Principal Components and Factor Analysis

Select the Explained variance tab of the Factor Analysis Results dialog box to access the options described here.

Eigenvalues. Click the Eigenvalues button to produce a spreadsheet containing the eigenvalues. The relative (percent) and cumulative eigenvalues are also reported. The interpretation of eigenvalues is described in the Introductory Overviews. In general, the eigenvalues reflect the amount of common variance accounted for by the respective number of factors.

Scree plot. Click the Scree plot button to produce a scree plot (Cattell, 1966). Specifically, the successive eigenvalues will be shown in a simple line plot. Cattell suggests to find the place where the smooth decrease of eigenvalues appears to level off to the right of the plot. To the right of this point, presumably, one finds only "factorial scree." Scree is the geological term referring to the debris that collects on the lower part of a rocky slope. Thus, no more than the number of factors to the left of this point should be extracted. Refer to the Introductory Overviews for a more detailed discussion of the number-of-factors issue in factor analysis.

Reproduced/residual corrs. Click the Reproduced/residual corrs. button to produce spreadsheets with the reproduced and residual (observed minus reproduced) correlation matrices.

Highlight residuals greater than. Use the Highlight residuals greater than box to enter the cut-off value for the residual correlations that are to be highlighted in the spreadsheet.

Communalities. Click the Communalities button to produce a spreadsheet with the communalities, computed from the respective number of factors extracted in the current analysis. Refer to the Introductory Overviews for a discussion of communalities. In general, the communalities can be interpreted as the proportion of variance accounted for in the respective variables by the current number of factors.

Goodness of fit test. Click the Goodness of fit test button to produce a spreadsheet containing the U test (Lawley, 1940) of fit for the current number of factors. In general, when the variables in the analysis come from a multivariate normal distribution, then the distribution of covariances follows the Wishart distribution (Wishart, 1928). Based on these assumptions, a Chi-square test can be constructed testing whether all residual correlations are equal to zero, that is, whether or not the residual correlation matrix is a diagonal matrix. If statistically significant, it can be concluded that the residual correlation matrix is significantly different from the diagonal matrix, and therefore, that significant correlations between variables are still not accounted for. Also, Chi-square tests of incremental models (with increasing numbers of factors) can be performed, allowing for statistical significance testing of the number of factors "contained" in the correlation matrix. However, note that the sensitivity of the Chi-square test is greatly affected by the sample size (n); refer to Harman (1976), Mulaik (1972), or Wherry (1984) for additional details. The Goodness of fit test button is only available if the Maximum likelihood factors option button was selected on the Define Method of Factor Extraction dialog box - Advanced tab.