The VARCLUS procedure divides a set of numeric variables into disjoint or hierarchical clusters. Associated with each cluster is a linear combination of the variables in the cluster. This linear combination can be either the first principal component (the default) or the centroid component (if you specify the CENTROID option). The first principal component is a weighted average of the variables that explains as much variance as possible. See ChapterĀ 79: The PRINCOMP Procedure, for further details. Centroid components are unweighted averages of either the standardized variables (the default) or the raw variables (if you specify the COVARIANCE option). PROC VARCLUS tries to maximize the variance that is explained by the cluster components, summed over all the clusters.
The cluster components are oblique, not orthogonal, even when the cluster components are first principal components. In an ordinary principal component analysis, all components are computed from the same variables, and the first principal component is orthogonal to the second principal component and to every other principal component. In PROC VARCLUS, each cluster component is computed from a set of variables that is different from all the other cluster components. The first principal component of one cluster might be correlated with the first principal component of another cluster. Hence, the PROC VARCLUS algorithm is a type of oblique component analysis.
As in principal component analysis, either the correlation or the covariance matrix can be analyzed. If correlations are used, all variables are treated as equally important. If covariances are used, variables with larger variances have more importance in the analysis.
PROC VARCLUS displays a dendrogram (tree diagram of hierarchical clusters) by using ODS Graphics. PROC VARCLUS can also can create an output data set that can be used by the TREE procedure to draw the dendrogram. A second output data set can be used with the SCORE procedure to compute component scores for each cluster.
PROC VARCLUS can be used as a variable-reduction method. A large set of variables can often be replaced by the set of cluster components with little loss of information. A given number of cluster components does not generally explain as much variance as the same number of principal components on the full set of variables, but the cluster components are usually easier to interpret than the principal components, even if the latter are rotated.
For example, an educational test might contain 50 items. PROC VARCLUS can be used to divide the items into, say, five clusters. Each cluster can then be treated as a subtest, with the subtest scores given by the cluster components. If the cluster components are centroid components of the covariance matrix, each subtest score is simply the sum of the item scores for that cluster.
The VARCLUS algorithm is both divisive and iterative. By default, PROC VARCLUS begins with all variables in a single cluster. It then repeats the following steps:
A cluster is chosen for splitting. Depending on the options specified, the selected cluster has either the smallest percentage of variation explained by its cluster component (using the PROPORTION= option) or the largest eigenvalue associated with the second principal component (using the MAXEIGEN= option).
The chosen cluster is split into two clusters by finding the first two principal components, performing an orthoblique rotation (raw quartimax rotation on the eigenvectors; Harris and Kaiser 1964), and assigning each variable to the rotated component with which it has the higher squared correlation.
Variables are iteratively reassigned to clusters to try to maximize the variance accounted for by the cluster components. You can require the reassignment algorithms to maintain a hierarchical structure for the clusters.
The procedure stops splitting when either of the following conditions holds:
The number of clusters is greater than or equal to the maximum number of clusters as specified by the MAXCLUSTERS= option is reached.
Every cluster satisfies the stopping criteria specified by the PROPORTION= option (percentage of variation explained) or the MAXEIGEN= option (second eigenvalue) or both.
By default, VARCLUS stops splitting when every cluster has only one eigenvalue greater than one, thus satisfying the most popular criterion for determining the sufficiency of a single underlying dimension.
The iterative reassignment of variables to clusters proceeds in two phases. The first is a nearest component sorting (NCS) phase, similar in principle to the nearest centroid sorting algorithms described by Anderberg (1973). In each iteration, the cluster components are computed, and each variable is assigned to the component with which it has the highest squared correlation. The second phase involves a search algorithm in which each variable is tested to see if assigning it to a different cluster increases the amount of variance explained. If a variable is reassigned during the search phase, the components of the two clusters involved are recomputed before the next variable is tested. The NCS phase is much faster than the search phase but is more likely to be trapped by a local optimum.
If principal components are used, the NCS phase is an alternating least squares method and converges rapidly. The search phase can be very time-consuming for a large number of variables. But if the default initialization method is used, the search phase is rarely able to substantially improve the results of the NCS phase, so the search takes few iterations. If random initialization is used, the NCS phase might be trapped by a local optimum from which the search phase can escape.
If centroid components are used, the NCS phase is not an alternating least squares method and might not increase the amount of variance explained; therefore it is limited, by default, to one iteration.
You can have VARCLUS do the clustering hierarchically by restricting the reassignment of variables such that the clusters maintain a tree structure. In this case, when a cluster is split, a variable in one of the two resulting clusters can be reassigned to the other cluster that results from the split but not to a cluster that is not part of the original cluster (the one that is split).