Most of the strategies that have been proposed to determine the number of components that account for the most variation in a principal components analysis of a correlation matrix rely on the analysis of the eigenvalues and on numerical solutions. The Cattell’s scree test is a graphical strategy with a nonnumerical solution to determine the number of components to retain. Like Kaiser’s rule, this test is one of the most frequently used strategies for determining the number of components to retain. However, the graphical nature of the scree test does not definitively establish the number of components to retain. To circumvent this issue, some numerical solutions are proposed, one in the spirit of Cattell’s work and dealing with the scree part of the eigenvalues plot, and one focusing on the elbow part of this plot. A simulation study compares the efficiency of these solutions to those of other previously proposed methods. Extensions to factor analysis are possible and may be particularly useful with many low-dimensional components.
The authors examined relations among motivational styles and school adjustment in a sample of 786 7th and 8th grade U.S. students. Specifically, the authors tested the hypothesis that agency beliefs mediate relations between styles of motivational self-regulation (i.e., intrinsic, identified, introjected, and extrinsic) and school adjustment (school grades, school well-being, and positive and negative affect). A structural equation model testing this hypothesis indicated that agency beliefs about one's effort mediate the relations between the styles and positive school adjustment. By contrast, the extrinsic style was not mediated by agency beliefs but reflected adverse low-magnitude direct effects on all of the outcomes except positive affect. Overall, the model strongly predicted school adjustment, and adherence to the identified motivational style was particularly important.
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