# The VARCLUS Procedure

### Example 120.1 Correlations among Physical Variables

The data in this example 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 display a tree diagram. The following statements create the data set and perform the analysis:

```data phys8(type=corr);
title 'Eight Physical Measurements on 305 School Girls';
title2 'Harman: Modern Factor Analysis, 3rd Ed, p22';
label ArmSpan='Arm Span'             Forearm='Length of Forearm'
LowerLeg='Length of Lower Leg' BitDiam='Bitrochanteric Diameter'
Girth='Chest Girth'            Width='Chest Width';
input _Name_ \$ 1-8
(Height ArmSpan Forearm LowerLeg Weight BitDiam
Girth Width)(7.);
_Type_='corr';
datalines;
Height     1.0    .846   .805   .859   .473   .398   .301   .382
ArmSpan    .846   1.0    .881   .826   .376   .326   .277   .415
Forearm    .805   .881   1.0    .801   .380   .319   .237   .345
LowerLeg   .859   .826   .801   1.0    .436   .329   .327   .365
Weight     .473   .376   .380   .436   1.0    .762   .730   .629
BitDiam    .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 using principal components.

As displayed in Output 120.1.1, when there is only one cluster, the cluster component (by default, the first principal component) explains 58.41% of the total variation of the eight 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`, `ArmSpan`, `Forearm`, and `LowerLeg` have been assigned to the first cluster, and that the variables `Weight`, `BitDiam`, `Girth`, and `Width` have been assigned to the second cluster.

The standardized scoring coefficients in Output 120.1.1 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.

The intercluster correlation table shows that the two cluster components have a moderate correlation of 0.44513.

Output 120.1.1: Principal Component Clusters

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

Oblique Principal Component Cluster Analysis

 Observations Proportion 10000 0 8 1

 Clustering algorithm converged.

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 because it has the largest second eigenvalue, 1.770983, which is greater than the MAXEIGEN=1 value.

 Clustering algorithm converged.

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

2 Clusters R-squared with 1-R**2
Ratio
Variable
Label
Cluster Variable Own
Cluster
Next
Closest
Cluster 1 ArmSpan 0.9002 0.1658 0.1196 Arm Span
Forearm 0.8661 0.1413 0.1560 Length of Forearm
LowerLeg 0.8652 0.1829 0.1650 Length of Lower Leg
Height 0.8777 0.2088 0.1545
Cluster 2 BitDiam 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
Weight 0.8477 0.1974 0.1898

Standardized Scoring Coefficients
Cluster   1 2
ArmSpan Arm Span 0.270377 0.000000
Forearm Length of Forearm 0.265194 0.000000
LowerLeg Length of Lower Leg 0.265057 0.000000
BitDiam Bitrochanteric Diameter 0.000000 0.294591
Girth Chest Girth 0.000000 0.286407
Width Chest Width 0.000000 0.272710
Height   0.266977 0.000000
Weight   0.000000 0.315597

Cluster Structure
Cluster   1 2
ArmSpan Arm Span 0.948813 0.407210
Forearm Length of Forearm 0.930624 0.375865
LowerLeg Length of Lower Leg 0.930142 0.427715
BitDiam Bitrochanteric Diameter 0.366201 0.859404
Girth Chest Girth 0.304779 0.835529
Width Chest Width 0.402430 0.795572
Height   0.936881 0.456908
Weight   0.444281 0.920686

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

 No cluster meets the criterion for splitting.

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

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;
```

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 shown previously in Output 120.1.1.

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. Since the centroid component for the one-cluster solution explains only 57.89% of the variation as shown in Output 120.1.2, the variables are split into two clusters. The resulting clusters are the same two clusters created by the principal component method. Recall that this outcome was suggested by the similar standardized scoring coefficients in the principal cluster component solution.

In the two-cluster solution, the centroid component of the second cluster explains only 72.75% of the total variation of the cluster. Since this percentage is less than 75%, the second cluster is split.

In the R-square 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.

Each cluster component 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.

The centroid method stops at the three-cluster solution. 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 squared correlation between the variables and their own cluster component is 0.7482.

If the PROPORTION= option were set to a value between 0.5789 (the proportion of variance explained in the one-cluster solution) and 0.7275 (the minimum proportion of variance explained in the two-cluster solution), PROC VARCLUS would stop at the two-cluster solution, and the centroid solution would find the same clusters as the principal components solution, although the cluster components would be slightly different.

Output 120.1.2: Centroid Component Clusters

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

Oblique Centroid Component Cluster Analysis

 Observations Proportion 10000 0.75 8 0

 Clustering algorithm converged.

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 1 will be split because it has the smallest proportion of variation explained, 0.578875, which is less than the PROPORTION=0.75 value.

 Clustering algorithm converged.

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

2 Clusters R-squared with 1-R**2
Ratio
Variable
Label
Cluster Variable Own
Cluster
Next
Closest
Cluster 1 ArmSpan 0.8994 0.1669 0.1208 Arm Span
Forearm 0.8663 0.1410 0.1557 Length of Forearm
LowerLeg 0.8658 0.1824 0.1641 Length of Lower Leg
Height 0.8778 0.2075 0.1543
Cluster 2 BitDiam 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
Weight 0.8368 0.1975 0.2033

Standardized Scoring Coefficients
Cluster   1 2
ArmSpan Arm Span 0.266918 0.000000
Forearm Length of Forearm 0.266918 0.000000
LowerLeg Length of Lower Leg 0.266918 0.000000
BitDiam Bitrochanteric Diameter 0.000000 0.293105
Girth Chest Girth 0.000000 0.293105
Width Chest Width 0.000000 0.293105
Height   0.266918 0.000000
Weight   0.000000 0.293105

Cluster Structure
Cluster   1 2
ArmSpan Arm Span 0.948361 0.408589
Forearm Length of Forearm 0.930744 0.375468
LowerLeg Length of Lower Leg 0.930477 0.427054
BitDiam Bitrochanteric Diameter 0.366212 0.856453
Girth Chest Girth 0.304821 0.835936
Width Chest Width 0.402246 0.804574
Height   0.936883 0.455485
Weight   0.444419 0.914781

Inter-Cluster Correlations
Cluster 1 2
1 1.00000 0.44484
2 0.44484 1.00000

 Cluster 2 will be split because it has the smallest proportion of variation explained, 0.7275, which is less than the PROPORTION=0.75 value.

 Clustering algorithm converged.

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

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

Standardized Scoring Coefficients
Cluster   1 2 3
ArmSpan Arm Span 0.26692 0.00000 0.00000
Forearm Length of Forearm 0.26692 0.00000 0.00000
LowerLeg Length of Lower Leg 0.26692 0.00000 0.00000
BitDiam Bitrochanteric Diameter 0.00000 0.37398 0.00000
Girth Chest Girth 0.00000 0.37398 0.00000
Width Chest Width 0.00000 0.00000 1.00000
Height   0.26692 0.00000 0.00000
Weight   0.00000 0.37398 0.00000

Cluster Structure
Cluster   1 2 3
ArmSpan Arm Span 0.94836 0.36613 0.41500
Forearm Length of Forearm 0.93074 0.35004 0.34500
LowerLeg Length of Lower Leg 0.93048 0.40838 0.36500
BitDiam Bitrochanteric Diameter 0.36621 0.87698 0.57700
Girth Chest Girth 0.30482 0.86501 0.53900
Width Chest Width 0.40225 0.65259 1.00000
Height   0.93688 0.43830 0.38200
Weight   0.44442 0.93196 0.62900

Inter-Cluster Correlations
Cluster 1 2 3
1 1.00000 0.41716 0.40225
2 0.41716 1.00000 0.65259
3 0.40225 0.65259 1.00000

 No cluster meets the criterion for splitting.

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

In the following statements, the MAXC= option computes all clustering solutions, from one to eight clusters, and the SUMMARY option suppresses all output except the final cluster quality table:

```ods graphics on;

proc varclus data=phys8 maxc=8 summary;
run;
```

The results from PROC VARCLUS are shown in Output 120.1.3.

Output 120.1.3: Hierarchical Clusters and the SUMMARY Option

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

Oblique Principal Component Cluster Analysis

 Observations Proportion 10000 1 8 0

 Clustering algorithm converged.

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. 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. The tree diagram in Output 120.1.4 displays the cluster hierarchy.

Output 120.1.4: Dendrogram

It appears 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.