You can use SAS clustering procedures to cluster the observations or the variables in a SAS data set. Both hierarchical and disjoint clusters can be obtained. Only numeric variables can be analyzed directly by the procedures, although the DISTANCE procedure can compute a distance matrix that uses character or numeric variables.

The purpose of cluster analysis is to place objects into groups, or clusters, suggested by the data, not defined a priori, such that objects in a given cluster tend to be similar to each other in some sense, and objects in different clusters tend to be dissimilar. You can also use cluster analysis to summarize data rather than to find "natural" or "real" clusters; this use of clustering is sometimes called *dissection* (Everitt; 1980).

Any generalization about cluster analysis must be vague because a vast number of clustering methods have been developed in several different fields, with different definitions of clusters and similarity among objects. The variety of clustering techniques is reflected by the variety of terms used for cluster analysis: botryology, classification, clumping, competitive learning, morphometrics, nosography, nosology, numerical taxonomy, partitioning, Q-analysis, systematics, taximetrics, taxonorics, typology, unsupervised pattern recognition, vector quantization, and winner-take-all learning. Good (1977) has also suggested aciniformics and agminatics.

Several types of clusters are possible:

Disjoint clusters place each object in one and only one cluster.

Hierarchical clusters are organized so that one cluster can be entirely contained within another cluster, but no other kind of overlap between clusters is allowed.

Overlapping clusters can be constrained to limit the number of objects that belong simultaneously to two clusters, or they can be unconstrained, allowing any degree of overlap in cluster membership.

Fuzzy clusters are defined by a probability or grade of membership of each object in each cluster. Fuzzy clusters can be disjoint, hierarchical, or overlapping.

The data representations of objects to be clustered also take many forms. The most common are as follows:

a square distance or similarity matrix, in which both rows and columns correspond to the objects to be clustered. A correlation matrix is an example of a similarity matrix.

a coordinate matrix, in which the rows are observations and the columns are variables, as in the usual SAS multivariate data set. The observations, the variables, or both can be clustered.

The SAS procedures for clustering are oriented toward disjoint or hierarchical clusters from coordinate data, distance data, or a correlation or covariance matrix. The following procedures are used for clustering:

- CLUSTER
performs hierarchical clustering of observations by using eleven agglomerative methods applied to coordinate data or distance data and draws tree diagrams, which are also called

*dendrograms*or*phenograms*.- FASTCLUS
finds disjoint clusters of observations by using a

*k*-means method applied to coordinate data. PROC FASTCLUS is especially suitable for large data sets.- MODECLUS
finds disjoint clusters of observations with coordinate or distance data by using nonparametric density estimation. It can also perform approximate nonparametric significance tests for the number of clusters.

- VARCLUS
performs both hierarchical and disjoint clustering of variables by using oblique multiple-group component analysis and draws tree diagrams, which are also called

*dendrograms*or*phenograms*.- TREE
draws tree diagrams, also called

*dendrograms*or*phenograms*, by using output from the CLUSTER or VARCLUS procedure. PROC TREE can also create a data set indicating cluster membership at any specified level of the cluster tree.

The following procedures are useful for processing data prior to the actual cluster analysis:

- ACECLUS
attempts to estimate the pooled within-cluster covariance matrix from coordinate data without knowledge of the number or the membership of the clusters (Art, Gnanadesikan, and Kettenring; 1982). PROC ACECLUS outputs a data set containing canonical variable scores to be used in the cluster analysis proper.

- DISTANCE
computes various measures of distance, dissimilarity, or similarity between the observations (rows) of a SAS data set. PROC DISTANCE also provides various nonparametric and parametric methods for standardizing variables. Different variables can be standardized with different methods.

- PRINCOMP
performs a principal component analysis and outputs principal component scores.

- STDIZE
standardizes variables by using any of a variety of location and scale measures, including mean and standard deviation, minimum and range, median and absolute deviation from the median, various M-estimators and A-estimators, and some scale estimators designed specifically for cluster analysis.

Massart and Kaufman (1983) is the best elementary introduction to cluster analysis. Other important texts are Anderberg (1973), Sneath and Sokal (1973), Duran and Odell (1974), Hartigan (1975) Titterington, Smith, and Makov (1985), McLachlan and Basford (1988), and Kaufman and Rousseeuw (1990). Hartigan (1975) and Spath (1980) give numerous FORTRAN programs for clustering. Any prospective user of cluster analysis should study the Monte Carlo results of Milligan (1980), Milligan and Cooper (1985), and Cooper and Milligan (1988). Important references on the statistical aspects of clustering include MacQueen (1967), Wolfe (1970), Scott and Symons (1971), Hartigan (1977, 1978, 1981, 1985a), Symons (1981), Everitt (1981) Sarle (1983), Bock (1985), and Thode, Mendell, and Finch (1988). Bayesian methods have important advantages over maximum likelihood; see Binder (1978, 1981), Banfield and Raftery (1993), and Bensmail et al. (1997). For fuzzy clustering, see Bezdek (1981) and Bezdek and Pal (1992). The signal-processing perspective is provided by Gersho and Gray (1992). See Blashfield and Aldenderfer (1978) for a discussion of the fragmented state of the literature on cluster analysis.