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Paul E. Green, Frank J. Carmone, Jr., and Scott M. Smith. Multidimensional Scaling. Concepts and Applications. Boston: Allyn and Bacon, I989. Pp. viii + 407, $60. Compared to other books on multidimensional scaling (MDS), this text represents a practical attempt to spread the use of MDS and related models. The applied approach of the book makes it specially useful for students and teachers alike, as well as for other researchers not already familiar with MDS. The subtitle (Concepts and applications) forthrightly notes this emphasis. After reading the book, our prediction is the impediments Ramsay (1978, p. 13) saw the use of MDS in empirical research will almost certainly wane because the influence of this book. Now computer programs will be readily available (two diskettes containing an important set of PC-MDS programs are included with the book), and potential MDS users are exposed to the concepts underlying MDS models, algorithms and computer programs. The book is an update to Green and Carmone (1970), but unlike the earlier version, the authors now avoid referring to "marketing research" in the title, although its applications are in fact only in the marketing research field. While this implicit emphasis may be regarded as a limitation by some readers (e.g., psychologists), we nevertheless find it fortunate because it allows the authors to offer valuable details regarding data collection, which were not usually described in the earlier literature. The book is divided into four parts, with the first being a brief, nontechnical overview of the field. Part II is the longest of the four and consists of five sections, where some basic (Coombsian) theory of data is provided and various MDS models are explained. In so doing, the authors place the theories in logical and historical. Another interesting and innovative feature of the book is that both consonance and dominance data (Coombs, I964) or, more specifically, proximity and preferential choice data (Carroll, 1980) are included. Most recent books have concentrated mainly (but not exclusively) on proximity data (e.g., Arabie, Carroll, & DeSarbo, 1987; Kruskal & Wish, 1978; Schiffman, Reynolds, & Young, 1981). In contrast, the inclusion of both these kinds of data allows for the coverage of (a) various MDS models (for proximity data), and (b) vector and unfolding models (for preferential choice data). Additionally, some MDS-related models such as principal component analysis, cluster analysis, and correspondence analysis are comparatively and briefly introduced. The authors have made a laudable attempt to present a considerable number of models, but one feels they might have fruitfully gone deeper. We specially miss more background on the INDSCAL model (Carroll & Chang, 1970), which is the most important contribution made to MDS theory since the earlier version of this book (1970), as shown by the number of computer programs which have implemented the model (ALSCAL, INDSCAL, MULTISCALE, SINDSCAL, SMACOF) and the large amount of applications of the model utilized over the last two decades (see the annual bibliografic survey service). The occasional lack of precision is frustrating. For example (p. 50 et seq.), the distinction between the SINDSCAL program (Pruzansky, I975) and its predecessor, the INDSCAL program is often overlooked, as is the difference between KYST2A (Kruskal, Young, & Seery, 1977) and its predecessor. Because the software supplied in sample form with the book (and marketed by Smith independently of the book's pub351

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lication) uses the earlier names--presumably because the latter ones are still claimed as proprietary by AT&T Bell Labs--, much confusion is likely to result from the careless use of programs names. Moreover, such other programs in common use today as MULTISCALE-II (Ramsay, 1977, 1982) or SAMCOF1B (de Leeuw, 1977; de Leeuw & Heiser, 1977; Stoop & de Leeuw, 1982) are not mentioned, while the MDSCAL program is introduced (p. 51 et seq.), but not referred to in Table 3.1 (implicitly assuming is not in common use today). Part III is a selection of literature reviews on MDS (and related models), and is divided into three sections corresponding to MDS, cluster analysis, and correspondence analysis. Each contains two papers. In the MDS section, the papers enclosed are " A Review of Multidimensional Scaling in Marketing Research" by Cooper (1983), and "Multidimensional Scaling" by Carroll and Arabie (1980). The papers selected for cluster analysis are "Cluster Analysis in Marketing Research: Review and Suggestions for Application" by Punj and Stewart (I983), and "Overlapping Clustering: A New Method for Product Positioning" by Arabie, Carroll, DeSarbo, and Wind (1981). The papers for correspondence analysis are "Correspondence Analysis: Graphical Representation of Categorical Data in Marketing Research" by Hoffman and Franke (1986), and "Comparing Interpoint Distances in Correspondence Analysis: A Clarification" by Carroll, Green, and Schaffer (1987). A reader of these sections has the opportunity to benefit from experts' theoretical knowledge, which nicely complements the empirical approach of the book. Particularly interesting is the Carroll and Arabie (1980) paper, where a very important theory of data for MDS is proposed. In part IV the authors make the most innovative contributions with regard to the earlier version of the book. A PC version of a wide range of MDS (and related models) programs is provided. The programs offered are the following: KYST for two-way (metric and nonmetric) MDS and unfolding (Kruskal, Young, & Seery, 1973), INDSCAL for INdividual Differences SCALing, a weighted three-way (metric) MDS (Chang & Carroll, 1969a), MDPREF (Carroll, 1972; Carroll & Chang, 1964; Chang & Carroll, 1969b) and PREFMAP (Carroll, 1972; Chang & Carroll, 1972) for internal and external preference analysis respectively, PROFIT for property fitting (Chang & Carroll, 1968), CORRESP for two-way contingency tables correspondence analysis (Carroll, Green, & Schaffer, 1987), and Howard-Harris for hierarchical cluster analysis (Harris, 1981). Although not explicitly stated in the book, the version of the programs provided in the diskettes is unfortunately a student version; that is, only very small data sets can be run. For example, KYST program does not accept runs with more than ten rows (or columns) and solutions cannot be obtained in more than three dimensions. The remaining programs have similar limitations. Regarding the selection of the programs, some comments are also needed. The MDPREF and PREFMAP programs may overlap. MDPREF was designed to carry out the (metric) vector model, whereas PREFMAP is capable of carrying out both vector and unfolding models, in metric and nonmetric cases. Carroll (I972, 1980) showed that the vector model is a particular case of the unfolding model. In fact, phase IV in PREFMAP corresponds to the vector model. The only justification for including both programs is that MDPREF performs an internal analysis, whereas PREFMAP only performs an external analysis. In the former case, the program estimates both the coordinates of the stimuli and the subjects' ideals or vectors, whereas in the latter the user has to provide the coordinates of the stimuli to allow the program to estimate the (subjects) ideal or vector. Providing the coordinates of the stimuli is not a problem when good programs such as KYST or INDSCAL are available. Obviously, because of the different nature of the models and methods involved in internal versus external

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analysis, the corresponding solutions are in general not identical. However, this difference between solutions is probably of more theoretical than practical interest, and minimizing such differences is consistent with the approach of the book. No program is provided for two-way additive clustering (Shepard & Arabie, 1979; Arabie, & Carroll, 1980), although the model is introduced in part III by Arabie, Carroll, DeSarbo, and Wind (1981, pp. 235-246). Also, applied research workers would benefit from inclusion in the package of some additional software to accompany INDSCAL analyses, for implementing the power metric model (Ramsay, 1977), a nonmetric version of the INDSCAL model (Takane, Young, & de Leeuw, 1977), a discrete counterpart of INDSCAL (Carroll, & Arabie, 1982, 1983), or a version of INDSCAL allowing for solutions in spaces other than Euclidean (Heiser, 1989). Citations are always an important indicator of the quality of a book, both regarding its form and its content. In this case, there is no doubt that references were judiciously selected; however an unfortunate chain of misidentifications (mainly although not exclusively found in part IV) casts a shadow on a book which is otherwise highly appealing. For example, with respect to the Howard-Harris algorithm, it is first identified as "Howard-Harris algorithm (1981)" (p. 123), then attributed to N. Howard and B. Harris (without date) in the heading of p. 332, and, finally, in a footnote on the same page documentation for the program is said to be taken from "Carroll (1973)". Nevertheless in the reference section, "Carroll (1973)" is omitted, and the only related available reference is Harris (1981). Citations to Carroll (1964, pp. 312-313), Chang and Carroll (1968, p. 304), or Carroll and Chang (1968, pp. 288-289) are not included in the reference section (or if they are included, either the citations or the references are confused). The order of the authors is occasionally interchanged (e.g., PROFIT program is attributed to Chang and Carroll, undated, in the heading of p. 318, but then attributed to Carroll and Chang, 1970, in a footnote on the same page; note that the date is also confused). Finally, two references from Chang and Carroll in the year 1969 are identified in the reference section as 1969a and 1969b, but in the text it is never specified which is a or b (see, e.g., pp. 84, 287). In spite of such confusions that could have been avoided by more careful editing, our general impression of the book is highly positive. It is well organized, its empirical approach fills a gap on MDS, and the PC programs provided may play an important future role in helping students, teachers and researchers unfamiliar with MDS to obtain a better understanding of models, what they are for, and how to use them in the applied research. UNIVERSITY OF SANTIAGO DE COMPOSTELA, SPAIN

Constantino Arce

References Arabie, P., & Carroll, J. D. (1980). MAPCLUS: A mathematical programming approach to fitting the ADCLUS model. Psychometrika, 45, 211-235. Arabie, P., Carroll, J. D., & DeSarbo, W. S. (1987). Three-way scaling and clustering. Newbury Park, CA: Sage. Arabie, P., Carroll, J. D., DeSarbo, W. S., & Wind, J. (1981). Overlapping clustering; A new method for product positioning. Journal of Marketing Research, 18, 310-317. Carroll, J. D. (1972). Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & S. Nerlove (Eds.), Multidimensional scaling: Theory and applications in the social sciences, Volume 1: Theory (pp. 105-155). New York: Seminaar Press. Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance) data. In E. D. Lantermann, & H. Feger (Eds.), Similarity and choice (pp. 234-289). Viena: Hans Huber. Carroll, J. D., & Arabie, P. (1980). Multidimensional scaling. In M. R. Rosensweig, & L. W. Porter (Eds.), Annual Review of Psychology (Vol. 31, pp. 607--649). Palo Alto, CA: Annual Reviews.

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Carroll, J. D., & Arabic, P. (1982). How to use INDCLUS: A computer program for fitting the individual differences generalization of the ADCLUS model and the MAPCLUS algorithm. Murray Hill, NJ: AT&T Bell Laboratories. Carroll, J. D., &Arabie, P. (1983). INDCLUS. An individual differences generalization of the ADCLUS model and the MAPCLUS algorithm. Psychometrika, 48, 157-169. Carroll, J. D., & Chang, J. J. (1964, October). Non-parametric multidimensional analysis of paired-comparisons data. Paper presented at the joint meeting of the Psychometric and Psychonomic Societies, Niagara Falls. Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via N-way generalization of Eckart-Young decomposition. Psychometrika, 35, 283-319. Carroll, J. D., Green, P. E., & Schaffer, C. M. (1987). Comparing interpoint distances in correspondence analysis: A clarification. Journal of Marketing Research, 24, 445-450. Chang, J. J., & Carroll, J. D. (1968). How to use PROFIT, a computer program for property fitting by optimizing nonlinear or linear correlation. Murray Hill, N J: AT&T Bell Laboratories. Chang, J. J., & Carroll, J. D. (1969a). How to use INDSCAL, a computer program for canonical decomposition of N-way tables and individual differences in multidimensional scaling. Murray Hill, NJ: AT&T Bell Laboratories. Chang, J. J., & Carroll, J. D. (1969b). How to use MDPREF, a computer program for multidimensional analysis o f preference data. Murray Hill, N J: AT&T Bell Laboratories. Chang, J. J., & Carroll, J. D. (1972). How to use PREFMAP and PREFMAP2--Programs which relate preference data to multidimensional scaling solution. Murray Hill, NJ: AT&T Bell Laboratories. Coombs, C. H. (1964). A theory of data. New York: John Wiley & Sons. Cooper, L. G. (1983). A review of multidimensional scaling in marketing research. Applied Psychological Measurement, 7, 427-450. de Leeuw, J. (1977). Applications of convex analysis to multidimensional scaling. In J. R. Barra et al. (Eds.), Recent development in statistics (pp. 133-145). Amsterdam: North-Holland. de Leeuw, J., & Heiser W. J. (1977). Convergence of correction matrix algorithms for multidimensional scaling. In J. Lingoes (Ed.), Geometric representations of relational data (pp. 735-752). Ann Arbor, MI: Mathesis press. Green, P. E., & Carmone, F. J. (1970). Multidimensional scaling and related techniques in marketing research. Boston: ~Allyn and Bacon. Harris, B. (1981). Howard-Harris hierarchical clustering. Unpublished note. University of Pennsylvania, Philadelphia. Heiser, W. J. (1989). The city-block model for three-way multidimensional scaling. In R. Coppi, & S. Bolasco (Eds.), Multiway data analysis (pp. 395-404). Amsterdam: Elsevier. Hoffman, D. L., & Franke, G. R. (1986). Correspondence analysis: Graphical representation of categorical data in marketing research. Journal of Marketing Research, 23,213-227. Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling. Newbury Park, CA: Sage. Kruskal, J. B., Young, F. W., & Seery J. B. (1973). How to use KYST, a very flexible program to do multidimensional scaling and unfolding. Murray Hill, N J: AT&T Bell Laboratories. Kruskal, J. B., Young, F. W., & Seery, J. B. (1977). How to use KYST2A, a very flexible program to do multidimensional scaling and unfolding. Murray Hill, NJ: AT&T Bell Laboratories. Pruzansky, S. (1975). How to use SINDSCAL: A computer program for individual differences in multidimensional scaling. Murray Hill, N J: AT&T Bell Laboratories. Punj, G., & Stewart, D. W. (1983). Cluster analysis in marketing research: Review and suggestions for application. Journal of Marketing Research, 20, 134-148. Ramsay, J. O, (1977). Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42,241246. Ramsay, J. O. (1978). MULT1SCALE: Four programs for multidimensional scaling by the method of maximum likelihood. Chicago: National Educational Resources. Ramsay, J. O. (1982). MULTISCALE-H manual. Mooresville, IN: International Educational Services. Schiffman, S. S., Reynolds, M. L., & Young, F. W. (1981). Introduction to multidimensional scaling: Theory, methods, and applications. New York: Academic Press. Shepard, R. N., &Arabie, P. (1979). Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psychological Review, 86, 2, 87-123. Stoop, I., & de Leeuw, J. (t982). How to use SMACOF-IB. Department of Data Theory, University of Leiden, The Netherlands. Takane, Y., Young, F. W., & de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least-squares method with optimal scaling features. Psychometrika, 42, 7-67.

PSYCHOMETRIKA--VOL.56, NO. 2, 355--358 JUNE 1991 REVIEWS

Patrick Suppes, David H. Krantz, R. Duncan Luce, and Amos Tversky. Foundations

o f Measurement, Volume H: Geometrical, Threshold, and Probabilistic Representations. San Diego: Academic Press, 1990, XV + 493 pp. $79.50. R. Duncan Luce, David H, Krantz, Patrick Suppes, and Amos Tversky. Foundations of Measurement, Volume 11I: Representation, Axiomatization, and Invariance. San Diego: Academic Press, 1990, XV + 356 pp. $53.55. Here are the long-awaited sequels to Foundations of Measurement, Volume I (Krantz, Luce, Suppes, & Tversky, 1971), and well worth the wait, too! It is a great pleasure to be able to reiterate (Ramsay, 1975) my enormous enthusiasm and admiration for the remarkable intellectual adventure that is so well recounted in these three volumes. Although fundamental measurement began with H61der (1901) and Campbell (1920), the first rigorous and systematic accounts were the papers by Suppes (1951) and Scott and Suppes (1958) on extensive and difference structures. Stevens' (1946) paper defining scales of measurement was a critical point of departure, as was the flowering of the work on utility theory in the mid fifties. On the mathematical side Tarski's work on the theory of models (1954) played a central role. Luce joined the enterprise in his startling and controversial paper on possible psychophysical laws (Luce 1959). It was the chapter by Suppes and Zinnes (1963) on basic measurement theory in the Handbook of Mathematical Psychology (Luce, Bush, & Galanter, 1963) that provided the first widely read and easily accessible account, and this was followed quickly by the first paper on conjoint measurement (Luce & Tukey, 1964) which served to well launch both a new field and a new journal. The first full monographs in the field were by Ellis (1966) and Pfanzagl (1968). Extensions such as polynomial conjoint measurement by Tversky (1967a, 1967b) also followed rapidly, and the appearance of the first volume in the present series (referred to in this review as FMI) not only added further new material, but made obvious the fact that a single book could no longer capture the scope and detail in the discipline at that point. At that time a second volume was promised, and chapters were outlined in FMI. Measurement, in a well defined sense as opposed to innumerable corruptions of the idea, consists of the use of numbers or vectors along with operations such as addition and multiplication to represent, not only objects or other entities themselves, but relations holding among them. Of these, the most basic is order, but measurement in the physical sciences has used addition to mirror an associative commutative solvable empirical operation referred to in FMI as concatenation. Examples of such operations in the behavioral sciences are extremely rare, except where physical measurements are used as indices or pointers to subjective properties, as in psychophysics. There, however, it is argued that the physical empirical relation is of no substantive relevance. In conjoint measurement, on the other hand, addition turns out to represent a structure in which two independent variables combine so as to satisfy the double cancellation axiom, a condition that at first sight appears innocuous, but that turns out to be very strong in its implications. FMI left us with many questions: 1. Are there more general classes of operations outside of the bisymmetric class of 355

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adding, differencing, and averaging that might prove more appropriate for representing certain empirical relations? 2. Is there some way to combine the axiomatic approach in FMI with an explicit recognition of the error and imprecision that is a part of all actual data? 3. What has fundamental measurement to say about multidimensional situations, and about representations based on proximity in particular? 4. What will be the impact on empirical research of this work? Are we witnessing a birth of a new branch of mathematics, of logic, or of philosophy of science? Where will the points of contact with subdisciplines such as psychometrics and mathematical modeling be? Responses to these and other questions (such as the logical status of dimensional analysis) were left to the next volume, which was expected imminently. These are hard questions, and this is no doubt sutficient by itself to explain the delay of nearly two decades. But the field has burgeoned, and all four authors have also gone on to distinguished contributions in other fields. Books have also been written, such as Fishburn (1985), Narens (1985), and Roberts (1979). But the scope of FMI was unique, and the appearance of the sequels (referred to here as FMII and FMIII) is a happy event. Actually FMII and FMIII play rather complementary roles. The former covers ground that was more or less promised in the chapter titles in FMI, and thus tends to contain much material that was already extant at the end of the sixties. Of course, there is some new material, but much more of the presentation is retrospective and with the careful exposition that one finds in good textbooks. FMIII, on the other hand, although dealing with topics identified in FM !, tends to blaze new ground, and to contain rather technical accounts of much that is new. Axiomatic treatments of plane geometry are of course very old indeed, and chapters 12 and 13 on analytic and axiomatic geometry serve as very nice textbook level introductions. In addition, they cover special topics such as G-spaces, in which geodesics with an additive structure are to be found, and Riemannian spaces in which these geodesics are parameterized. These are digressions since they are about mathematical rather than empirical relational structures. The applications to relativistic space-time and Luneberg's theory of binocular vision are interesting, although of limited psychological relevance. Chapter 14, by contrast, deals with a topic of much interest to readers of Psychometrika, the representation of proximity structures. Here the results by Beals and Krantz (1967) and Beals, Krantz, and Tversky (1968) showing that a metric representation of a proximity relation with a collinear betweenness relation exists, and in that certain cases it has an additive factorial structure. Also discussed is Tversky's (1977) modeling of proximity in terms of intersections of sets of features. An axiomatic treatment of color and force fields in chapter 15 is something of a sidelight since these structures, which include the real numbers, are only partially empirical. In psychophysical situations the laws of weak order break down when stimuli are close to threshold, and chapter 17 covers axiom systems that incorporate an indifference relation that lies between equality and strict inequality. Choice probabilities are the subject of the next chapter, as they were in chapter 8 in FMI. However, it might be argued that axiomatic treatments of probability lie outside of fundamental measurement since we are representing one mathematical abstraction (a measure space) by another. Perhaps an independent volume on choice systems is called for. Treatments of extensive and conjoint measurement in FMI provided existence, uniqueness, and characterization results for a specific numerical system, that involving addition. Chapter 19 raises the level of abstraction dramatically by showing that essentially the same results hold if one drops associativity as an empirical condition. In

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this case, for structures referred to as positive concatenation structures, one can show that representations exist, and that these are also either interval or ratio scales. Of course the question of precisely what numerical operation is involved depends on other specific characteristics of the empirical operation. These results highlight the central theme of FMIII: the study of the invariances or symmetries in relational structures and their representations. It is remarkable that one learns from investigation of the automorphisms and endomorphisms of these more general structures that Stevens' classification was essentially exhaustive. Chapter 20 demonstrates the need for a distinction between two indices: (a) the degree of homogeneity M such that there always exists an automorphism taking any M distinct elements into any other M distinct elements, and (b) uniqueness N such that if the values resulting from applying any automorphism to N elements is known, the automorphism is uniquely defined. FMIII goes on to demonstrate, whatever the numerical operation that winds up representing positive or intensive concatenation structures, the representation will be on an interval, log-interval, or ratio scale. The final chapter 22 goes on to describe invariant or meaningful relations within such structures, and completes the treatment of dimensional analysis begun in Chapter 10. This is very new work, begun by Narens (1985), and not surprisingly the exposition has less than the high level of polish and neatness that characterizes most of the three volumes. Readers of FMI may find themselves out of their depth in Chapter 21, which applies the theory of models, a branch of mathematical logic, to a discussion of the logical status of the axioms in relational structures. However, the chapter aims to be self-contained. The various versions of the Archimedean axiom come in for particular attention. Common to all three volumes is a strongly axiomatic approach, and it is perhaps the greatest achievement of the authors--they have shown just how much is possible by a rigorous attack on some of the very old problems posed by the scientific use of numbers. The great effort in FMI to keep the axioms as sparse as possible meant that stronger essentially topological properties leading to real numerical representations were avoided. FMII and FMIII tend to bring these in fairly freely, as did Pfanzagl (1968) and as does the functional equation literature (Aczrl, 1966; Aczrl & Dhombres, 1989), and the gain in realism appears to this reviewer to be worth the loss in elegance. Each volume begins with a very nice overview chapter which situates subsequent chapters with respect to previous coverage in FMI and the field as a whole, and excellent introductions are given within each chapter. Indeed, the entire layout of the volumes is as kind as possible to those making first contacts with the area, with definitions and theorems set apart in italics and proofs reserved to special sections. Senior undergraduate or graduate students can be recommended to many sections and chapters, where they will find a variety of useful problems and examples. The fact that the axioms have to date been little used in tests of the applicability of the structures seems of little importance, since there are other approaches using modern statistics and data analysis that work better. The spectacular growth in techniques for transforming data, which may begin simply as representations of empirical order, to additive and other relationships is properly not alluded to in these volumes, but nevertheless is in the same spirit. Instead we learn something fundamental: the lack of empirical relations other than order in the behavioral sciences severely limits the scope of interpretation of numerical data. It would be a good thing, in my opinion, if we could restrain our use of terms such "measurement" and "instrument" in the context of fields such as mental testing in deference to this fact. Index or pointer variables are the stuffof social science, and their usefulness has been so well demonstrated that we have

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no need to apologize for them. But the distinction between fundamental and index measure is critical, and it should be drawn forcefully and early in the education of psychologists among others. The Foundations of Measurement volumes are quite simply a triumph, and I recommend them without reservation to anyone with a taste for mathematics and reflection.

J. O. Ramsay

McGILL UNIVERSITY References

Acz61, J. (1966). Lectures on functional equations and their applications. New York: Academic Press. Acz61, J. & Dhombres, J. (1989). Functional equations in several variables with applications to mathematics, information theory and to the natural and social sciences. Cambridge: Cambridge University Press. Beals, R., & Krantz, D. (1967). Metrics and geodesics induced by order relations. Mathematische Zeitschrift, 101,285-298. Beals, R., Krantz, D., & Tversky, A. (1968). Foundations of multidimensional scaling. Psychological Review, 75, 127-142. Campbell, N. R. (1920). Physics: The elements. London: Cambridge University Press. (Reprinted as Foundations of science: The philosophy of theory and experiment. New York: Dover, 1957) Ellis, 13. (1966). Basic concepts in measurement. London: Cambridge University Press. Fishburn, P. C. (1985). Interval orders and interval graphs. New York: Wiley. H61der, O. (1901). Die Axiome der quantitiit und die tehre von mass [The axioms of quantity and the theory of mass]. Ber. Siich., Gesellsch. Wiss., Math-Phy. Klasse, 53, 1-64. Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of measurement, Volume L New York: Academic Press. Luce, R. D. (1959). On the possible psychophysical laws. Psychological Review, 66, 81-95. Luce, R. D., 13ush, R. R., & Galanter, E. (1963). Handbook of mathematical psychology, Volume L New York: Wiley. Luce, R. D., & Tukey, J. W. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement. Journal of Mathematical Psychology, 1, 1-27. Narens, L. (1985). Abstract measurement theory. Cambridge, MA: MIT Press. Pfanzagl, J. (1968). A theory of measurement. New York: Wiley. Ramsay, J. O. (1975) Review of "Foundations of measurement, Volume I." Psychometrika, 40, 257-262. Roberts, F. S. (1979). Measurement theory with applications to decision making, utility, and the social sciences. Reading, MA: Addison-Wesley. Scott, D., & Suppes, P. (1958). Foundationalaspects of theories of measurement. Journal of Symbolic Logic, 23, 113-128. Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103, 677-680. Suppes, P. (1951). A set of independent axioms for extensive quantities. Portugaliae Mathematica, 10, 163-172. Suppes, P., & Zinnes, J. (1963). Basic measurement theory. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology, Volume I. New York: Wiley, 1-76. Tarski, A. (1954). Contributions to the theory of models, I and II. lndagationes Mathematicae, 16, 572-588. Tversky, A. (1967a). A general theory of polynomial conjoint measurement. Journal of Mathematical Psychology, 4, 1-20. Tversky, A. (1967b). Additivity, utility, and subjective probability. Journal of Mathematical Psychology, 4, 175-201. Tversky, A. (1977). Features of similarity. Psychological Review, 84, 327-352.