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The VARCLUS Procedure

Example 68.1: Correlations among Physical Variables

The following data are correlations among eight physical variables as given by Harman (1976). The first PROC VARCLUS run clusters on the basis of principal components, the second run clusters on the basis of centroid components. The third analysis is hierarchical, and the TREE procedure is used to print a tree diagram. The results of the analyses follow.

   data phys8(type=corr);
      title 'Eight Physical Measurements on 305 School Girls';
      title2 'Harman: Modern Factor Analysis, 3rd Ed, p22';
      label height='Height'
            arm_span='Arm Span'
            forearm='Length of Forearm'
            low_leg='Length of Lower Leg'
            weight='Weight'
            bit_diam='Bitrochanteric Diameter'
            girth='Chest Girth'
            width='Chest Width';

      input _name_ $ 1-8
            (height arm_span forearm low_leg weight bit_diam 
             girth width)(7.);
      _type_='corr';
      datalines;
   height  1.0    .846   .805   .859   .473   .398   .301   .382
   arm_span.846   1.0    .881   .826   .376   .326   .277   .415
   forearm .805   .881   1.0    .801   .380   .319   .237   .345
   low_leg .859   .826   .801   1.0    .436   .329   .327   .365
   weight  .473   .376   .380   .436   1.0    .762   .730   .629
   bit_diam.398   .326   .319   .329   .762   1.0    .583   .577
   girth   .301   .277   .237   .327   .730   .583   1.0    .539
   width   .382   .415   .345   .365   .629   .577   .539   1.0
   ;

   proc varclus data=phys8;
   run;

The PROC VARCLUS statement invokes the procedure. By default, PROC VARCLUS clusters on the basis of principal components.

Output 68.1.1: Principal Cluster Components: Cluster Summary

Eight Physical Measurements on 305 School Girls
Harman: Modern Factor Analysis, 3rd Ed, p22

Oblique Principal Component Cluster Analysis

Cluster summary for 1 cluster
Cluster Members Cluster
Variation
Variation
Explained
Proportion
Explained
Second
Eigenvalue
1 8 8 4.67288 0.5841 1.7710

Total variation explained = 4.67288 Proportion = 0.5841

Cluster 1 will be split.

Cluster summary for 2 clusters
Cluster Members Cluster
Variation
Variation
Explained
Proportion
Explained
Second
Eigenvalue
1 4 4 3.509218 0.8773 0.2361
2 4 4 2.917284 0.7293 0.4764

Total variation explained = 6.426502 Proportion = 0.8033

Cluster Variable R-squared with 1-R**2
Ratio
Variable
Label
Own
Cluster
Next
Closest
Cluster 1 height 0.8777 0.2088 0.1545 Height
  arm_span 0.9002 0.1658 0.1196 Arm Span
  forearm 0.8661 0.1413 0.1560 Length of Forearm
  low_leg 0.8652 0.1829 0.1650 Length of Lower Leg
Cluster 2 weight 0.8477 0.1974 0.1898 Weight
  bit_diam 0.7386 0.1341 0.3019 Bitrochanteric Diameter
  girth 0.6981 0.0929 0.3328 Chest Girth
  width 0.6329 0.1619 0.4380 Chest Width

No cluster meets the criterion for splitting.


As displayed in Output 68.1.1, the cluster component (by default, the first principal component) explains 58.41% of the total variation in the 8 variables.

The cluster is split because the second eigenvalue is greater than 1 (the default value of the MAXEIGEN option).

The two resulting cluster components explain 80.33% of the variation in the original variables. The cluster summary table shows that the variables height, arm_span, forearm, and low_leg have been assigned to the first cluster; and that the variables weight, bit_diam, girth, and width have been assigned to the second cluster.

Output 68.1.2: Standard Scoring Coefficients and Cluster Structure Table

Oblique Principal Component Cluster Analysis

Standardized Scoring Coefficients
Cluster   1 2
height Height 0.266977 0.000000
arm_span Arm Span 0.270377 0.000000
forearm Length of Forearm 0.265194 0.000000
low_leg Length of Lower Leg 0.265057 0.000000
weight Weight 0.000000 0.315597
bit_diam Bitrochanteric Diameter 0.000000 0.294591
girth Chest Girth 0.000000 0.286407
width Chest Width 0.000000 0.272710

Cluster Structure
Cluster   1 2
height Height 0.936881 0.456908
arm_span Arm Span 0.948813 0.407210
forearm Length of Forearm 0.930624 0.375865
low_leg Length of Lower Leg 0.930142 0.427715
weight Weight 0.444281 0.920686
bit_diam Bitrochanteric Diameter 0.366201 0.859404
girth Chest Girth 0.304779 0.835529
width Chest Width 0.402430 0.795572


The standardized scoring coefficients in Output 68.1.2 show that each cluster component has similar scores for each of its associated variables. This suggests that the principal cluster component solution should be similar to the centroid cluster component solution, which follows in the next PROC VARCLUS run.

The cluster structure table displays high correlations between the variables and their own cluster component. The correlations between the variables and the opposite cluster component are all moderate.

Output 68.1.3: Inter-Cluster Correlations

Oblique Principal Component Cluster Analysis

Inter-Cluster Correlations
Cluster 1 2
1 1.00000 0.44513
2 0.44513 1.00000


The intercluster correlation table shows that the cluster components are moderately correlated with \rho = 0.44513.

In the following statements, the CENTROID option in the PROC VARCLUS statement specifies that cluster centroids be used as the basis for clustering.

   proc varclus data=phys8 centroid;
   run;

Output 68.1.4: Centroid Cluster Components: Cluster Summary

Oblique Centroid Component Cluster Analysis

Cluster summary for 1 cluster
Cluster Members Cluster
Variation
Variation
Explained
Proportion
Explained
1 8 8 4.631 0.5789

Total variation explained = 4.631 Proportion = 0.5789

Cluster summary for 2 clusters
Cluster Members Cluster
Variation
Variation
Explained
Proportion
Explained
1 4 4 3.509 0.8773
2 4 4 2.91 0.7275

Total variation explained = 6.419 Proportion = 0.8024

Cluster Variable R-squared with 1-R**2
Ratio
Variable
Label
Own
Cluster
Next
Closest
Cluster 1 height 0.8778 0.2075 0.1543 Height
  arm_span 0.8994 0.1669 0.1208 Arm Span
  forearm 0.8663 0.1410 0.1557 Length of Forearm
  low_leg 0.8658 0.1824 0.1641 Length of Lower Leg
Cluster 2 weight 0.8368 0.1975 0.2033 Weight
  bit_diam 0.7335 0.1341 0.3078 Bitrochanteric Diameter
  girth 0.6988 0.0929 0.3321 Chest Girth
  width 0.6473 0.1618 0.4207 Chest Width


The first cluster component, which, in the centroid method, is an unweighted sum of the standardized variables, explains 57.89% of the variation in the data. This value is near the maximum possible variance explained, 58.41%, which is attained by the first principal component (Output 68.1.1).

The centroid clustering algorithm splits the variables into the same two clusters created in the principal component method. Recall that this outcome was suggested by the similar standardized scoring coefficients in the principal cluster component solution.

The default behavior in the centroid method is to split any cluster with less than 75% of the total cluster variance explained by the centroid component. In the next step, the second cluster, with a component that explains only 72.75% of the total variation of the cluster, is split.

In the R-squared table for two clusters, the width variable has a weaker relation to its cluster than any other variable; in the three cluster solution this variable is in a cluster of its own.

Output 68.1.5: Standardized Scoring Coefficients

Oblique Centroid Component Cluster Analysis

Standardized Scoring Coefficients
Cluster   1 2
height Height 0.266918 0.000000
arm_span Arm Span 0.266918 0.000000
forearm Length of Forearm 0.266918 0.000000
low_leg Length of Lower Leg 0.266918 0.000000
weight Weight 0.000000 0.293105
bit_diam Bitrochanteric Diameter 0.000000 0.293105
girth Chest Girth 0.000000 0.293105
width Chest Width 0.000000 0.293105


Each cluster component (Output 68.1.5) is an unweighted average of the cluster's standardized variables. Thus, the coefficients for each of the cluster's associated variables are identical in the centroid cluster component solution.

Output 68.1.6: Cluster Summary for Three Clusters

Oblique Centroid Component Cluster Analysis

Cluster summary for 3 clusters
Cluster Members Cluster
Variation
Variation
Explained
Proportion
Explained
1 4 4 3.509 0.8773
2 3 3 2.383333 0.7944
3 1 1 1 1.0000

Total variation explained = 6.892333 Proportion = 0.8615

Cluster Variable R-squared with 1-R**2
Ratio
Variable
Label
Own
Cluster
Next
Closest
Cluster 1 height 0.8778 0.1921 0.1513 Height
  arm_span 0.8994 0.1722 0.1215 Arm Span
  forearm 0.8663 0.1225 0.1524 Length of Forearm
  low_leg 0.8658 0.1668 0.1611 Length of Lower Leg
Cluster 2 weight 0.8685 0.3956 0.2175 Weight
  bit_diam 0.7691 0.3329 0.3461 Bitrochanteric Diameter
  girth 0.7482 0.2905 0.3548 Chest Girth
Cluster 3 width 1.0000 0.4259 0.0000 Chest Width


The centroid method stops at the three cluster solution. As displayed in Output 68.1.6 and Output 68.1.7, the three centroid components account for 86.15% of the variability in the eight variables, and all cluster components account for at least 79.44% of the total variation in the corresponding cluster. Additionally, the smallest correlation between the variables and their own cluster component is 0.7482.

Output 68.1.7: Cluster Quality Table

Oblique Centroid Component Cluster Analysis

Number
of
Clusters
Total
Variation
Explained
by Clusters
Proportion
of Variation
Explained
by Clusters
Minimum
Proportion
Explained
by a Cluster
Minimum
R-squared
for a
Variable
Maximum
1-R**2 Ratio
for a
Variable
1 4.631000 0.5789 0.5789 0.4306  
2 6.419000 0.8024 0.7275 0.6473 0.4207
3 6.892333 0.8615 0.7944 0.7482 0.3548


Note that, if the proportion option were set to a value between 0.5789 (the proportion of variance explained in the 1-cluster solution) and 0.7275 (the minimum proportion of variance explained in the 2-cluster solution), PROC VARCLUS would stop at a two cluster solution, and the centroid solution would find the same clusters as the principal components solution.

In the following statements, the MAXC= option computes all clustering solutions, from one to eight clusters. The SUMMARY option suppresses all output except the final cluster quality table, and the OUTTREE= option saves the results of the analysis to an output data set and forces the clusters to be hierarchical. The TREE procedure is invoked to produce a graphical display of the clusters.

   proc varclus data=phys8 maxc=8 summary outtree=tree;
   run;

   goptions ftext=swiss; 
   axis2 minor=none;
   axis1 label=('Proportion of Variation Explained') minor=none;
   proc tree horizontal vaxis=axis2 haxis=axis1 lines=(width=2);
      height _propor_;
   run;

Output 68.1.8: Hierarchical Clusters and the SUMMARY Option

Oblique Principal Component Cluster Analysis

Number
of
Clusters
Total
Variation
Explained
by Clusters
Proportion
of Variation
Explained
by Clusters
Minimum
Proportion
Explained
by a Cluster
Maximum
Second
Eigenvalue
in a Cluster
Minimum
R-squared
for a
Variable
Maximum
1-R**2 Ratio
for a
Variable
1 4.672880 0.5841 0.5841 1.770983 0.3810  
2 6.426502 0.8033 0.7293 0.476418 0.6329 0.4380
3 6.895347 0.8619 0.7954 0.418369 0.7421 0.3634
4 7.271218 0.9089 0.8773 0.238000 0.8652 0.2548
5 7.509218 0.9387 0.8773 0.236135 0.8652 0.1665
6 7.740000 0.9675 0.9295 0.141000 0.9295 0.2560
7 7.881000 0.9851 0.9405 0.119000 0.9405 0.2093
8 8.000000 1.0000 1.0000 0.000000 1.0000 0.0000


The principal component method first separates the variables into the same two clusters that were created in the first PROC VARCLUS run. Note that, in creating the third cluster, the principal component method identifies the variable width. This is the same variable that is put into its own cluster in the preceding centroid method example.

Output 68.1.9: TREE Diagram from PROC TREE
vcle3b.gif (3490 bytes)

The tree diagram in Output 68.1.9 displays the cluster hierarchy. It is clear from the diagram that there are two, or possibly three, clusters present. However, the MAXC=8 option forces PROC VARCLUS to split the clusters until each variable is in its own cluster.

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