoretical constructs in terms of observable data. It is the purpose of this thesis to examine this view of science, and specifically, to apply the discussion of measurement toward the development of a more rigorous application of the principle of iso— morphism in science. In the following chapters, the notion of high level scaling in the behavioral sciences will be considered and an application to communica- tion science developed. The thesis will make two main arguments: 1) that the complex and multi-faceted nature of communication phenomena requires a multidimensional approach to measurement, and 2) that the processual character of communication phenomena requires interval or higher level scaling. Finally it will contend that these two require- ments can be met by metric multidimensional scaling techniques. One such system, Galileo, along with complete computer software, will be presented as an example. Chapter II Historical Development of Multidimensional Scaling The development of multidimensional scaling appears to be culmina- ting in the form of a broad and powerful technique for measuring self- conception and influences upon the self-conception on a micro level, and cultural processes on a larger scale. It is a technique which specifically for communication study, holds great potential for under- standing the effects of communicative acts, and their subsumed compo- nents, on cognitive aspects of the act. Historically, many attempts have been made to achieve this goal, and these attempts have contrib- uted directly, and indirectly, to the development of multidimensional scaling. It will be useful to a more complete understanding and apprecia- tion of how MDS can provide such a powerful interpretation of data to examine the development of the mathematical and theoretical components involved. Further, it will be briefly placed into the context of the more fundamental sciences from which it derives. Pre-Psychophysical Influences on Multidimensional Scaling Multidimensional scaling as a psychological measuring technique can be attributed primarily to the work of Torgerson (1951, 1952, 1958). It also draws heavily on the theoretical construction of Gulliksen (19u6) and Thurstone (1927a), and the mathematic contributions of Hotelling (1933), Young and Householder (1938), and Garnett (1919a). However, it is useful to examine the more basic scientific roots of the 21+ 25 technique and place it into the perspective of science as a whole before considering the more technical aspects of the multidimensional "model". The mathematical history of multidimensional scaling is derived, appropriately, from the mathematics of astronomy and specifically celestial mechanics. Since MDS attempts to treat the self-concept or culture as an analogy to Euclidean real space, the structure and genera- tion of such space may be examined with regard to the context from which it developed. The notion of "space" was originally treated, philosophically, by Plato, and developed conceptually by Aristotle, as the universe of ob- jects and abstractions (intelligence) in which man functions. As a pure scientific construct, Euclid, in the third century B.C., proposed space as the context for objects in relationship to one another. To express this, Euclid proposed the geometry, a formulation of mathemat— ical rules to define physical relationships according to distance and direction (angle). The most cogent presentation of Euclid's geometry for deducing relationships from information that is incomplete or not in its most representative form, was its application to celestial description by Aristarchus of Samos (310-230 B.C.). It was Aristarchus' contention that the universe was heliocentric, and that despite the observable motion of the sun and other bodies in the sky the earth's motion would describe a circle around the sun. To argue his model, Aristarchus used a viewpoint outside of this system, where he could "see" the earth and the other known celestial bodies as comparable globes in space. He 26 then intersected these bodies and their shadow cones with planes, drew intersections as circles and triangles, and applied Euclid's rigid methods described in the Elements to demonstrate the relative positions and the most likely paths of movement. In addition, by treating the problem as a geometric lemma, Aristarchus established that the hypothet- ical structure of the spatial relationships could be derived from a known subset of the interrelationships of the celestial bodies (which would intuitively suggest a different solution). The Greek views of science, particularly astronomy, mathematics, and philosophy, dominated Western thought until the fifteenth century and later. Among the first of the major challenges to this trend, it was Descartes' notion of separating the abstract reality of the entel- fishy from physical reality that provided much of the stimulus to change. In his Principia PhiloSophiae (1685), Descartes stated, "I will explain the results by their causes, and not the causes by their results." With this he initiated a set of theories of space and forces of motion and maintenance in space which made it possible to predict observable phenomena. It was this theorizing by Descartes which Newton was responding to when he introduced the laws of motion and initiated the classical mechanics (Pannekoek, 1961). Both Newton and Descartes, however, had in common the notion of a Euclidean model as the reference system in which to examine and describe the forces and effects that they postu- lated. From this point in scientific evolution, the disciplines of astro- nomy and mathematics began to develop simultaneously and mutually, in 27 the form of celestial mechanics (Hagihara, 1970). It was also at this juncture that the science of the mind began to develop separately, first in the form of philosophy, and later in the forms of psychology and psychophysics. To express his theories of mechanics, Newton simultaneously and independently with Leibniz also invented the first forms of differen- tial and integral calculus; these mathematical aspects were refined by others to perpetuate the study of mechanics and developed to provide working algorithms for manipulating spatial concepts. For the later psychometric techniques of factor analysis and multidimensional scaling this meant that the relationship of data expressed by points would be interpretable by the mathematics of mechanics. Theoretically, the notion of representing points in space was formalized by Pierre Simon de Laplace who is credited with the founding of celestial mechanics as a major function of astronomy. Laplace, with Lagrange is responsible for the validation of Newtonian mechanics at a point in time when the model was about to be abandoned (Pannekoek, 1961). In the late eighteenth century, they measured the accelaration and retardation of planetary motion, explaining perturbations and ap- parent inconsistencies of force as functions of distance and elliptical motion. This action gave rise to the more extensive mathematical de- velopments which followed in the nineteenth century. During the nineteenth century, much of the mathematics of psychom- etrics was developed in the context of astronomy. Following Laplace's advances and publications (1796), numerous computational methods were developed by Gauss, Jacobi, Bravias, and Seidel (Wilkinson, 1965; 28 Jacobi, 1846; Harmon, 1967; Pannekoek, 1961) and refined by others (notably astronomers and mathematicians such as Bessel, Encke, Olbers, Cauchy, Hansen, Leverier, Poincare, and many others). Several important contributions which affect the conception of multidimensional scaling were presented by: l. C. F. Gauss. In 1804 he originated the "method of least squares" which, "by the condition that the sum total of the squares of the remaining errors shall be a minimum, the 'most probable' value of the unknown quantity is found (Pannekoek, 1961)." Geometrically and statistically, this is interpreted to mean the sum of the squared distances from the projections of points on a line when minimized yields the line of best fit to the configuration of values. It is this foundation upon which equations for factors and dimensions are based. 2. C. G. J. Jacobi. The problems inherent in interpreting ma- trices (Jacobi, 1846; Wilkinson, 1965) led to the development of the diagonalization method for deriving the eigen roots. This not only provided the technique for generating the latent structure for an un- limited number of points in space; it also provided a rapid convergence algorithm upon which many high speed computer routines are based. 3. A. L. Cauchy. In his work on determinants (Kowalewski, 1909), Cauchy provides a proof of the reality of roots and the determination of the number of underlying dimensions. This treatment of dimension— ality, which is basically dealt with in discussion of quadratic sur— faces in analytic geometry, influenced Hotelling (1933) in his develop- ment of factor analytic procedures and the restriction of interpretations 29 made from results of those procedures. Near the end of the nineteenth century, pure mathematics and the mathematics of celestial mechanics began to diverge. Computational devices, in conjunction with improved technology for astronomical measurement allowed the astronomer to develop theory and concentrate less on improving and developing more accurate algorithms. Conversely, mathematics ceased to be devoted exclusively to the development of computational methods, and began to gain recognition as the language of scientific theory. Figure 5 represents these developments graphical- 1y. Early Factor Analysis At the same time that mathematics began to diverge from its roots in astronomy, psychology began to seek and develop more rigorous meth- ods of measuring ability and behavior and expressing theory. Early twentieth century psychologists such as Spearman and Thomson began to draw on statistical science to express their theories probabilistically. In 1901, Karl Pearson, a mathematician and statistician, published the paper "0n Lines and Planes of Closest Fit to Systems of Points in Space." In it he presents the method of principal axes as a technique fOr deriving the line of best-fit through a system of points in two, three, or n dimensions. This method was significant because it allowed points to be represented by a vector from the origin of a coordinate system to some point in the space defined by the coordinate system. (Frequently the vector is represented only by its endpoint). Further, this vector is a function of the configuration itself. It is obtained Ol—PYTHAGORAS | O2-PLATO : 03-ARISTOTLE 041EUCLID 05- ARISTARCHUS .~-_—_ (Astronom Mathematics, Philosophy) Ya : 06- DESCARTES I O7-NEWTON (Astronomy, Mathematics) (Philosophy) 08-LAPLACE OQ-LAGRANGE l 10-GAUSS ll-JACOBI (Astronomy) (Mathematics) l2-POINCARE 13-PEARSON V i Figure 5. (Behavioral sciences) Ol—foundations of mathematics 02-philosophy of science 03—scientific principles 04—principles of geometry OS-application of geometry to celestial arrangement 06-causality; philosophical and scientific revision 07-classica1 mechanics 08-celestial mechanics 09-with Laplace, validation of Newtonian mechanics lO—method of least squares ll-numerical methods for solving spatial manifolds l2-theoretical astronomy lS—statistical mathematics Chronological development of pre-factor analytic contribu- tions to the development of multidimensional scaling. 3O 31 by minimization of the sum of squares of the perpendiculars, or pro- jections (Figure 6) of a set of points representing any data configura- tion. projection Figure 6. Line of least squares best-fit. Earlier astronomical techniques allowed for the representation of celestial points on an arbitrarily designated coordinate system derived from Euclidean distances. By Pearson's technique, the endpoint of the vector (the least squares line of best-fit) could be placed in a co- ordinate system or the line itself could be designated as one axis of the coordinate system. This was useful, statistically, when the purpose of the representation was not to predict from one variable to another but to observe the interrelationships of a number of variables. By this, Pearson notes, "we observe §_and y and seek a unique functional rela- tionship between them." Utilizing this geometric representation of data, and the means, standard deviations, and correlation coefficients used to generate it, 32 Spearman (1904) initiated his theory of the general factor. With his theory he posited that ability was correlated with intelligence and that from the measurement of a set of abilities that a factor, or com- ponent, common to all abilities could be identified. To test his theory, Spearman developed the "criterion of hier- archy" or method of tetrad differences. If for four tests, or all com- binations of four tests, the intercorrelations could be accounted for by a single source of variation (or factor) then the general factor was identified for those tests. This was achieved when the coefficients in any combination of two columns of the intercorrelation matrix were found to be proportional; that is, the following equations were sat- isfied: / / P r r13 23 r14 24 r12/r32 Plu/rsu r12/P42 = r13/ru3. Spearman found that for a considerable number of the psychophysical tests the intercorrelations satisfied the proportionality criterion and could be accounted for by the general factor. Additionally, he postu- lated that every test which satisfies the proportionality criterion also contains a second factor that is specific to a given test and, statistically, represents that portion of the variance that does not correlate with the other tests. Thus by Spearman's theory, a per- fectly reliable test would have two components, g (the general factor) and s (the specific factor), such that g2 + s2 = 1.00. Fruchter (1954) 33 .866 .5 6 l 07 .71.” .917 .u s g 2 .5 .866 '8 u 3 '3 .954 .600 Figure 7. The "two-factor theory" applied to six tests. The values of the overlapping g factor are the correlations of each test with that factor. The 8 factor is represented by the residuals (s2 = 1.0 - g2). Table l. Coefficients for the six tests represented in Figure 7. g * s1 s2 33 S4 s5 S6 h2 l .7 .71” .49 2 .5 .866 .25 3 .3 .954 .09 H .8 .600 .64 5 .4 .917 .16 6 .5 .866 .25 suggested the schematic presentation of this "two-factor theory" shown in Figure 7 and Table 1. In a number of articles (Spearman, 1904, 1914a, 1914b, 1920, 1922, 1923; Krueger and Spearman, 1906; Hart and Spearman, 1912), the two factor theory was considered, defined, and developed. Further statisti- cal tests for the general factor, such as satisfaction of the equation: 34 r - r r ac ag cg ac.g k k 38 CS where k = 1.0 - r 2 and k = 1.0 - r 2, were proposed to strengthen a8_ a8 cg cg the argument for a two—factor theory. However, it became readily apparent to Spearman that the two-factor theory was insufficient. Spearman's adaptation of Pearson's (1901) method for deriving the projections of points on a least squares line of best-fit involved de- riving the communality, h2, such that an individual test loading on the general factor, an’ was equal to he2, where e is the element of the factor being computed. Given the criterion of proportionality, the equation for the loading is: 2 2 2:(r'ejpek; j, k = 19 29.0.9 n; j: k # e; j < k) a : h = . 2(rjk; j, k = l, 2,..., n; j, k f e; j < k) For example, the square of the general factor coefficient (loading) for the first variable in a set of five would be: 2 r12"13 + r12r14 + r12115 + r13"19 I r13115 + r14r15 a10 = ' ° r23 + I24 I P25 + r34 + P35 + ”as Harmon (1967) suggests the more convenient computational fOrm: n 2 n 2 rej — Z rej2 a 2 = j:l j=l 80 n n ~—fi (e is fixed, j # e). 2 r. - Z r . jonocnzcnwocnzm cm 9 £940 09 HHHJ-‘MHD u u nun II II 9‘9 22 nzzxomzuawqonoa N2 1224 ‘6'. U... END CV CC- MCM‘WD at. “0 MMLLDMWDMNQODO IO. U ZNDONO xnm vac-Hm» w‘O Dmm4- 9 O o 99 9K9'HA'RIN‘J9K’9772 J KT9HATRIX|J,KD9"2 NSO 50’ 7209720960 T-NATRIXGJgK‘) N (J SQ T( *8 N H UCUUOUUCUCOCCIOOUCUUICUOUC......O¥.l....5.UUUCUUUIUC.CO.Q.¥l F GALILEO SUBROUTINE FILE U’UU...UUOUICU.'¥.U.UUUOUUUI¥§U..U§.UOUUUCOU.OUUUOUUO'UUUUUU IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII“ 0984