Empirical Comparison of Six Cessation Tests for use in Principal Components Factor Analysis
THE DETERMINATION of the "best stopping place" in extracting factors has long beenaper plexing problem for factor analysts. In keeping with the scientific ideal of parsimony in re search, the factor analyst desires to take out as many factors as may be necessary to de scribe a given set of variables, but he does not want to extract too many or too few factors. If he takes out more factors than are necessary, he spends valuable research time producing fig ures which will add little to his understanding of the variables being studied; if he extracts too few factors, he is likely to miss an important part of the factor constitution of the variables. Hence a cessation test is very important in fac tor analysis. A test of significance of factors should have certain desirable-characteristics. The test should be one which can be applied while the factoring process progresses?it should be se quential. [Although if the complete factoriza tion of a large matrix can be accomplished in a few short hours by the new computers, as indi cated by Wrigley and Neuhaus (14), sequential ity is less necessary. ] The test should be such that its application is relatively simple; compu tational requirements should not be laborious or too involved. It seems logical that a suitable criterion will be some function of the size of the sample. It may also be expected to be a func tion of the number of variables. As far as pos sible, the estimator should be consistent and efficient. It is of some advantage if the criter ion is also unbiased, but this is not crucial if the bias is recognized. The above properties of a model cessation test should be kept in mind while comparing the results of the tests d e scribed in the following pages. Many tests have been devised to measure the significance of a given factor, or of a given number of factors. Investigators have used for criteria residuals, sample size, number of var iables, number of sign changes, reliabilities, factor variances, etc. Thomson discusses many tests in the latest edition of his book on fac to r analysis (12). Burt, in a recent article (2), an alyzes the different tests that have been suggest ed and describes problems involved with each. Some cessation tests provide liberal interpreta tions, resulting in the extraction of more fac tors than are later found to be necessary or in terpretable. Others are too conservative, result ing in a loss of information. Previous investigat ors (11,13) have compared different tests used to determine when to stop factoring when using the centroid method, but to the author's knowledge, no previous empirical comparison has been made of cessation tests specifically applicable to principal components factor analysis. The present study has been restricted toa comparison of six cessation tests: an adaptation of Kelley's test (8) to the Hotelling iterative meth od of factoring (7); the respective tests invented by Bartlett (1) and Hoel (6); the test devised by Guilford and Lacey (5); Burt's formula (3); and the test proposed by McNemar (10). Several oth er tests were considered for inclusion in this study, but they were discarded either beca use they were applicable only to the centroid method of factoring, or because their use appeared to be too laborious or non-sequential.