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Scaling and Dimensional Analysis

William Jacoby Michigan State University

The terms “scaling” and “dimensional analysis” refer to a wide variety of research strategies and procedures. The common element among them is that they all seek to provide quantitative and/or geometric representations of the internal structure in a set of data. Researchers apply these techniques for three main reasons: (1) Simple data reduction— summarizing a large set of variables with a smaller number of composite measures; (2) examining dimensionality— testing the underlying sources of variation in a dataset; and (3) measurement— obtaining empirical representations of the underlying (and usually unobservable) dimensions, which can be employed as analytic variables in other statistical procedures. On a less formal note, researchers will often find that dimensional analysis is very beneficial for “conceptualizing” the contents of their data. In addition, these techniques usually provide visual displays that are very useful for presenting analytical results to other people. Thus, for a variety of reasons, scaling and dimensional analysis are useful additions to the social scientist's “repertoire” of research strategies.

READING MATERIAL Unfortunately, there is no single textbook that covers all of the topics in this course. In addition, many of the texts that are available have certain drawbacks that limit their usefulness for our purposes: They tend to be very expensive; they usually assume a high level of mathematical sophistication; they often contain sections that are out of date. Because of these considerations, we will rely primarily on several shorter works. Students should consider purchasing at least some of the following texts (although I strongly recommend waiting until after the first workshop session before doing so):

Arabie, Phipps; J. Douglas Carroll; Wayne S. DeSarbo (1987) Three-Way Scaling and Clustering. Sage University Paper.

Bailey, Kenneth D. (1994) Typologies and Taxonomies: An Introduction to Classification Techniques. Sage University Paper.

Bartholomew, David J.; Fiona Steele; Irini Moustaki; Jane I. Galbraith. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. Chapman and Hall/CRC.

Lattin, James; J. Douglas Carroll; Paul E. Green (2003) Analyzing Multivariate Data. Brooks/Cole— Thomson Learning.

Dunteman, George H. (1989) Principal Components Analysis. Sage University Paper.

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Jacoby, William G. (1991) Data Theory and Dimensional Analysis. Sage University Paper.

Kim, Jae-On and Charles W. Mueller (1978a) Introduction to Factor Analysis. Sage University Paper.

Kim, Jae-On and Charles W. Mueller (1978b) Factor Analysis: Statistical Methods and Practical Issues. Sage University Paper.

Kruskal, Joseph B. and Myron Wish (1978) Multidimensional Scaling. Sage University Paper.

McIver, John and Edward G. Carmines (1981) Unidimensional Scaling. Sage University Paper.

Weller, Susan C. and A. Kimball Romney (1990) Metric Scaling: Correspondence Analysis. Sage University Paper

Readings will also be taken from the following works:

Asher, Herbert B., Herbert F. Weisberg, John H. Kessel, W. Phillips Shively, Editors (1984) Theory-Building and Data Analysis in the Social Sciences. University of Tennessee Press.

Basilevsky, Alexander (1994) Statistical Factor Analysis and Related Methods: Theory and Applications. Wiley-Interscience.

Bollen, Kenneth A. (1989) Structural Equation Models with Latent Variables. John Wiley.

Borg, Ingwer and Patrick Groenen (2005) Modern Multidimensional Scaling: Theory and Applications (Second Edition). Springer-Verlag.

Carroll, J. Douglas; Paul E. Green; Anil Chaturvedi. (1997) Mathematical Tools for Applied Multivariate Analysis (Revised Edition). Academic Press.

Coombs, Clyde H. (1964) A Theory of Data. John Wiley (Reprinted 1976, Mathesis Press).

Cox, Trevor F. and Michael A. A. Cox (2001) Multidimensional Scaling (Second Edition). Chapman and Hall.

Davies, P.M. and A.P.M. Coxon, Editors (1982) Key Texts in Multidimensional Scaling. Heinemann.

Davison, Mark L. (1983) Multidimensional Scaling. John Wiley.

DeVellis, Robert F. (1991) Scale Development: Theory and Applications. Sage.

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Gorsuch, Richard L. (1983) Factor Analysis (Second Edition). Lawrence Erlbaum Associates.

Greenacre, Michael J. (1984) Theory and Applications of Correspondence Analysis. Academic Press.

Harman, Harry H. (1976) Modern Factor Analysis (Third Revised Edition). University of Chicago Press.

Mokken, Robert J. (1971) A Theory and Procedure of Scale Analysis with Applications in Political Research. Mouton.

Poole, Keith T. (2005) Spatial Models of Parliamentary Voting. Cambridge University Press.

Schiffman, Susan S.; M. Lance Reynolds; Forrest W. Young (1981) Introduction to Multidimensional Scaling. Academic Press.

Sijtsma, Klaas and Ivo W. Molenaar (2002) Introduction to Nonparametric Item Response Theory. Sage.

Torgerson, Warren S. (1958) Theory and Methods of Scaling. John Wiley.

Wickens, , Thomas D. (1995) The Geometry of Multivariate Statistics. Lawrence Erlbaum,

Young, Forrest W. and Robert M. Hamer (1987) Multidimensional Scaling: History, Theory, and Applications. Lawrence Erlbaum.

SOFTWARE CONSIDERATIONS With very few exceptions, the methods covered in this workshop are computationally intensive. Therefore, appropriate software is required to perform most of the analyses. Fortunately, most of the widely-available statistical packages (e.g., STATA, SAS, SPSS, SYSTAT) contain routines for carrying out the major techniques (e.g., factor analysis, multidimensional scaling, correspondence analysis). But, there are a few special-purpose programs that will also be used for particular applications. These will be introduced as necessary, in class, and they will all be available on the ICPSR Summer Program network. Handouts and examples will generally present analyses in STATA (with some exceptions where necessary). But, participants are also encouraged to try to perform the analyses using the R statistical computing environment.

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TOPICS AND READING ASSIGNMENTS The topics covered in this workshop fall within three major sections, although they are not identified as such in the syllabus. The first section covers several scaling strategies appropriate for analyzing data that are obtained from a “dominance” process. That is, each datum indicates the extent to which a stimulus object exceeds (or fails to exceed) some standard of comparison (e.g., another stimulus object, a unit mark on a measurement scale, etc.). Specific procedures to be covered in this part of the workshop include summated rating scales, cumulative scaling, and factor analysis. The second section moves on to methods for dealing with “proximity” data. This is information that indicates how “close” or “similar” one object is to another. Here, the workshop will focus on unfolding models, multidimensional scaling, and correspondence analysis. The third section covers data theory and some general considerations related to dimensional analysis. Here, there is less focus on specific scaling techniques and more attention to a general framework for integrating the material that has already been covered. This last part of the workshop is particularly important for understanding when different scaling techniques can and should be employed in empirical research. And hopefully, it will leave workshop participants with a relatively optimistic view of the nature, quality, and potential for accurate measurement of important concepts in the social and behavioral sciences. In the outline, entries marked with a double asterisk should be considered essential readings. Those with a single asterisk are recommended works. Unmarked entries are supplemental readings which generally cover specific aspects of the respective topics in greater detail. I. Introduction and Some General Considerations

** Jacoby (1991), Chapters 1 and 2. ** Jacoby, William G. (1999) “Levels of Measurement and Political Research: An

Optimistic View.” American Journal of Political Science 43: 271-301.

** Young and Hamer (1987), Chapter 3. ** Weller and Romney (1990), Chapter 1. ** Lattin et al. (2003), Chapter 1. ** Bartholomew et al. (2002), Chapter 1. II. Classification and Clustering: A Very Brief Introduction ** Bartholomew et al. (2002), Chapter 2. ** Bailey (1994), Chapters 1-3.

Gordon, A.D. (1999) Classification (Second Edition).

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III. Summated Rating Scales A. Scale Construction ** Jacoby (1991), pp. 38-41.

** McIver and Carmines (1981), pp. 22-26. * DeVellis (1991), Chapters 2 and 5.

Spector, Paul E. (1992) Summated Rating Scale Construction. Sage University Paper. B. Scale Assessment and Reliability ** McIver and Carmines (1981), pp. 26-40. * DeVellis (1991), Chapter 3. * Bollen (1989), Chapter 6, especially pp. 206-223. * Traub, Ross E. (1994) Reliability for the Social Sciences: Theory and Applications.

Sage. Chapters 3-6, 10.

Nunnally, Jum C. and Ira Bernstein (1994) Psychometric Theory (Third Edition). McGraw-Hill: Chapters 6 and 7.

Greene, V.L. and E.G. Carmines (1979) “Assessing the Reliability of Linear Composites.” In Karl Schuessler (Editor) Sociological Methodology 1980. Jossey-Bass.

Niemi, Richard G.; Edward G. Carmines; John P. McIver (1986) “The Impact of Scale Length on Reliability and Validity.” Quality and Quantity 20: 371-376.

C. Magnitude Scaling

** Lodge, M. and B. Tursky (1981) “On the Magnitude Scaling of Political Opinion in

Survey Research.” American Journal of Political Science 25: 376-419. ** Jacoby (1991), pp. 53-58.

Lodge, M. (1981) Magnitude Scaling. Sage University Paper.

Saris, Willem E. (1988) “A Measurement Model for Psychophysical Scaling.” Quality and Quantity 22: 417-433.

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IV. The Cumulative Scaling Model in a Single Dimension A. Guttman Scaling ** Jacoby (1991), pp. 41-43 ** McIver and Carmines (1981), Chapters 4-5. * Coombs (1964), Chapters 10-11.

Torgerson (1958), Chapter 12.

Mokken (1971), Chapter 2.

Proctor, C.H. (1970) “A Probabilistic Formulation and Statistical Analysis of Guttman Scaling.” Psychometrika 35: 73-78.

Clogg, C.C. and D.O. Sawyer (1981) “A Comparison of Alternative Models for Analyzing the Scalability of Response Patterns.” In S. Leinhardt (Editor) Sociological Methodology 1981. Jossey-Bass

Zwick, Rebecca (1987) “Some Properties of the Correlation Matrix of Dichotomous Guttman Items.” Psychometrika 52: 515-520.

Meijer, Rob R. (1994) “The Number of Guttman Errors as a Simple and Powerful Person-Fit Statistic.” Applied Psychological Measurement 18: 311-314.

B. Mokken Scaling ** van Schuur, Wijbrandt H. (2003) “Mokken Scale Analysis: Between the Guttman

Scale and Parametric Item Response Theory.” Political Analysis 11: 139-163. ** Jacoby (1991), pp. 44-46. ** Mokken, Robert J. and Charles Lewis (1982) “A Nonparametric Approach to the

Analysis of Dichotomous Item Responses.” Applied Psychological Measurement 7: 45-55.

** Sijtsma, K.; P. Debets; I.W. Molenaar (1990) “Mokken Scale Analysis for

Polychotomous Items: Theory, A Computer Program, and an Empirical Application.” Quality and Quantity 24: 173-188.

* Gillespie, Michael; Elizabeth M. Tenvergert; Johannes Kingsma (1987) “Using

Mokken Analysis to Develop Unidimensional Scales.” Quality and Quantity 21: 393-408.

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* Sijtsma and Molenaar (2002).

Hemker, Bas. T.; Klaas Sijtsma; Ivo W. Molenaar. (1995) “Selection of Unidimensional Scales from a Multidimensional Item Bank in the Polytomous Mokken IRT Model.” Applied Psychological Measurement 19: 337-352.

Meijer, Rob; Klaas Sijtsma; Ivo W. Molenaar. (1995) “Reliability Estimation for Single Dichotomous Items Based on Mokken’s IRT Model.” Applied Psychological Measurement 19: 323-335.

Mokken (1971), Chapter 4.

Roskam, Edward E.; Arnold L. van den Wollenberg; Paul G.W. Jansen (1986) “The Mokken Scale: A Critical Discussion.” Applied Psychological Measurement 10: 265-277. Also see the Rejoinder by Mokken, Lewis, and Sijtsma.

Sijtsma, Klaas. (1998) “Methodology Review: Nonparametric IRT Approaches to the analysis of Dichotomous Item Scores.” Applied Psychological Measurement 22: 3-32.

Kingsma, Johannes and Terry Taerum (1988) “MOKSCAL: A Program for a Nonparametric Item Response Theory Model.” Applied Psychological Measurement 14: 188.

C. Rasch Models ** Bartholomew et al. (2002), Chapter 7. ** Andrich, David (1985) “An Elaboration of Guttman Scaling with Rasch Models for

Measurement.” In N. Brandon-Tuma (Editor), Sociological Methodology, 1985. Jossey-Bass.

* Meijer, Rob R.; Klaas Sijstma; Nico G. Smid (1988) “Theoretical and Empirical

Comparison of the Mokken and the Rasch Approach to IRT.” Applied Psychological Measurement 12: 283-298.

Andrich, David (1988) Rasch Models for Measurement. Sage University Paper.

Fischer, Gerhard H. and Ivo W. Molenaar (Editors) (1995) Rasch Models: Foundations, Recent Developments, and Applications. Springer-Verlag.

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V. Preparation for Multidimensional Models A. Brief Overview of Matrix Algebra ** Davison (1983), Chapter 2.

* Carroll, Green, Chaturvedi, Chapter 2.

Borg and Groenen (2005), pp. 137-145. B. Vector Geometry and Linear Models

** Lattin et al. (2003), pp. 19-32. ** Wickens (1995), Chapters 1-5.

* Carroll, Green, Chaturvedi, Chapter 3.

Greenacre (1984), Chapter 2.

C. The Basic Structure of a Matrix ** Lattin et al. (2003), pp. 32-36. ** Weller and Romney (1990), Chapter 2.

* Carroll, Green, Chaturvedi, Chapter 5.

Borg and Groenen (2005), pp. 146-163.

Greenacre (1984), pp. 340-351. VI. Multidimensional Summaries of Multivariate Data

A. Principal Component Analysis ** Bartholomew et al. (2002), Chapter 5. ** Lattin et al. (2003), Chapter 4. ** Weller and Romney (1990), Chapter 3. * Dunteman (1989), Chapters 1-6, 8.

Basilevsky (1994), Chapter 3.

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B. The Biplot: Simultaneous Graphical Representation of Variables and Observations ** Gabriel, K.R. (1971) “The Biplot Graphic Display of Matrices with Application to

Principal Components Analysis.” Biometrics 58: 453-467.

** Jacoby, William G. (1998) Statistical Graphics for Visualizing Multivariate Data. Sage Publications. Sage. Chapter 7.

* Cox and Cox (2001), Chapter 7. * Gower, J. C. and D. J. Hand (1996) Biplots. Chapman and Hall. VII. Factor Analysis A. The Common Factor Model ** Bartholomew et al. (2002), pages 143-151. ** Lattin et al. (2003), pp. 127-131. ** Jacoby (1991), pp. 47-53. ** Kim and Mueller (1978a), pp. 1-46. ** Wickens (195), Chapter 9. * Loehlin, John C. (1992) Latent Variable Models: An Introduction to Factor, Path,

and Structural Analysis (Second Edition). Lawrence Erlbaum, Chapter 1. * Harman (1976), Chapters 1-4. * Gorsuch (1983), Chapters 1-4.

MacCallum, Robert C. and Ledyard R. Tucker (1991) “Representing Sources of Error in the Common Factor Model: Implications for Theory and Practice.” Psychological Bulletin 97: 85-93.

B. Estimation of the Factor Model ** Bartholomew et al. (2002), pages 151-156. ** Lattin et al. (2003), pp. 131-153 ** Kim and Mueller (1978a), pp. 46-70.

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** Kim and Mueller (1978b), pp. 7-29, 41-46. * Harman (1976), Chapters 5 and 8 (Sections 1-3 only). * Gorsuch (1983), Chapters 6 and 8.

Bartholomew, David J. (1987) Latent Variable Models and Factor Analysis. Oxford University Press. Especially Chapters 3 and 4.

Basilevsky (1994), pp. 351-395.

Wood, James M.; Douglas A. Tataryu; Richard L. Gorsuch. (1996) “Effects of Under- and Overextraction on Principal Axis Factor Analysis with Varimax Rotation.” Psychological Methods 1: 354-365. Kano, Yutaka (1990) “Noniterative Estimation and the Choice of the Number of Factors in Exploratory Factor Analysis.” Psychometrika 55: 277-291.

Yanai, Haruo and Masanori Ichikawa (1990) “New Lower and Upper Bounds for Communality in Factor Analysis.” Psychometrika 55: 405-410.

Lambert, Zarrel V.; Albert R. Wildt; Richard M. Durand (1991) “Approximating Confidence Intervals for Factor Loadings.” Multivariate Behavioral Research 26: 421-434.

Ichikawa, Masanori and Sadanori Konishi (1995) “Application of the Bootstrap Methods in Factor Analysis.” Psychometrika 60: 77-93.

Sinha, Atanu; John E. Anderson; Bruce S. Buchanon. (1995) “Assessing the Stability of Principal Components Using Regression.” Psychometrika 60: 355-369.

C. Rotation ** Bartholomew et al. (2002), pages 156-160. ** Lattin et al. (2003), pp. 153-156. ** Kim and Mueller (1978b), pp. 29-41. * Harman (1976), Chapter 13. * Gorsuch (1983), Chapters 9-10.

Koschat, Martin A. and Deborah F. Swayne (1991) “A Weighted Procrustes Criterion.” Psychometrika 56: 229-239.

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Basilevsky (1994), pp. 402-414.

Kiers, Henk A. L. (1994) “SIMPLIMAX: Oblique Rotation to an Optimal Target with Simple Structure.” Psychometrika 59: 567-579.

D. Construction of Factor Scales ** Bartholomew et al. (2002), pages 160-167. ** Lattin et al. (2003), pp. 156-166 ** Kim and Mueller (1978b), pp. 60-73.

Harman (1976), Chapter 16. Gorsuch (1983), Chapter 12.

Basilevsky (1994), pp. 395-400.

E. Principal Components Analysis Compared to Factor Analysis ** Bartholomew et al. (2002), pages 167-174.

Gangemi, Giuseppe (1986) “Epistemological Reasons for Preferring Component Analysis to Factor Analysis.” Quality and Quantity 20: 75-84.

Velicer, Wayne F. and Douglas N. Jackson (1990) “Component Analysis versus Common Factor Analysis: Some Issues in Selecting an Appropriate Procedure.” Multivariate Behavioral Research 25: 1-28. Also see the Comments that follow.

Widaman, Keith F. (1993) “Common Factor Analysis Versus Principal Component Analysis: Differential Bias in Representing Model Parameters.” Multivariate Behavioral Research 28: 263-311.

F.Introduction to Confirmatory Factor Analysis ** Lattin et al. (2003), Chapter 6. ** Kim and Mueller (1978b), pp. 46-60. * Gorsuch (1983), Chapter 7. * Bollen (1989), Chapter 7.

Basilevsky (1994), pp. 414-417.

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VIII. Vector Analysis of Preferences ** Lattin et al. (2003), pp. 244-252

** Weller and Romney (1990), pp. 44-54. * Carroll, J.D. (1972) “Individual Differences and Multidimensional Scaling.” In

Shepard, Roger N., A. Kimball Romney, Sara Beth Nerlove (Editors), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Volume I, Theory). Seminar Press: pp. 105-155, especially pages 123-130. Chapter is reprinted in Davies and Coxon, pp. 267-301.

IX. Spatial Distance Models for Analyzing Proximity Data ** Lattin et al. (2003), pp. 206-211. ** Kruskal and Wish (1978), pp. 1-19. * Davison (1983), Chapter 1. * Poole (2005), Chapter 1.

Young and Hamer (1987), Chapter 4.

Schiffman et al. (1981), Chapter 1.

Jacoby (1991), pp. 58-62.

Arabie, Phipps (1991) “Was Euclid an Unnecessarily Sophisticated Psychologist?” Psychometrika 56: 419-431.

Cox and Cox (2001), Chapter 1.

Borg and Groenen (2005), Chapters 1, 17-19.

X. The Unidimensional Unfolding Model and Related Approaches A. Unfolding Analysis ** McIver and Carmines (1981), pp. 71-86. ** Coombs (1964), pp. 80-140.

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Poole, Keith T. (1984) “Least Squares Metric, Unidimensional Unfolding.” Psychometrika 49: 311-323.

Poole, Keith T. (1990) “Least-Squares, Metric, Unidimensional Scaling of Multivariate Linear Models.” Psychometrika 55: 123-149.

B. Proximity and Parallelogram Scaling ** Jacoby (1991), pp. 70-72. ** Coombs (1964), pp. 305-312, Chapter 4. * Poole (2005), Chapter 2. * Van Schuur, W.H. (1988) “Stochastic Unfolding.” In W.E. Saris and I.N. Gallhofer

(Editors), Sociometric Research, Volume 1: Data Collection and Scaling. MacMillan Press.

* Cliff, Norman; Linda M. Collins; Judith Zatken; Dannie Gallipeau; Douglas J.

McCormick (1988) “An Ordinal Scaling Method for Questionnaire and Other Ordinal Data.” Applied Psychological Measurement 12: 83-97.

* Hoijtink, Herbert (1991) “The Measurement of Latent Traits by Proximity Items.”

Applied Psychological Measurement 15: 153-169.

* Hoijtink, Herbert and Ivo W. Molenaar (1994) “An Item Response Model with Single Peaked Item Characteristic Curves: The PARELLA Model.” Quality & Quantity 28: 99-116.

* Roberts, James S. and James S. Laughlin. (1996) “A Unidimensional Response

Model for Unfolding Responses from a Graded Disagree-Agree Response Scale.” Applied Psychological Measurement 20: 231-255.

Roberts, James S. (1998) “GUMJML: A Program to Estimate Parameters in the Graded Unfolding Model Using a Joint Maximum Likelihood Technique.” Applied Psychological Measurement 22: 70.

Hoijtink, Herbert (1990) “A Latent Trait Model for Dichotomous Choice Data.” Psychometrika 55: 641-656.

Weisberg, H.F. (1972) “Scaling Models for Legislative Roll-Call Analysis.” American Political Science Review 66: 1306-1315.

Dawes, R.M. (1972) Fundamentals of Attitude Measurement. John Wiley: Chapter 4.

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Coombs, C.H. and J.E.K. Smith (1973) “On the Detection of Structure in Attitudes and Developmental Processes.” Psychological Review 80: 337-351. Torgerson (1958), pp. 410-412.

XI. Multidimensional Scaling: Basic Models and Procedures A. Classical Multidimensional Scaling ** Bartholomew et al. (2002), Chapter 3. ** Lattin et al. (2003), pp. 211-235. ** Kruskal and Wish (1978), pp. 19-30, 48-60. ** Davison (1983), Chapters 4-5. ** Rabinowitz, G.B. (1984) “An Introduction to Nonmetric Multidimensional Scaling.”

In Asher et al., pp. 391-410.

Schiffman et al. (1981), Chapter 4.

Young and Hamer (1987), Chapter 5.

Cox and Cox (2001), Chapters 2 and 3.

Borg and Groenen (2005), Chapters 2, 3, 8, 9, 11-13.

Kruskal, Joseph B. (1964) “Multidimensional Scaling by Optimizing Goodness of Fit to a Non-Metric Hypothesis.” Psychometrika 29: 1-27. Reprinted in Davies and Coxon, pp. 59-83.

Kruskal, Joseph B. (1964) “Non-Metric Multidimensional Scaling: A Numerical Method.” Psychometrika 29: 115-129. Reprinted in Davies and Coxon, pp. 84-97.

B. Weighted Multidimensional Scaling

** Lattin et al. (2003), pp. 235-243. ** Kruskal and Wish (1978), pp. 60-73. ** Arabie, Carroll, and DeSarbo (1987), pp. 1-53. * Davison (1983), Chapter 6.

Cox and Cox (2001), Chapters 10 and 11.

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Borg and Groenen (2005), Chapter 22.

Carroll, J.D. and J.J. Chang (1970) “An Analysis of Individual Differences in Multidimensional Scaling via an N-Way Generalization of 'Eckart-Young' Decomposition.” Psychometrika 35: 283-319. Reprinted in Davies and Coxon, pp. 229-252.

Takane, Y., F.W. Young, J. DeLeeuw (1977) “Nonmetric Individual Differences Multidimensional Scaling: An Alternating Least Squares Method with Optimal Scaling Features.” Psychometrika 42: 7-67.

Young and Hamer (1987), Chapter 6.

Weinberg, Sharon L. and Violeta C. Menil (1993) “The Recovery of Structure in Linear and Ordinal Data: INDSCAL Versus ALSCAL.” Multivariate Behavioral Research 28: 215-233.

Young, Forrest W. (1979) “Principal Directions Scaling.” Unpublished Notes.

Jacoby, William G. (1988) “The General Euclidean Model: Applications in Political Science.” Unpublished Paper.

Kiers, Henk A.L. (1989) “An Alternating Least Squares Algorithm for Fitting the Two- and Three-Way DEDICOM Model and the IDIOSCAL Model.” Psychometrika 54: 515-521.

ten Berge, Jos M.F. and Henk A.L. Kiers (1991) “Some Clarifications of the CANDECOMP Algorithm Applied to INDSCAL.” Psychometrika 56: 317-326.

Rodgers, Joseph Lee (1991) “Matrix and Stimulus Sample Sizes in the Weighted MDS Model: Empirical Metric Recovery Functions.” Applied Psychological Measurement 15: 71-77.

ten Berge, Jos M.F.; Paul A. Bekker; Henk A. L. Kiers (1994) “Some Clarifications of the TUCKALS2 Algorithm Applied to the IDIOSCAL Problem.” Psychometrika 59: 193-201.

XII. Multidimensional Scaling: Additional Considerations A. Interpretation of Multidimensional Scaling Results ** Kruskal and Wish (1978), pp. 30-48. * Davison (1983), pp. 189-195.

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* Rabinowitz (1984), pp. 415-426. * Schiffman et al. (1981), Chapters 12-13.

Cox and Cox (2001), Chapter 6.

Young, F.W. (1970) “Nonmetric Multidimensional Scaling: Recovery of Metric Information.” Psychometrika 35: 455-473.

Best, A.M., F.W. Young, R.G. Hall (1979) “On the Precision of a Euclidean Structure.” Psychometrika 44: 395-408.

Reynolds, T.J. and K.H. Sutrick (1986) “Assessing the Correspondence of One or More Vectors to a Symmetric Matrix Using Ordinal Regression.” Psychometrika 51: 101-112.

Reynolds, Thomas J.; David Weeks; Steve Perkins (1987) “CDASCAL: An Algorithm for Assessing the Correspondence of One or More Vectors to a Symmetric Matrix Using Ordinal Regression.” Psychometrika 52: 293-301.

Reynolds, Thomas J. and Kenneth H. Sutrick (1988) “Cognitive Differentiation Analysis: A Regression Extension of the Reynolds-Sutrick Model.” Multivariate Behavioral Research 23: 451-467.

B. Data for Multidimensional Scaling Analyses ** Jacoby (1991), pp. 63-66. ** Kruskal and Wish (1978), pp. 73-82. * Davison (1983), Chapter 3. * Jones, B.D. (1974) “Some Considerations in the Use of Nonmetric Multidimensional

Scaling.” Political Methodology 1: 1-31. * Rabinowitz, G.B. (1976) “A Procedure for Ordering Object Pairs Consistent with the

Multidimensional Unfolding Model.” Psychometrika 41: 349-373.

Borg and Groenen (2005), Chapter 6.

Jacoby, William G. (1992) “A SAS Macro for Calculating the Line-of-Sight Measure of Interobject Dissimilarity.” Psychometrika 58: 511-512.

Schiffman et al. (1981), Chapter 2.

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Thompson, Paul (1983) “Some Missing Data Patterns for Multidimensional Scaling.” Applied Psychological Measurement 7: 45-55.

C. Hypothesis Testing and Confirmatory Multidimensional Scaling

** Kruskal and Wish (1978), pp. 89-92. ** Davison (1983), pp. 195-201. * Heiser, W.J. and J. Meulman (1983) “Constrained Multidimensional Scaling,

Including Confirmation.” Applied Psychological Measurement 7: 381-404.

* Lingoes, James C. and Ingwer Borg (1986) “On Evaluating the Equivalency of Alternative MDS Representations.” Quality and Quantity 20: 249-256.

Peay, Edmund R. (1988) “Multidimensional Rotation and Scaling of Configurations to Optimal Agreement.” Psychometrika 53: 199-208.

* Ramsay, J.O. “MULTISCALE,” in Schiffman et al. (1981), pp. 309-405.

Ramsay, J.O. (1977) “Maximum Likelihood Estimation in Multidimensional Scaling.” Psychometrika 42: 241-266. Reprinted in Davies and Coxon, pp. 303-326.

Cox and Cox (2001), Chapters 4 and 5.

Borg and Groenen (2005), Chapters 10, 20, 21.

Weinberg, Sharon L.; J. Douglas Carroll; Harvey S. Cohen (1984) Confidence Regions for INDSCAL Using the Jackknife and Bootstrap Techniques.” Psychometrika 49: 475-491.

Rodgers, Joseph Lee (1985) “Statistical Tests of Group Differences in ALSCAL-Derived Subject Weights: Some Monte Carlo Results.” Applied Psychological Measurement 9: 241-248.

Brady, Henry E. (1985) “Statistical Consistency and Hypothesis Testing for Nonmetric Multidimensional Scaling.” Psychometrika 50: 509-537.

XIII. Multidimensional Unfolding ** Jacoby (1991), pp. 66-70. ** Davison (1983), Chapter 7. ** Weller and Romney, Chapter 4.

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** Poole (2005), Chapters 2-5. * Carroll, J.D. (1972) “Individual Differences and Multidimensional Scaling.” In

Shepard, Roger N., A. Kimball Romney, Sara Beth Nerlove (Editors), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Volume I, Theory). Seminar Press: pp. 105-155. Reprinted in Davies and Coxon, pp. 267-301.

* Poole, Keith T. (2000) “Nonmetric Unfolding of Binary Choice Data.” Political

Analysis 8: 211-238. * Cox and Cox (2001), Chapter 8.

Political Analysis 9 (Summer 2001). A Special Issue on “Estimating Legislators’ Preferences with Roll Call Data.”

Borg and Groenen (2205), Chapters 14-16.

Thompson, Paul (1989) “A Monte Carlo Examination of External Unfolding.” Applied Psychological Measurement 13: 167-176.

Rodgers, J.L. and F.W. Young (1981) “Successive Unfolding of Family Preferences.” Applied Psychological Measurement 5: 51-62.

Brannick, Michael T. and June Hahn (1985) “A Comparison of Internal, External, and Successive Unfolding Based on Occupational Judgments.” Applied Psychological Measurement 9: 199-208.

Davison, Mark L. (1988) “A Reformulation of the General Euclidean Model for The External Analysis of Preference Data.” Psychometrika 53: 305-320.

DeSarbo, Wayne S. and J. Douglas Carroll (1985) “Three-Way Metric Unfolding via Alternating Weighted Least Squares.” Psychometrika 50: 275-300.

Greenacre, Michael J. and Michael W. Browne (1986) “An Efficient Alternating Least Squares Algorithm to Perform Multidimensional Unfolding.” Psychometrika 51: 241-250.

Brady, Henry E. (1989) “Factor and Ideal Point Analysis for Interpersonally Incomparable Data.” Psychometrika 54: 181-202.

Brady, Henry E. (1990) “Dimensional Analysis of Ranking Data.” American Journal of Political Science 34: 1017-1048.

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