This is a statistical method that merges attributes into clusters based on the residual error within the differences of the instance attributes from those of an other instance or group. This method tries to minimize the variance of the differences in attributes within a cluster using the distance algorithm based on the sum of squares of the difference of the attributes. It joins cluster pairs whose merger minimizes the increase in the total sum of squares within-group error.

We have not found a description on how SAS handles nominal variables in this method. It is our assumption that each attribute value is one distance away from the next in ascending order as specified in the data.

A more scientific description of this method is shown below, taken from the SAS documentation:

The following method is obtained by specifying METHOD=WARD. The distance between two clusters is defined by:

If d(x,y) = (1/2) | x - y |², then the combinatorial formula is:

D_JM = [((N_J + N_K) D_JK + (N_J + N_L) D_JL - N_JD_KL) / (N_J + N_M)]

In Ward's minimum-variance method, the distance between two clusters is the ANOVA sum of squares between the two clusters added up over all the variables. At each generation, the within-cluster sum of squares is minimized over all partitions obtainable by merging two clusters from the previous generation. The sums of squares are easier to interpret when they are divided by the total sum of squares to give proportions of variance (squared semipartial correlations).

Ward's method joins clusters to maximize the likelihood at each level of the hierarchy under the following assumptions:

• multivariate normal mixture
• equal spherical covariance matrices
• equal sampling probabilities

Ward (1963) describes a class of hierarchical clustering methods including the minimum variance method.