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Extensions of Factorial Solutions

Published online by Cambridge University Press:  01 January 2025

Harry H. Harman
Harry H. Harman*
Affiliation:
The University of Chicago
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Abstract

A method is developed for extending any type of factor solution to new tests. The theoretical basis for this approximating scheme is thoroughly investigated, and then a simplification in the technique is introduced for practical purposes. An example is presented which illustrates the procedure of extending a factor solution to three new tests simultaneously.

Type
Original Paper
Information
Psychometrika , Volume 3 , Issue 2 , June 1938 , pp. 75 - 84
Copyright
Copyright © 1938 The Psychometric Society

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Footnotes

*

Dwyer, P. S., “The Determination of the Factor Loadings of a Given Test from the Known Factor Loadings of Other Tests,” Psychometrika, 1937, 2, PP. 173-178.

†

Thurstone, L. L., The Vectors of Mind. Chicago: The University of Chicago Press, 1935, p. 155.

References

* A general discussion of the maxima and minima of functions of a single independent variable is presented in Granville, W. A., Smith, P. F., and Longley, W. R., Elements of the Differential and Integral Calculus. Boston: Ginn and Company, 1934, pp. 182-184.

† The derivative of a determinant of the nth order is the sum of the n determinants obtained by differentiating each column, in turn, and leaving all the remaining columns unchanged.

* If A FF, — Δ = 0 then RF(12 .... n) = 0 and there is no maximum or minimun.

* The data of this example are taken from Preliminary Report on Spearman- Holzinger Unitary Trait Study, No. 9. Prepared at the Statistical Laboratory, Department of Education, The University of Chicago, 1936, Tables 3 and 8. The factor weights of the intelligence tests as given in Table 8 were not obtained directly by bi-factor analysis (the procedure is described on p. 3 of Report 9).

† By a factor pattern is meant the set of linear equations expressing the tests in terms of the factors, and by a factor structure is meant the table of correlations of the tests with the facors. The elexnents of a factor structure are identical with the coefficients in the factor pattern if, and only if, the factors are orthogonal or uncorrelated.

* For a description of these tests see Preliminary Repart on Spearman-Holzinger Unitary Trait Study, No. 1.

† For a discussion of the Doolittle method see Holzinger, K. J., Swlneford, F., and Harman, H. H., Student Manual of Factor Analysis (Planographed). Prepared at the Statistical Laboratory, Department of Education, The University of Chicago, 1937, pp. 32-36.

* Except for the three columns headed O, K, and M.

* See Holzinger, K. J., and H,arman, H. H., “Relationships between Factors Obtained from Certain Analyses.” The Journal of Educational Psychology, Vol. XXVIII, 1937, pp. 339-341.

† To illustrate how the technique developed in this paper differs from Dwyer’s technique, the present method was applied to his example (op. cit., pp. 174-175). His values rsΛ1 ═ .3708 and rsΛ2 ═ ─.7826 become rsΛ1 ═ .3347 and rsΛ2 ═ ─.6317, Dwyer’s values yield zero residual correlations of the new test, while the values obtained by the method of this paper yield small negligible residuals as normally appear in a factor analysis.