Distortion as a Function of the Number of Factors Rotated under Varying Levels of Common Variance and Error

Abstract

This study examined the effects of under and overrotation on common factor loading stability under three levels of common variance and three levels of error. Four representative factor matrices were selected. In each case, the factor matrix was adjusted to account for 30%, 45%, and 60% of common variance. Each of the adjusted matrices was postmultiplied by its transpose and the intercorrelation matrix obtained was factor analyzed to obtain the criterion matrix. Using Fisher's z transformations, randomly-generated error based on 100, 200, and 500 subjects was added to the similarly transformed intercorrelation matrix. Each error-laden matrix of z's was then transformed back to r 's and factor analyzed. Ten replications were completed for each experimental condition. Several rotations were tried below, equal to, and above the correct number of factors for each problem matrix chosen. Root-mean-square (RMS) mean values were obtained between the first few factors of each criterion matrix and the corresponding factors from the successive rotations. The RMS mean deviation values were plotted against the number of factors rotated and a multifactor repeated measures ANOVA performed. The number of factors rotated, the interaction of rotations with common variance, and the interaction of rotations with error were found significant at the .01 level in all four problems. The interaction of rotations with common variance and error was found significant at the .01 level for Problems Three and Four only. Although these findings must be considered within the logistical limitations imposed upon this study, it appears, nevertheless, that matrices which account for large amounts of common variance are less susceptible to the vagaries of overrotation. Hence, these matrices tend to have stable factor loadings. Furthermore, common-factor space is clearly distorted in the case of underrotation.