Factor Analysis 5 factors, no rotation and using COVARIANCE matrix

In this section an alternative method for PCA and Factor Analysis is described using SPSS. However, these notes are only presented for completeness, it is rarely advisable to use this method. In Minitab the same analysis can be completed by selecting the Covariance matrix as the matrix to factor in the Options window.

The previous analyses have been based on the standardised correlation matrix. This means that all variables have the same mean and standard deviation, thus each variable makes an equal contribution to the overall variance. Indeed the variance is simply the number of variables since each has a variance of 1.0 in standardised form.

The initial communalities are the variances of the five variables. They are also rescaled to make the communalties equal 1.0. This does not mean that they have been standardised, a communality of 1.0 means that 100% of a variables variance (whatever that may be) is shared in common.

Communalities

Raw Rescaled
Initial Extraction Initial Extraction
V1 0.436 0.436 1.000 1.000
V2 1008.544 1008.544 1.000 1.000
V3 0.054 0.054 1.000 1.000
V4 0.055 0.055 1.000 1.000
V5 3.261 3.261 1.000 1.000
Extraction Method: Principal Component Analysis.

Because the variables have retained their initial variances almost all of the total variance is due to V2, the other four variables make a minor contribution to the total. Thus, we extract components that explain the maximum amount of variability the first component is almost entirely composed of V2. Indeed 99.65% of total variance is retained by this component, which (in terms of the raw factor laodings) is almost entirely V2

Total Variance Explained
  Initial Eigenvalues(a) Extraction Sums of Squared Loadings
  Component Total % of Variance Cumulative % Total % of Variance Cumulative %
Raw 1 1008.768 99.646 99.646 1008.768 99.646 99.646
2 3.354 0.331 99.977 3.354 0.331 99.977
3 0.182 0.018 99.995 0.182 0.018 99.995
4 0.036 0.004 99.999 0.004 0.004 99.999
5 0.010 0.001 100.000 0.001 0.001 100.000
Rescaled 1 1008.768 99.646 99.646 2.261 45.216 45.216
2 3.354 0.331 99.977 1.478 29.566 74.781
3 0.182 0.018 99.995 0.420 8.407 83.188
4 0.036 0.004 99.999 0.652 13.049 96.237
5 0.011 0.001 100.000 0.188 3.763 100.000
Extraction Method: Principal Component Analysis.
a When analyzing a covariance matrix, the initial eigenvalues are the same across the raw and rescaled solution.

However, if we examine the rescaled component loadings we observe the same pattern that we have seen in the other analyses. Although there are some differences in the loadings we would interpret the pattern of loadings in the same way as in the other analyses.

Component Matrix(a)

Raw Rescaled
Component Component
1 2 3 4 5 1 2 3 4 5
V1 0.417 -0.292 0.420 -0.008 0.004 0.631 -0.443 0.637 -0.012 0.005
V2 31.758 0.000 -0.006 0.000 -0.001 1.000 0.000 0.000 0.000 0.000
V3 0.208 -0.017 -0.016 -0.005 0.101 0.895 -0.072 -0.070 -0.020 0.434
V4 0.057 0.124 0.022 0.190 0.003 0.244 0.529 0.094 0.807 0.012
V5 0.057 1.804 0.066 -0.014 0.001 0.032 0.999 0.037 -0.008 0.001
Extraction Method: Principal Component Analysis.
a 5 components extracted.

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