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 The VARCLUS Procedure

## Background

The VARCLUS procedure attempts to divide a set of variables into nonoverlapping clusters in such a way that each cluster can be interpreted as essentially unidimensional. For each cluster, PROC VARCLUS computes a component that can be either the first principal component or the centroid component and tries to maximize the sum across clusters of the variation accounted for by the cluster components. PROC VARCLUS is a type of oblique component analysis related to multiple group factor analysis (Harman 1976). multiple group (VARCLUS) The VARCLUS procedure 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 fifty 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.

By default, PROC VARCLUS begins with all variables in a single cluster. It then repeats the following steps:

1. 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 PERCENT= option) or the largest eigenvalue associated with the second principal component (using the MAXEIGEN= option).
2. 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), and assigning each variable to the rotated component with which it has the higher squared correlation.
3. Variables are iteratively reassigned to clusters to maximize the variance accounted for by the cluster components. The reassignment may be required to maintain a hierarchical structure.

The procedure stops when each cluster satisfies a user-specified criterion involving either the percentage of variation accounted for or the second eigenvalue of each cluster. By default, PROC VARCLUS stops when each cluster has only a single eigenvalue greater than one, thus satisfying the most popular criterion for determining the sufficiency of a single underlying factor 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.

You can have the iterative reassignment phases restrict the reassignment of variables such that hierarchical clusters are produced. In this case, when a cluster is split, a variable in one of the two resulting clusters can be reassigned to the other cluster resulting from the split but not to a cluster that is not part of the original cluster (the one that is split). If principal components are used, the NCS phase is an alternating least-squares method and converges rapidly. The search phase is very time consuming for a large number of variables and is omitted by default. If the default initialization method is used, the search phase is rarely able to improve the results of the NCS phase. If random initialization is used, the NCS phase may 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 may not increase the amount of variance explained; therefore, it is limited, by default, to one iteration.

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