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Whatever Happened to Broad Perspective? William W. Rozeboom Published online: 10 Jun 2010.
To cite this article: William W. Rozeboom (1990) Whatever Happened to Broad Perspective?, Multivariate Behavioral Research, 25:1, 61-65, DOI: 10.1207/ s15327906mbr2501_7 To link to this article: http://dx.doi.org/10.1207/s15327906mbr2501_7
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
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This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Multivariate Behavioral Research, 25 (1), 6 1-65 Copyright 0 1990, Lawrence Erlbaum Associates, Inc.
Whatever Happened to Broad Perspective? William W. Rozeboom
Downloaded by [Central Michigan University] at 06:32 03 February 2015
University of Alberta
No matter how exhaustively researchers explore their topic, it is always possible for a critic to complain that their conclusions are vitiated by insufficient attention to this-or-that facet of the enterprise. Yet such demurrers often have validity; so I hope that Velicer and Jackson (1990, hereafter referred to as V&J) will forgive me for suggesting that their extensive review suffers from constricted vision. When factoring a covariance matrix Cyy,there are four primary method choices to be made: (a) what diagonal matrix U2of uniquenesses to subtract from Cyy;(b) how much of Cyy- U2 to throw away as noise when approximating this by a common parts covariance matrix C,, of lower rank; (c) what differential weighting to give the variables when appraising how acceptably C,, approximates Cyy- U2,that is, what diagonal rescaling matrix Dwto use when measuring the - U2 - Chh)Dw; goodness of solution by how small are the elements of Dw(Cyy and finally, (d) if C,, has been judged meritorious, which of its decompositions C,, = AC&' into factor pattern A and factor covariances C,, to prefer for interpretation. In practice, of course, selections (a) through (c) are tightly interwoven; but as V&J's (1990) mathematical review makes plain, one's choice of factoring style -principal components versus principal factors versus image factors versus least-squares (Minres) versus generalized least-squares versus MLFA versus, and so forth- is most strongly distinguished by its choice on (a) and (c), with (b) remaining negotiable within the style and (d), though crucial for interpretation, standing apart as a methodological afterthought largely independent of what has gone before. I shall now argue that dubiously restricted preferences on (c) have blinkered V&J's view of both our natural prospects at (a) and certain importantly unsolved problems thereof. And those issues also relate to choice (d) insufficiencies in their inventory of comparison studies. It was once taken pretty much for granted that variance-standardized data variables should be given equal weight when solved for common factors, that should be a correlation matrix. But Maximum Likelihood is, that DwCyyDw factor analysis and several less prominent mid-Century factor models have found it mathematically expedient to stipulate Dw= U-',which in effect rescales JANUARY 1990
61
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W. Rozeboorn
to equalize uniquenesses and thereby makes a variable's influence on the factor solution an increasing function of its standard-scale communality. This is not qualitatively counter-intuitive, and moreover we can expect that within rather broad limits, choice of Dwshouldn't really matter much. But because the style of common factoring used in the comparison studies surveyed by V&J (1990) has been almost exclusively MLFA, ,hey provide no confirmation of that expectation. More importantly, the MLFA weighting method becomes degenerate if any uniqueness in U approaches zero; and although V&J clearly recognize this as a boundary problem, they lump it with Heywood cases as an uninterpretable anomaly. But in practice, we may well want uniquenesses of zero, or nearly so. This is true in particular when some of the datavariables in our study are dummycoded experimental-treatment contrasts or other prima facie causal antecedents of our study's output measures. If the latter are numerous enough to be factorable, we can analyze our data most informatively by factoring the dependent and independent variables jointly, subject to the constraint that the independent variables have essentially zero uniquenesses and their true-parts are distinguished as axes of our initial common factor solution that subsequent rotation to oblique simple structure is not allowed to reposition. (See Rozeboom. in press-a, for operational details.) So long as we idealize the uniquenesses of such manifest source variables as strict/) zero, they can be admitted by MLFA as easily as by any other method: We simply partial them out of the output variables as our first subset of initial factors, and common factor the output residuals by MLFA or any other favored method to obtain the remaining initial axes. (See Mulaik, 1972. p. 167: indeed, see also Velicer, Peacock, & Jackson, 1982, p. 375.) However, we also want license to discover that our empirical measures of manifest source variables are almost but not quite errorless: and in that case MLFA resumes its dubious near-boundary performance for which V&J prefer to fault the application rather than the method. There is a major and a minor point to be taken from this. The minor one registers astonishment that MLFA should be esteemed as the common factor style of choice. Its theoretical superiority is debatable, to put it gently. and V&J (1990) offer no evidence that any uniqueness-equalizing weighting method achieves better solutions than do such weighting orthodoxies as standard-scale principal factoring and unweighted least-squares even well away from boundaries much less near them. But the larger point is that theory and practice on communality estimation still remain deeply problematic. Were our troubles here merely the boundary transgressions that dominate V&J's account of impr.oper solution.^, they would be almost benign; for I have already argued (and Velicer, Peacock, & Jackson, 1982, implicitly concurred) that zero unique-
62
MULTIVARIATE BEHAVIORAL RESEARCH
Downloaded by [Central Michigan University] at 06:32 03 February 2015
nesses are no problem at all, either interpretively or computationally, whereas a sensibly simple and effective treatment of variables whose uniquenesses keep trying to go negative is to partial them out first as factor axes that seemingly lie in data space. (If the computer programs to which you now have access don't allow that, be apprised that my MODAIHYBALL package, now ready for distribution, makes it routine.) Where our competence with communalities falters is not at the boundary of interpretable factor solutions, but in our ability to select wisely from the infinitude of solutions available well within the interpretable region. The well-known indeterminateness of uniquenesses for any common factor solution containing doublet factors (which V&J, 1990, seem to find disturbing only insofar as it promotes wandering into a boundary) can easily be resolved by imposing additional model constraints, such as that the two variables loading on a doublet factor are to have equal uniquenesses. (Technically that is not always admissible, but it illustrates the point.) You will, of course, protest that auxiliary constraints this arbitrary do nothing to enhance the meaningfulness of results so obtained. But the very sameprotest applies to our rock-of-ages custom of seeking communalities that minimize the rank ofCY).- U2. There is nothing in the hypothesis that the covariances among m data variables are due to their mutual linear dependence on r < m common sources to imply that r is the smallest rank to which the data-covariance matrix can be reduced by subtraction of uniquenesses. Indeed, the arguments of Thurstone (1 935,p.76) and Ledermann ( 1937) can be extended to show that whatever the interpretively optimal solution for U, there is a fair chance that the rank of Cyy- U2 can be reduced further by more tinkering with its diagonal. So solving for U to minimize rank is not factoranalytically ideal; rather, it is just an expedient awaiting replacement by more interpretively cogent criteria for U. Unhappily, we have been giving little thought to what those might be, much less how to operationalize them. Yet if we can't trust rank-minimizing uniquenesses to yield the solutions we want, what should we be looking for? The answer that V&J (1990) may now favor more strongly than ever, namely, to obviate the communalities problem by opting for data-space factors, can be brushed aside with the old joke about the drunk looking for his lost keys under the streetlight. But those of us who regard common factoring as our best way to search out the causal origins of manifest covariances still have serious work ahead: (1) For openers, we need a clearer mathematical understanding of the conditions under which, when C,, contains the causally correct common parts covariances in Cyy,its rank cannot be further decreased by adjustments on its diagonal. Can we show, for example, that if the rank of C,, is less than half that
JANUARY 1990
Downloaded by [Central Michigan University] at 06:32 03 February 2015
W. Rozeboorn
of CY). (well below the Thurstone/Ledermann mark), spurious additional reducibility is no longer a serious threat? (Although my own unpublished inquiry into this matter has gone farther than the Thurstone/Ledermann analyses, I have been unable to reach conclusions that are illuminatingly firm. ') (2) We need to ascertain, both mathematically and by simulation studies. how severely common factor solutions tend in fact to be degraded by suboptimal uniqueness estimation. Chances are that within limits, accurate U does not really matter all that much. Even so, we want to know the shape of those limits; whether one direction of error is more detrimental than another; and whether. when we arrive at a terminal factor solution which is good but not great, we can improve its quality by fine-tuning the uniquenesses. (The last can be endeavored now with Lisrel, but not necessarily to the extent in the manner we want.) (3) If minimal common parts dimensionality is to be dethroned as common factoring's regent goal, what target do we put in its place? In grandly qualitative terms, the answer is obvious: We want to optimize whatever features of a factor solution are most clearly diagnostic of the data's causal origins. But what such features may be and how to optimize their manifest presence plainly deserves much thoughtful debate. Even so, we do have an established standard for this. one that despite susceptibility to artifact embodies the authentic logic of inductive scientific explanation (Rozeboom, 1972). I refer, of course, to the Simple Structure criterion for terminal axis placement; and this brings me to the third respect in which V&J's ( 1990)survey seems questionably comprehensive. When comparing assorted factoring styles for their similarity of results, what criterion we choose for axis positioning and what features of the rotated solution we pick for comparison must be expected to have considerable say in how the comparisons come out. If those don't show much method difference. we still need to ask whether rotating to optimize other features more significanl for interpretation might not establish a clearer preference ordering on factor methods. And two important deficiencies in the studies surveyed by V&J ( 1990) are candidates for exemplification of this point. The first is that all but one of the cited studies compared factor puttrrns rather than data-space estimators of the factors on which the variables have that pattern. Because an obliquely rotated factor can incur quite large shifts of position with only modest changes in the pattern thereon, it seems far more appropriate to measure similarity of factor recovery by correlation of factor estimates than by congruence of patterns. And secondly, V&J's idealized simulation studies are structured so simplistically that they scarcely begin to test any method's prowess at source recovery in the face of adversity. Serious adjudication of that needs to use simple-structured ' C'or.r.ec~rion.Although I have neglected this issue for some tinie, present commentary has goadedme to study it anew - with resulrs that I think you will find instructive even ifunseltlinp. Preprints arc available on request. 64
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W. Rozeboom
source configurations rather more obliquely complex than independent clusters on orthogonal factors. And comparing how crisply different methods recover pattern features that visual inspection of pattern plots finds most interpretively provocative requires tests whose criterion for terminal axis positioning emulates subjective rotation's emphasis on hyperplane prominence rather thanextremitized loading distributions. (Eber, 1966, and Katz & Rohlf, 1974, provide such rotation algorithms, and my MODA/HYBALL package unleashes others with the theory thereof detailed in Rozeboom, in press-b.) Preciosities in one's simple-structure criterion probably matter little in practice; but that still awaits demonstration. References Eber, H. W. (1966). Toward oblique simple structure: Maxplane. Multivariate Behavioral Rrsear-ch, 1 , 1 12- 125. Katz. J. O., & Rohlf, F. J. (1974). Functionplane -a new approach to simple structure rotation. Psychometrika, 39, 37-5 I. Ledermann, W. (1937). On the rank of the reduced correlational matrix in multiple-factor analysis. Psychometrika, 3, 85-93. Mulaik, S. A. (1972). The foundations of far.tor. unalysis. New York: McGraw-Hill. Rozeboom, W. W. (1972). Scientific inference: The myth and the reality. In S. R. Brown & D. J. Brenner (Eds.), Science, psychology, and c.ommunication: Essays honoring William Stephenson. New York: Teachers College Press. Rozeboom, W. W. (in press-a). HYBALL: A method for subspace-constrained oblique factor rotation. Multivariate Beha~ior-a1Researc.h. Rozeboom, W .W .(in press-b). Theory &practice of analytic hyperplane detection. Multi~ariate Behavioral Research. Thurstone, L. L. (1935). The vectors oj'the mind. Chicago: University of Chicago Press. Velicer, W. F., Peacock, A. C., &Jackson, D. W. ( I 982). A comparison of component and factor patterns: A Monte Carlo approach. Multi~'ariuteBehavioral Research, 17, 37 1-388. Velicer, W. F., & Jackson, D. N. (1990). Component analysis versus common factor analysis: Some isues in selecting an appropriate procedure. Multivariate Behavioral Research, 25, 1-28.
JANUARY 1990
Multivariate Behavioral Research Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/hmbr20
Whatever Happened to Broad Perspective? William W. Rozeboom Published online: 10 Jun 2010.
To cite this article: William W. Rozeboom (1990) Whatever Happened to Broad Perspective?, Multivariate Behavioral Research, 25:1, 61-65, DOI: 10.1207/ s15327906mbr2501_7 To link to this article: http://dx.doi.org/10.1207/s15327906mbr2501_7
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
Downloaded by [Central Michigan University] at 06:32 03 February 2015
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Multivariate Behavioral Research, 25 (1), 6 1-65 Copyright 0 1990, Lawrence Erlbaum Associates, Inc.
Whatever Happened to Broad Perspective? William W. Rozeboom
Downloaded by [Central Michigan University] at 06:32 03 February 2015
University of Alberta
No matter how exhaustively researchers explore their topic, it is always possible for a critic to complain that their conclusions are vitiated by insufficient attention to this-or-that facet of the enterprise. Yet such demurrers often have validity; so I hope that Velicer and Jackson (1990, hereafter referred to as V&J) will forgive me for suggesting that their extensive review suffers from constricted vision. When factoring a covariance matrix Cyy,there are four primary method choices to be made: (a) what diagonal matrix U2of uniquenesses to subtract from Cyy;(b) how much of Cyy- U2 to throw away as noise when approximating this by a common parts covariance matrix C,, of lower rank; (c) what differential weighting to give the variables when appraising how acceptably C,, approximates Cyy- U2,that is, what diagonal rescaling matrix Dwto use when measuring the - U2 - Chh)Dw; goodness of solution by how small are the elements of Dw(Cyy and finally, (d) if C,, has been judged meritorious, which of its decompositions C,, = AC&' into factor pattern A and factor covariances C,, to prefer for interpretation. In practice, of course, selections (a) through (c) are tightly interwoven; but as V&J's (1990) mathematical review makes plain, one's choice of factoring style -principal components versus principal factors versus image factors versus least-squares (Minres) versus generalized least-squares versus MLFA versus, and so forth- is most strongly distinguished by its choice on (a) and (c), with (b) remaining negotiable within the style and (d), though crucial for interpretation, standing apart as a methodological afterthought largely independent of what has gone before. I shall now argue that dubiously restricted preferences on (c) have blinkered V&J's view of both our natural prospects at (a) and certain importantly unsolved problems thereof. And those issues also relate to choice (d) insufficiencies in their inventory of comparison studies. It was once taken pretty much for granted that variance-standardized data variables should be given equal weight when solved for common factors, that should be a correlation matrix. But Maximum Likelihood is, that DwCyyDw factor analysis and several less prominent mid-Century factor models have found it mathematically expedient to stipulate Dw= U-',which in effect rescales JANUARY 1990
61
Downloaded by [Central Michigan University] at 06:32 03 February 2015
W. Rozeboorn
to equalize uniquenesses and thereby makes a variable's influence on the factor solution an increasing function of its standard-scale communality. This is not qualitatively counter-intuitive, and moreover we can expect that within rather broad limits, choice of Dwshouldn't really matter much. But because the style of common factoring used in the comparison studies surveyed by V&J (1990) has been almost exclusively MLFA, ,hey provide no confirmation of that expectation. More importantly, the MLFA weighting method becomes degenerate if any uniqueness in U approaches zero; and although V&J clearly recognize this as a boundary problem, they lump it with Heywood cases as an uninterpretable anomaly. But in practice, we may well want uniquenesses of zero, or nearly so. This is true in particular when some of the datavariables in our study are dummycoded experimental-treatment contrasts or other prima facie causal antecedents of our study's output measures. If the latter are numerous enough to be factorable, we can analyze our data most informatively by factoring the dependent and independent variables jointly, subject to the constraint that the independent variables have essentially zero uniquenesses and their true-parts are distinguished as axes of our initial common factor solution that subsequent rotation to oblique simple structure is not allowed to reposition. (See Rozeboom. in press-a, for operational details.) So long as we idealize the uniquenesses of such manifest source variables as strict/) zero, they can be admitted by MLFA as easily as by any other method: We simply partial them out of the output variables as our first subset of initial factors, and common factor the output residuals by MLFA or any other favored method to obtain the remaining initial axes. (See Mulaik, 1972. p. 167: indeed, see also Velicer, Peacock, & Jackson, 1982, p. 375.) However, we also want license to discover that our empirical measures of manifest source variables are almost but not quite errorless: and in that case MLFA resumes its dubious near-boundary performance for which V&J prefer to fault the application rather than the method. There is a major and a minor point to be taken from this. The minor one registers astonishment that MLFA should be esteemed as the common factor style of choice. Its theoretical superiority is debatable, to put it gently. and V&J (1990) offer no evidence that any uniqueness-equalizing weighting method achieves better solutions than do such weighting orthodoxies as standard-scale principal factoring and unweighted least-squares even well away from boundaries much less near them. But the larger point is that theory and practice on communality estimation still remain deeply problematic. Were our troubles here merely the boundary transgressions that dominate V&J's account of impr.oper solution.^, they would be almost benign; for I have already argued (and Velicer, Peacock, & Jackson, 1982, implicitly concurred) that zero unique-
62
MULTIVARIATE BEHAVIORAL RESEARCH
Downloaded by [Central Michigan University] at 06:32 03 February 2015
nesses are no problem at all, either interpretively or computationally, whereas a sensibly simple and effective treatment of variables whose uniquenesses keep trying to go negative is to partial them out first as factor axes that seemingly lie in data space. (If the computer programs to which you now have access don't allow that, be apprised that my MODAIHYBALL package, now ready for distribution, makes it routine.) Where our competence with communalities falters is not at the boundary of interpretable factor solutions, but in our ability to select wisely from the infinitude of solutions available well within the interpretable region. The well-known indeterminateness of uniquenesses for any common factor solution containing doublet factors (which V&J, 1990, seem to find disturbing only insofar as it promotes wandering into a boundary) can easily be resolved by imposing additional model constraints, such as that the two variables loading on a doublet factor are to have equal uniquenesses. (Technically that is not always admissible, but it illustrates the point.) You will, of course, protest that auxiliary constraints this arbitrary do nothing to enhance the meaningfulness of results so obtained. But the very sameprotest applies to our rock-of-ages custom of seeking communalities that minimize the rank ofCY).- U2. There is nothing in the hypothesis that the covariances among m data variables are due to their mutual linear dependence on r < m common sources to imply that r is the smallest rank to which the data-covariance matrix can be reduced by subtraction of uniquenesses. Indeed, the arguments of Thurstone (1 935,p.76) and Ledermann ( 1937) can be extended to show that whatever the interpretively optimal solution for U, there is a fair chance that the rank of Cyy- U2 can be reduced further by more tinkering with its diagonal. So solving for U to minimize rank is not factoranalytically ideal; rather, it is just an expedient awaiting replacement by more interpretively cogent criteria for U. Unhappily, we have been giving little thought to what those might be, much less how to operationalize them. Yet if we can't trust rank-minimizing uniquenesses to yield the solutions we want, what should we be looking for? The answer that V&J (1990) may now favor more strongly than ever, namely, to obviate the communalities problem by opting for data-space factors, can be brushed aside with the old joke about the drunk looking for his lost keys under the streetlight. But those of us who regard common factoring as our best way to search out the causal origins of manifest covariances still have serious work ahead: (1) For openers, we need a clearer mathematical understanding of the conditions under which, when C,, contains the causally correct common parts covariances in Cyy,its rank cannot be further decreased by adjustments on its diagonal. Can we show, for example, that if the rank of C,, is less than half that
JANUARY 1990
Downloaded by [Central Michigan University] at 06:32 03 February 2015
W. Rozeboorn
of CY). (well below the Thurstone/Ledermann mark), spurious additional reducibility is no longer a serious threat? (Although my own unpublished inquiry into this matter has gone farther than the Thurstone/Ledermann analyses, I have been unable to reach conclusions that are illuminatingly firm. ') (2) We need to ascertain, both mathematically and by simulation studies. how severely common factor solutions tend in fact to be degraded by suboptimal uniqueness estimation. Chances are that within limits, accurate U does not really matter all that much. Even so, we want to know the shape of those limits; whether one direction of error is more detrimental than another; and whether. when we arrive at a terminal factor solution which is good but not great, we can improve its quality by fine-tuning the uniquenesses. (The last can be endeavored now with Lisrel, but not necessarily to the extent in the manner we want.) (3) If minimal common parts dimensionality is to be dethroned as common factoring's regent goal, what target do we put in its place? In grandly qualitative terms, the answer is obvious: We want to optimize whatever features of a factor solution are most clearly diagnostic of the data's causal origins. But what such features may be and how to optimize their manifest presence plainly deserves much thoughtful debate. Even so, we do have an established standard for this. one that despite susceptibility to artifact embodies the authentic logic of inductive scientific explanation (Rozeboom, 1972). I refer, of course, to the Simple Structure criterion for terminal axis placement; and this brings me to the third respect in which V&J's ( 1990)survey seems questionably comprehensive. When comparing assorted factoring styles for their similarity of results, what criterion we choose for axis positioning and what features of the rotated solution we pick for comparison must be expected to have considerable say in how the comparisons come out. If those don't show much method difference. we still need to ask whether rotating to optimize other features more significanl for interpretation might not establish a clearer preference ordering on factor methods. And two important deficiencies in the studies surveyed by V&J ( 1990) are candidates for exemplification of this point. The first is that all but one of the cited studies compared factor puttrrns rather than data-space estimators of the factors on which the variables have that pattern. Because an obliquely rotated factor can incur quite large shifts of position with only modest changes in the pattern thereon, it seems far more appropriate to measure similarity of factor recovery by correlation of factor estimates than by congruence of patterns. And secondly, V&J's idealized simulation studies are structured so simplistically that they scarcely begin to test any method's prowess at source recovery in the face of adversity. Serious adjudication of that needs to use simple-structured ' C'or.r.ec~rion.Although I have neglected this issue for some tinie, present commentary has goadedme to study it anew - with resulrs that I think you will find instructive even ifunseltlinp. Preprints arc available on request. 64
MULTIVARIATE BEHAVIORAL RESEARCH
Downloaded by [Central Michigan University] at 06:32 03 February 2015
W. Rozeboom
source configurations rather more obliquely complex than independent clusters on orthogonal factors. And comparing how crisply different methods recover pattern features that visual inspection of pattern plots finds most interpretively provocative requires tests whose criterion for terminal axis positioning emulates subjective rotation's emphasis on hyperplane prominence rather thanextremitized loading distributions. (Eber, 1966, and Katz & Rohlf, 1974, provide such rotation algorithms, and my MODA/HYBALL package unleashes others with the theory thereof detailed in Rozeboom, in press-b.) Preciosities in one's simple-structure criterion probably matter little in practice; but that still awaits demonstration. References Eber, H. W. (1966). Toward oblique simple structure: Maxplane. Multivariate Behavioral Rrsear-ch, 1 , 1 12- 125. Katz. J. O., & Rohlf, F. J. (1974). Functionplane -a new approach to simple structure rotation. Psychometrika, 39, 37-5 I. Ledermann, W. (1937). On the rank of the reduced correlational matrix in multiple-factor analysis. Psychometrika, 3, 85-93. Mulaik, S. A. (1972). The foundations of far.tor. unalysis. New York: McGraw-Hill. Rozeboom, W. W. (1972). Scientific inference: The myth and the reality. In S. R. Brown & D. J. Brenner (Eds.), Science, psychology, and c.ommunication: Essays honoring William Stephenson. New York: Teachers College Press. Rozeboom, W. W. (in press-a). HYBALL: A method for subspace-constrained oblique factor rotation. Multivariate Beha~ior-a1Researc.h. Rozeboom, W .W .(in press-b). Theory &practice of analytic hyperplane detection. Multi~ariate Behavioral Research. Thurstone, L. L. (1935). The vectors oj'the mind. Chicago: University of Chicago Press. Velicer, W. F., Peacock, A. C., &Jackson, D. W. ( I 982). A comparison of component and factor patterns: A Monte Carlo approach. Multi~'ariuteBehavioral Research, 17, 37 1-388. Velicer, W. F., & Jackson, D. N. (1990). Component analysis versus common factor analysis: Some isues in selecting an appropriate procedure. Multivariate Behavioral Research, 25, 1-28.
JANUARY 1990
