PCA of construct correlations

Description
Principal component analysis (PCA) is a method to identify structures in data. It is a well known method of data reduction which can also be applied to any squared matrix. It is applied to the inter-construct correlations in order to explore structures in the relations between constructs in the case of grids. The default type of rotation used is Varimax. Other methods can be chosen (see ?constructPca). In comparison to the Root-mean-square statistic, a PCA accounts for the sign of the correlation thus allowing the.

R-Code
The following codes calculates a PCA with three factors (default) and varimax rotation (default). > constructPca(fbb2003)

Construct PCA

Number of components extracted: 3 Type of rotation: varimax

Loadings: RC1  RC2   RC3 clever - not bright                0.96  0.02  0.25 disorganized - organized          -0.82 -0.40 -0.17 listens - doesn't hear             0.92 -0.26  0.12 no clear view - clear view of life -0.45 -0.15 -0.76 understands me - no understanding  0.87 -0.07 -0.08 ambitious - no ambition            0.02  0.13  0.94 respected - not respected          0.91 -0.01  0.22 distant - warm                    -0.13  0.74  0.25 rather aggressive - not aggressive 0.07  0.96  0.00

RC1 RC2  RC3 SS loadings   4.27 1.74 1.69 Proportion Var 0.47 0.19 0.19 Cumulative Var 0.47 0.67 0.86

You can specify the number of components to extract. The following code yields the examples from Fransella et al. (2003, p.87). Two components are extracted using varimax rotation.

> constructPca(fbb2003, nf=2)

Construct PCA

Number of components extracted: 2 Type of rotation: varimax

Loadings: RC1  RC2 clever - not bright                0.98  0.13 disorganized - organized          -0.79 -0.40 listens - doesn't hear             0.95 -0.17 no clear view - clear view of life -0.57 -0.54 understands me - no understanding  0.84 -0.13 ambitious - no ambition            0.20  0.64 respected - not respected          0.93  0.09 distant - warm                    -0.16  0.75 rather aggressive - not aggressive -0.03 0.79

RC1 RC2 SS loadings   4.47 2.13 Proportion Var 0.50 0.24 Cumulative Var 0.50 0.73

To gain an easier overview of the data, a cutoff level can be set to surpress the printing of small loadings. > constructPca(fbb2003, nf=2, cut=.3)

Construct PCA

Number of components extracted: 2 Type of rotation: varimax

Loadings: RC1  RC2 clever - not bright                0.98 disorganized - organized          -0.79 -0.40 listens - doesn't hear             0.95 no clear view - clear view of life -0.57 -0.54 understands me - no understanding  0.84 ambitious - no ambition                  0.64 respected - not respected          0.93 distant - warm                           0.75 rather aggressive - not aggressive       0.79

RC1 RC2 SS loadings   4.47 2.13 Proportion Var 0.50 0.24 Cumulative Var 0.50 0.73

Different methods of rotation can be chosen: none, varimax, promax, cluster.

constructPca(fbb2003, rotate="none") constructPca(fbb2003, rotate="varimax") constructPca(fbb2003, rotate="promax") constructPca(fbb2003, rotate="cluster")

As a default the correlation matrix is calculated using product-moment correlation. The methods that can be selected are pearson, kendall, spearman,

constructPca(fbb2003, method="pearson")   # default setting constructPca(fbb2003, method="kendall") constructPca(fbb2003, method="spearman")

In case the results are needed for further processing you can surpress the console output pc <- constructPca(fbb2003, out=0) pc

Literature

 * Fransella, F., Bell, R. C., & Bannister, D. (2003). A Manual for Repertory Grid Technique (2. ed.). Chichester: John Wiley & Sons.