Having reasons and the factoring account

It’s natural to say that when it’s rational for me to φ, I have reasons to φ. That is, there are reasons for φ-ing, and moreover, I have some of them. Mark Schroeder calls this view The Factoring Account of the having reasons relation. He thinks The Factoring Account is false. In this paper, I defend The Factoring Account. Not only do I provide intuitive support for the view, but I also defend it against Schroeder’s criticisms. Moreover, I show that it helps us understand the requirements of substantive rationality, or what we are rationally required to do when responding to reasons.     

Canonical/redundancy factoring analysis

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation betweenlinear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.Wayne S. DeSarbo is a Member of Technical Staff at Bell Laboratories in the Mathematics and Statistics Research Group at Murray Hill, N.J. I wish to express my appreciation to J. Kettenring, J. Kruskal, C. Mallows, and R. Gnanadesikan for their valuable technical assistance and/or for comments on an earlier draft of this paper. I also wish to thank the editor and reviewers of this paper for their insightful remarks.     

General theory and methods for matric factoring

Methods are developed for factoring an arbitrary rectangular matrixS of rankr into the formFP, whereF hasr columns andP hasr rows. For the statistical problem of factor analysis,S may be the score matrix of a population of individuals on a battery of tests. ThenF is a matrix of factor loadings,P is a matrix of factor scores, andr is the number of factor variates. (As in current procedures, there remains a subsequent problem of rotation of axes and interpretation of factors, which is not discussed here.) Methods are also developed for factoring an arbitrary Gramian matrixG of rankr into the formFF, whereF hasr columns andF denotesF transposed. For the statistical problem of factor analysis,G may be the matrix of intercorrelations,R, of a battery of tests, with unity, communalities, or other parameters in the principal diagonal.R is proportional toSS, and it is shown thatS can be factored by factoringR. This may usually be the most economical procedure in practice; it should not be overlooked, however, thatS can be factored directly. The general methods build up anF (andP) in as many stages as desired; as many factors as may be deemed computationally practical can be extracted at a time. Perhaps it will usually be found convenient to extract not more than three factors at a time. Current procedures, like the centroid and principal axes, are special cases of a general method presented here for extracting one factor at a time.     

A necessary and sufficient formula for matric factoring

For the purpose of extracting factors from matrices, it is proved that a certain formula is both necessary and sufficient. In factor analysis, the formula may be applied either to the correlation matrix, or directly to the score matrix (assuming the communality problem is solved). As many factors as desired can be extracted in one operation. Having such a compact formulation is useful for teaching as well as computing purposes, since it includes all techniques of factor extraction as special cases.On leave from the Israel Institute for Applied Social Research. This research was facilitated in part by a grant from the Lucius N. Littauer Foundation to the American Committee for Social Research in Israel, Inc.     

A method for factoring large numbers of items

The computation of intercorrelation matrices involving large numbers of variables and the subsequent factoring of these matrices present a formidable task. A method for estimating factor loadings without computing the intercorrelation matrix is developed. The estimation procedure is derived from a theoretical model which is shown to be a special case of the multiple-group centroid method of factoring. Empirical checks have indicated that the model, even though it makes some stringent assumptions, can be applied to a variety of variables found in psychological factoring problems. It has been found to be particularly useful in factoring test items.     

A multiple group method of factoring the correlation matrix

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be calleda multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrixF which satisfies the fundamental relationFF'=R.This study is one of a series of investigations in the development of multiple factor analysis and application to the study of primary mental abilities. We wish to acknowledge the financial assistance from the Social Science Research Committee of The University of Chicago which has made possible the work of the Psychometric Laboratory.     

An improved procedure for the wherry-winer method for factoring large numbers of items

A technique is presented that differs from the previous one in that the use of variance terms is eliminated from the computations; thus some formulas are simplified. A rationale for the improved method is presented.     

Child behavior disorders by age and sex based on item factoring of the revised Bristol Guides

Standardized teacher observations of 2,527 schoolchildren, selected at random for the revised Bristol Social Adjustment Guides were partitioned into four subsamples consisting of 797 5- to 10-year-old boys, 758 5- to 10-year-old-girls; 508 11- to 15-year-old boys, and 464 11- to 15-year-old girls, respectively. The children were observed by over 900 teachers and rated on 104 indicators of maladaptive behavior. Item scores for each age/ sex sample were subjected to first- and second-order factor analysis, with varimax rotation yielding identical second-order models of behavior disorder across age and sex samples and somewhat different first-order models for each sample. Comparison of derived dimensions with dimensions emergent in other behavior problem research indicated considerable consistency. Moreover, the similarity of the factorially derived dimensions confirmed the cross-age and -sex generality of the syndromes known as unforthcomingness, hostility, and depression, and provided reasonable support for the utility of the syndrome of inconsequence, although it was apparent that inconsequence stands as more a composite of underlying factor dimensions reflecting hyperactive and attention-seeking behaviors. While the withdrawal syndrome found factorial support, its integrity was clearly specific to child age and sex.     

A definitive,large-sample factoring of personality structure in objective measures,as a basis for the high school objective-analytic battery

Eighty-two subtests chosen from a series of preceding researches as the best marker variables for 12 previously checked and interpreted personality factors (U.I. 16, 17, 19, 20, 21, 23–26, 28, 32, and 33) were administered to 2,522 boys of high school age. The ensuing factor analysis, aimed to check and precision the factor patterns, to lead to the definitive HSOA battery, and to yield scores for further age and genetic analyses, observed the following procedural requirements: (1) Scree test for number of factors, (2) iterated communalities, (3) blind automated rotation, carried further by rotoplot to a maximum simple structure, (4) a test of simple structure significance, and (5) matching of patterns for invariance against other studies. As before in this domain, 19–20 factors were found, and the 10 for which definite markers had been made available matched very well with former patterns. Although a small minority of tests failed to load as hypothesized, this results in some weakness of factor estimate only on U.I. 32, Exvia-Invia. For on all 10 factors for the planned HSOA battery, with this possible exception, the validities of weighted battery scores reached acceptable levels.     

Counterfactuals and the applications of mathematics

Conclusion It has been argued that the attempt to meet indispensability arguments for realism in mathematics, by appeal to counterfactual statements, presupposes a view of mathematical modality according to which even though mathematical entities do not exist, they might have existed. But I have sought to defend this controversial view of mathematical modality from various objections derived from the fact that the existence or nonexistence of mathematical objects makes no difference to the arrangement of concrete objects. This defense of the controversial view of mathematical modality obviously falls far short of a full endorsement of the counterfactual approach, but I hope my remarks may serve to help keep such an approach a live option.