A Feasible Method for Standard Errors of Estimate in Maximum Likelihood Factor Analysis.

Published on Jun 1, 1980in Psychometrika2.5
路 DOI :10.1007/BF02294078
Robert I. Jennrich33
Estimated H-index: 33
(UCLA: University of California, Los Angeles),
D. B. Clarkson1
Estimated H-index: 1
(MU: University of Missouri)
Sources
Abstract
A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.
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#1Douglas B. Clarkson (UMSL: University of Missouri鈥揝t. Louis)H-Index: 2
The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in whicht-statistics based upon these jackknife estimates of the standard errors are computed. The validity of thet-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackkni...
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The formulae which give the standard errors of factor loading estimates, while available and computable, are complicated and our understanding of them is limited. A non-technical description of their behaviour under favourable and unfavourable conditions is given. Of particular interest is their behaviour in the presence of singularities arising from equal eigenvalues and undefined rotation.
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#1Robert I. Jennrich (UCLA: University of California, Los Angeles)H-Index: 33
#2Dorothy T. ThayerH-Index: 21
Evidence is given to indicate that Lawley's formulas for the standard errors of maximum likelihood loading estimates do not produce exact asymptotic results. A small modification is derived which appears to eliminate this difficulty.
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#1Robert I. Jennrich (UCLA: University of California, Los Angeles)H-Index: 33
In a manner similar to that used in the orthogonal case, formulas for the aymptotic standard errors of analytically rotated oblique factor loading estimates are obtained. This is done by finding expressions for the partial derivatives of an oblique rotation algorithm and using previously derived results for unrotated loadings. These include the results of Lawley for maximum likelihood factor analysis and those of Girshick for principal components analysis. Details are given in cases including di...
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#1Claude O. ArcherH-Index: 2
#2Robert I. Jennrich (UCLA: University of California, Los Angeles)H-Index: 33
Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives...
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The problem of the statistical significance of factor loadings is attacked, using the 鈥榡ack-knife鈥. The technique, while not new, has curiously not been applied to factor analysis. The procedure is outlined and applied to two classes of data: (1) real data from which a clear pattern of significant factor loadings emerges, and (2) random data which do not fare well under 鈥榡ack-knifing鈥.
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#1E. B. AndersenH-Index: 1
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#1D. N. Lawley (Edin.: University of Edinburgh)H-Index: 6
Until now the chief obstacle to the application of the maximum likelihood method of estimation to factor analysis has been the lack of any really good numerical method of solution. In this paper we give a brief review of recent work which remedies this defect. Two factor analysis models are considered. In each case we derive results which are of use in connection with new methods of solution. Formulae are given for the large-sample variances and covariances of the estimates of parameters in the ...
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#1D. N. LawleyH-Index: 6
#2A. E. MaxwellH-Index: 8
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#1Guangjian Zhang (ND: University of Notre Dame)H-Index: 11
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