# The Revolution Essay

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• First Read Principal Components Analysis.

The methods we have employed so far attempt to repackage all of the variance in the p variables into principal components. We may wish to restrict our analysis to variance that is common among variables. That is, when repackaging the variables’ variance we may wish not to redistribute variance that is unique to any one variable. This is Common Factor Analysis. A common factor is an abstraction, a hypothetical dimension that affects at least two of the variables. We assume that there is also one unique factor for each variable, a factor that affects that variable but does not affect any other variables. We assume that the p unique factors are

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For an iterated principal axis solution SPSS first estimates communalities, with R2 ’s, and then conducts the analysis. It then takes the communalities from that first analysis and inserts them into the main diagonal of the correlation matrix in place of the R2 ’s, and does the analysis again. The variables’ SSL’s from this second solution are then inserted into the main diagonal replacing the communalities from the previous iteration, etc. etc., until the change from one iteration to the next iteration is trivial.

Look at the communalities after this iterative process and for a two-factor solution. They now sum to 5.60. That is, 5.6/7 = 80% of the variance is common variance and 20% is unique. Here you can see how we have packaged that common variance into two factors, both before and after a varimax rotation:

[pic] The final rotated loadings are:

[pic] These loadings are very similar to those we obtained previously with a principal components analysis.

Reproduced and Residual Correlation Matrices

Having extracted common factors, one can turn right around and try to reproduce the correlation matrix from the factor loading matrix. We assume that the correlations between variables result from their