Empirical Note on the “Number of Factors” Problem in Factor Analysis

This paper illustrates a practical method for determining the number of factors to be extracted from a matrix of sample correlations. Although a number of criteria have been proposed (cf. Cattell, 1952; Harman, 1960), many are cumbersome, while those that are not often lead to contradictory decisions. Tryon ( 1960) has shown that in the use of tests suggested by Coombs ( 1941 ) and Tucker (Thurstone, 1938), the extraction of an additional factor after the criterion indicates that factoring should end often leads to a decision to extract still further faccors. That is, in some cases [he criterion for stopping is reached after extracting four factors, but if a fifth faccor is extracted the test may indicate further extraction. It is not likely chat statistical tests, such as Lnwley's ( 1940), which cause the number of factors to be a function of sample size (and significance level) will satisfy all factor analysts, e.g., if the same battery of tests is administered to samples of different sizes, Lawley's test may indicate different numbers of faccors in each case, complicating the problem of factorial invariance. The general practice has been to rely upon rules of thumb. Among these is the recommendation that principal axis factoring end at the point when eigenvalues fall below 1.00 (Harmar,, 1960, p. 363). Another common practice is to cease factoring when there is a sharp break in [he amount of variance reduction between two adjacent factors. Recently, Wrigley (1960) has offered a practical method for limiting factor extraction to common factors. This paper presents an illustration and excension of Wrigley's method. Wrigley (1960) has proposed overfactoring (drawing more than the assumed number of factors) and then using Varimax (Kaiser, 1958) or Quartimax (Neuhaus & Wrigley, 1954) to rocate successive numbers of factors. In this procedure one searches among the rotated factors of a solution for one having the highest loading of only one variable. If such a specific factor is found, the last factor from the principal axis solution is omitted and Varimax (or Quartimax) is reapplied. The procedure is repeated until that solucion is found in which each faccor contains the highest loadings of at least rwo (experimentally independent) variables. This method serves to limit factoring to the extraction of common factors. However, [here may still be certain ambiguities associated with some of these factors. The following example illustrates this and suggests a way of handling such ambiguities. As part of a larger smdyl behavioral data were obcained on che activities