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'''Preface.''' Thurstone explains that this is an extended and more formal presentation of his Multiple Factor Analysis paper of 1931, and gives an overview of the book's chapters. He only recently learned of matrix theory and presumes that other psychologists had similar limitations to their training. He finds existing textbooks on the topic inadequate and begins with a presentation of matrix theory, written for those with conventional undergraduate instruction in analytic geometry and real number calculus. He is indebted to various professors in the mathematics department of the University of Chicago for helping him to develop his ideas about factor analysis. He expresses appreciation to his computer (a person), Leone Chesire, who also edited his manuscripts and wrote the appendix on the calculations used in the centroid method. He foresees a bright future for factor analysis, with simplification of the computational methods. Factor analysis will become an important technique for the early stages of science. The laws of classical mechanics might have been revealed by a factor analysis, by analyzing a great many attributes of objects that are dropped or thrown from an elevated point, with the time of fall factor uncorrelated with the weight factor. Work by Sewell Wright on path coefficients and Truman L. Kelley on multiple factors is quite different from his own approach, which is an extension of professor Spearman's work that does not conflict with it in any way. |
'''Preface.''' Thurstone explains that this is an extended and more formal presentation of his Multiple Factor Analysis paper of 1931, and gives an overview of the book's chapters. He only recently learned of matrix theory and presumes that other psychologists had similar limitations to their training. He finds existing textbooks on the topic inadequate and begins with a presentation of matrix theory, written for those with conventional undergraduate instruction in analytic geometry and real number calculus. He is indebted to various professors in the mathematics department of the University of Chicago for helping him to develop his ideas about factor analysis. He expresses appreciation to his computer (a person), Leone Chesire, who also edited his manuscripts and wrote the appendix on the calculations used in the centroid method. He foresees a bright future for factor analysis, with simplification of the computational methods. Factor analysis will become an important technique for the early stages of science. The laws of classical mechanics might have been revealed by a factor analysis, by analyzing a great many attributes of objects that are dropped or thrown from an elevated point, with the time of fall factor uncorrelated with the weight factor. Work by Sewell Wright on path coefficients and Truman L. Kelley on multiple factors is quite different from his own approach, which is an extension of professor Spearman's work that does not conflict with it in any way. |
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'''Mathematical Introduction''' |
'''Mathematical Introduction.''' A brief presentation of matrices, determinants, matrix multiplication, diagonal matrices, the inverse, the characteristic equation, summation notation, linear dependence, geometric interpretations, orthogonal transformations, and oblique transformations. |
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'''Chapter I. The Factor Problem.''' |
'''Chapter I. The Factor Problem.''' Natural phenomena are only comprehensible through constructs that are man-made inventions. A scientific law is not part of nature; it is but man's way of understanding nature. Examples are provided of such man-made constructs from physics. He responds to skepticism from the practitioners of "rigorous science" that human behavior can ever be brought into the fold of such science by pointing out that there is considerable individuality in physical events that are described by rigorous scientific laws, such as the fact that every explosion is unique, even though governed by scientific laws. Human abilities are the cause of individual differences in the "completion of a task". The science of psychology will reduce a large number of psychological abilities down to primary reference traits. Formal definitions are provided for trait, ability, test, score, linear independence, statistical independence, experimental independence, reference abilities, primary abilities, and unitary ability. These conceptions constitute a theory of measurement that defines factors common to all tests in a battery-the communality of the test battery-, a specific factor that is unique to one test-the specificity of the test-, and the error variance. Factor analysis can determine the communality of a test, but cannot separate the uniqueness into the specific factor and the error factors. The reliability coefficient is the sum of communality and the specificity of a test. |
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'''Chapter II. The Fundamental Factor Theorem.''' The |
'''Chapter II. The Fundamental Factor Theorem.''' The factor matrix post-multiplied by its transpose gives the reduced correlation matrix: this is the fundamental factor theorem. The task of factor analysis is to find a factor matrix of lowest possible rank that can reproduce the off-diagonal members of the observed correlation matrix as close as can be expected, allowing for sample variation. The bulk of the chapter considers mathematical issues, including the rank of a matrix and methods for estimating the commonalities of the correlation matrix (the diagonal elements). |
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'''Chapter III: The Centroid Method'''. A computation method is developed for factoring a correlation matrix, which is a symmetric matrix of real elements. After a conceptual presentation of the method, some worked example are provided, including one with eight variables, another with fifteen variables that are factored into four factors. The mechanics of the |
'''Chapter III: The Centroid Method'''. A computation method is developed for factoring a correlation matrix, which is a symmetric matrix of real elements. After a conceptual presentation of the method, some worked example are provided, including one with eight variables, another with fifteen variables that are factored into four factors. The mechanics of the calculations are given in Appendix I, which provides the specific steps in making the calculation (an algorithm). |
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'''Chapter IV: The Principal Axes.''' A method is presented for determining a desirable rotation of the factors extracted using the centroid method. |
'''Chapter IV: The Principal Axes.''' A method is presented for determining a desirable rotation of the orthogonal factors extracted using the centroid method. The mathematical foundations are provided, as well as worked examples. His conceptualization of principal axes is distinguished from Hotelling's method, which he finds to have limited utility. The unrotated solution for 15 psychological tests given in chapter III are rotated to their principal axes. |
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'''Chapter V: The Special Case of Rank One.''' A |
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== Historical Context == |
== Historical Context == |
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Revision as of 21:12, 16 August 2020
Übersicht
The Vectors of Mind is an exposition of Thurstone's method for multiple factor analysis, which was first presented in a paper in 1931[1] but further developed before the complete method was presented in this book. It relies heavily upon matrix algebra and gives a technical presentation of mathematical methods and worked examples. His paper of the same name, published a year before the book,[2] had introduced several fundamental concepts of factor analysis, including communality, factor loadings, and matrix rank. The book introduced concepts of uniqueness, rotation, oblique factors, and simple structure. It also defined test reliability in terms of factor loadings. Fast electronic computers were not even imagined in 1935, and Thurstone's presents the centroid method of factor extraction which requires calculations that, while arduous, can be completed by hand. All the methods necessary to conduct a factor analysis with multiple factors, including both orthogonal and oblique rotation.
Synopsis
Preface. Thurstone explains that this is an extended and more formal presentation of his Multiple Factor Analysis paper of 1931, and gives an overview of the book's chapters. He only recently learned of matrix theory and presumes that other psychologists had similar limitations to their training. He finds existing textbooks on the topic inadequate and begins with a presentation of matrix theory, written for those with conventional undergraduate instruction in analytic geometry and real number calculus. He is indebted to various professors in the mathematics department of the University of Chicago for helping him to develop his ideas about factor analysis. He expresses appreciation to his computer (a person), Leone Chesire, who also edited his manuscripts and wrote the appendix on the calculations used in the centroid method. He foresees a bright future for factor analysis, with simplification of the computational methods. Factor analysis will become an important technique for the early stages of science. The laws of classical mechanics might have been revealed by a factor analysis, by analyzing a great many attributes of objects that are dropped or thrown from an elevated point, with the time of fall factor uncorrelated with the weight factor. Work by Sewell Wright on path coefficients and Truman L. Kelley on multiple factors is quite different from his own approach, which is an extension of professor Spearman's work that does not conflict with it in any way.
Mathematical Introduction. A brief presentation of matrices, determinants, matrix multiplication, diagonal matrices, the inverse, the characteristic equation, summation notation, linear dependence, geometric interpretations, orthogonal transformations, and oblique transformations.
Chapter I. The Factor Problem. Natural phenomena are only comprehensible through constructs that are man-made inventions. A scientific law is not part of nature; it is but man's way of understanding nature. Examples are provided of such man-made constructs from physics. He responds to skepticism from the practitioners of "rigorous science" that human behavior can ever be brought into the fold of such science by pointing out that there is considerable individuality in physical events that are described by rigorous scientific laws, such as the fact that every explosion is unique, even though governed by scientific laws. Human abilities are the cause of individual differences in the "completion of a task". The science of psychology will reduce a large number of psychological abilities down to primary reference traits. Formal definitions are provided for trait, ability, test, score, linear independence, statistical independence, experimental independence, reference abilities, primary abilities, and unitary ability. These conceptions constitute a theory of measurement that defines factors common to all tests in a battery-the communality of the test battery-, a specific factor that is unique to one test-the specificity of the test-, and the error variance. Factor analysis can determine the communality of a test, but cannot separate the uniqueness into the specific factor and the error factors. The reliability coefficient is the sum of communality and the specificity of a test.
Chapter II. The Fundamental Factor Theorem. The factor matrix post-multiplied by its transpose gives the reduced correlation matrix: this is the fundamental factor theorem. The task of factor analysis is to find a factor matrix of lowest possible rank that can reproduce the off-diagonal members of the observed correlation matrix as close as can be expected, allowing for sample variation. The bulk of the chapter considers mathematical issues, including the rank of a matrix and methods for estimating the commonalities of the correlation matrix (the diagonal elements).
Chapter III: The Centroid Method. A computation method is developed for factoring a correlation matrix, which is a symmetric matrix of real elements. After a conceptual presentation of the method, some worked example are provided, including one with eight variables, another with fifteen variables that are factored into four factors. The mechanics of the calculations are given in Appendix I, which provides the specific steps in making the calculation (an algorithm).
Chapter IV: The Principal Axes. A method is presented for determining a desirable rotation of the orthogonal factors extracted using the centroid method. The mathematical foundations are provided, as well as worked examples. His conceptualization of principal axes is distinguished from Hotelling's method, which he finds to have limited utility. The unrotated solution for 15 psychological tests given in chapter III are rotated to their principal axes.
Chapter V: The Special Case of Rank One. A
Historical Context
In 1904 Charles Spearman published a paper that largely founded the field of psychometrics and included a crude form of factor analysis that attempted to determine if a single factor model was appropriate.[3] There was limited subsequent work on factor analysis until Thurstone published a paper in 1931 called Multiple Factor Analysis,[1] which expanded Spearman's single-factor analysis to include more than one factor. In 1932, Hotelling presented a more accurate method of extracting factors, which he called principal components analysis.[4] Thurstone rejected Hotelling's approach because it set the commonalities to 1.0, and Thurstone realized that will introduce distortions to the factor loadings when variables include unique components. Hotelling's method was also limited by the fact that it required too much calculation to be useable with more than about ten variables.[5] A year after Hotelling's paper, Thurstone presented a more efficient way of extracting factors, called the centroid method,[6] which allowed the factor analysis of a far larger number of variables. Later that year he gave his presidential address to the American Psychological Association wherein he presented the results of several factor analyses, including a factor analysis of 60 adjectives describing personality traits, showing how they could be reduced to five personality traits. He also presented analyses of 37 mental health symptoms, of attitudes towards 12 controversial social issues, and of 9 IQ tests.[2] In those analyses, Thurstone had made use of tetrachoric correlation coefficients, a method for estimating continuous variable correlations with dichotomous variables. Tetrachorics require extensive calculations but in early 1933, he and two colleagues at the University of Chicago published a set of computing diagrams that greatly reduce the calculations needed for those coefficients.[7] His 1933 presidential address was published in early 1934 with the title Vectors of the Mind. It lacked methodological and mathematical details of his technique, which is then the subject of this book. A 2004 conference called Factor Analysis at 100 produced a book with two chapters that document the historical importance Thurstone's contributions to factor analysis.[8][9] Thurstone's approach to factor analysis remains an important method in psychological research and it has since been used in numerous other fields of study.[10] It is now considered part of a family of methods for analyzing the covariance structure of variables, which includes principal components analysis, exploratory factor analysis, confirmatory factor analysis, and structural equation modeling.[11]
See also
References
- ^ a b Thurstone, Louis (1931). "Multiple factor analysis". Psychological Review. 38: 406–427.
- ^ a b Thurstone, Louis (1934). "The Vectors of Mind". The Psychological Review. 41: 1–32.
- ^ Spearson, Charles (1904). "General intelligence objectively determined and measured". American Journal of Psychology. 15: 201–293.
- ^ Hotelling, H. "Analysis of a complex of statistical variables into principal components". Journal of Educational Psychology. 24: 417–441, 498-520.
- ^ Harman, Harry (1976). Modern Factor Analysis. Third Edition Revised. Chicago, Illinois: The University of Chicago Press. p. 5. ISBN 0-226-31652-1.
- ^ Mulaik, Stanley (2010). Foundations of Factor Analysis. Second Edition. Boca Raton, Florida: CRC Press. pp. 147–151. ISBN 978-1-4200-9961-4.
- ^ Chesire, Leone; Saffir, Milton; Thurstone, L.L. (1933). Computing Diagrams for the Tetrachoric Correlation Coefficient. Chicago, Illinois: The University of Chicago Bookstore.
- ^ Bock, Darrell (2007). "Rethinking Thurstone". In Cudeck, Robert; MacCallum, Robert C. (eds.). Factor Analysis at 100. Historical Developments and Future Directions. Mahwah, New Jersey: Lawrence Erlbaum Associates. ISBN 978-0-8058-5347-6.
- ^ Bock, Darrell (2007). "Rethinking Thurstone". In Cudeck, Robert; MacCallum, Robert C. (eds.). Factor Analysis at 100. Historical Developments and Future Directions. Mahwah, New Jersey: Lawrence Erlbaum Associates. ISBN 978-0-8058-5347-6.
- ^ Harman, Harry H. (1976). Modern Factor Analysis. Third Edition Revised. Chicago, Illinois: University of Chicago Press. pp. 6–8. ISBN 0-226-31652-1.
- ^ Mulaik, Stanley A. (2010). Foundations of Factor Analysis. Second Edition. Boca Raton, Florida: CRC Press. pp. 1–3. ISBN 978-1-4200-9961-4.