Multivariate Measures of Profile Similarity for the Objective Stratification of
Multivariate Measures of Profile Similarity for the Objective Stratification of
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Correlation, of course, ordinarily is thought of as a measure of the angular separation between variables; spec-ifically, as the cosine of an angle that separates a pair of variables in observation space, or as the cosine of -=^ — For references to the classical literature on Q-Correlation see, C. Burfc, "Correlations Between Persons, " British Journal of Psychology, 28, 1937, For empirical applications, see W. Stephenson, The Study of Behavior ^ University of Chicago Press, Chicago, 1953^ and ...for a critical discussion, see J. L. Cronback and G. C. Gleser, op. Cit .
- 22 - the angle between colijmn vectors of Z. Shoxild Z be normalized (to zero mean, unit variance) by columns, the inner product (26) Z^Z - R defines a pxp matrix of zero order correlation coefficients between variables, Q-correlation is similarly defined. , except that angular separation is measured between observations in variables space, or between rows of Z^ rather than variables, or columns of Zo Accordingly; let us define a transforination J of Z such that | is normalized to zero mean unit variance l^ rows c A (per- haps very large) NxN matrix g of zero order correlation coefficients between observations now can be defined as the outer product, (27) I 1^' - go Developed largely by and for psychologists^, Q correlation's geometric properties, including its relation to Euclidean distance^ may be illustrated graphically through an example such as Fi, gijre 2; where vectors A and B repre- sent scores by persons on batteries of arithnetie and verbal aptitude testSj plotted in the space defined by the tests (or variables )o pearsonian dis- tance between A and B;, clearly, can be measured directly from the Figure as the lengt-h of a line (not shown) connecting the points, Q correlation measures angular distance between persons through the zero oirder correlation coefficient, q = cos 9, Normalizing observations to unit length, of course, is eq^jivalent to projecting A and B into A' and B' on a unit circle from the origin, 'j, ac- cordingly, can be measured by the perpendicular (canonical) projection of 23 Verbal Ability- Arithmetic Ability Figure 2 B' onto A (or, A' onto b); and can easily be seen from the figure to vary 17 inversely with the squared distance between normalized vectors A' and B', a (28) d^ ^ 21 qj 17 With Stephenson^ and Cronback and Gleser as notable exceptions, much of the psychometric literature surrounding Q correlation tends to play down the significance of the standardization (within observations^ or rows of Z), Burt in particular argiaes that similarities between persons and vari- ables, analyzed through Q and R correlation, respectively ;, "shovild lead to consistent, and in the end, to identical conclusions.
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You can find similar books in the "Read Also" column, or choose other free books by Donald Eugene Farrar to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Correlation, of course, ordinarily is thought of as a measure of the angular separation between variables; spec-ifically, as the cosine of an angle that separates a pair of variables in observation space, or as the cosine of -=^ — For references to the classical literature on Q-Correlation see, C. Burfc, "Correlations Between Persons, " British Journal of Psychology, 28, 1937, For empirical applications, see W. Stephenson, The Study of Behavior ^ University of Chicago Press, Chicago, 1953^ and ...for a critical discussion, see J. L. Cronback and G. C. Gleser, op. Cit .
- 22 - the angle between colijmn vectors of Z. Shoxild Z be normalized (to zero mean, unit variance) by columns, the inner product (26) Z^Z - R defines a pxp matrix of zero order correlation coefficients between variables, Q-correlation is similarly defined. , except that angular separation is measured between observations in variables space, or between rows of Z^ rather than variables, or columns of Zo Accordingly; let us define a transforination J of Z such that | is normalized to zero mean unit variance l^ rows c A (per- haps very large) NxN matrix g of zero order correlation coefficients between observations now can be defined as the outer product, (27) I 1^' - go Developed largely by and for psychologists^, Q correlation's geometric properties, including its relation to Euclidean distance^ may be illustrated graphically through an example such as Fi, gijre 2; where vectors A and B repre- sent scores by persons on batteries of arithnetie and verbal aptitude testSj plotted in the space defined by the tests (or variables )o pearsonian dis- tance between A and B;, clearly, can be measured directly from the Figure as the lengt-h of a line (not shown) connecting the points, Q correlation measures angular distance between persons through the zero oirder correlation coefficient, q = cos 9, Normalizing observations to unit length, of course, is eq^jivalent to projecting A and B into A' and B' on a unit circle from the origin, 'j, ac- cordingly, can be measured by the perpendicular (canonical) projection of 23 Verbal Ability- Arithmetic Ability Figure 2 B' onto A (or, A' onto b); and can easily be seen from the figure to vary 17 inversely with the squared distance between normalized vectors A' and B', a (28) d^ ^ 21 qj 17 With Stephenson^ and Cronback and Gleser as notable exceptions, much of the psychometric literature surrounding Q correlation tends to play down the significance of the standardization (within observations^ or rows of Z), Burt in particular argiaes that similarities between persons and vari- ables, analyzed through Q and R correlation, respectively ;, "shovild lead to consistent, and in the end, to identical conclusions.
What to read after Multivariate Measures of Profile Similarity for the Objective Stratification of?
You can find similar books in the "Read Also" column, or choose other free books by Donald Eugene Farrar to read onlineMoreLess
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