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 100.1.1, 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 100.1.1 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 100.1.1: Principal Component Clusters

Eight Physical Measurements on 305 School Girls |

Harman: Modern Factor Analysis, 3rd Ed, p22 |

Oblique Principal Component Cluster Analysis

Observations | 10000 | Proportion | 0 |
---|---|---|---|

Variables | 8 | Maxeigen | 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 100.1.1.

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 100.1.2, 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 100.1.2: Centroid Component Clusters

Eight Physical Measurements on 305 School Girls |

Harman: Modern Factor Analysis, 3rd Ed, p22 |

Oblique Centroid Component Cluster Analysis

Observations | 10000 | Proportion | 0.75 |
---|---|---|---|

Variables | 8 | Maxeigen | 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 100.1.3.

Output 100.1.3: 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 | 10000 | Proportion | 1 |
---|---|---|---|

Variables | 8 | Maxeigen | 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 100.1.4 displays the cluster hierarchy.

Output 100.1.4: 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.