  Rand (1971) proposed what has since become a well-known index for comparing
two partitions obtained on the same set of units. The index takes a value on
the interval between 0 and 1, where a higher value indicates more similar
partitions. Sometimes, e.g. when the units are observed in two time periods,
the splitting and merging of clusters should be considered differently,
according to the operationalization of the stability of clusters. The Rand
Index is symmetric in the sense that both the splitting and merging of clusters
lower the value of the index. In such a non-symmetric case, one of the Wallace
indexes (Wallace, 1983) can be used. Further, there are several cases when one
wants to compare two partitions obtained on different sets of units, where the
intersection of these sets of units is a non-empty set of units. In this
instance, the new units and units which leave the clusters from the first
partition can be considered as a factor lowering the value of the index.
Therefore, a modified Rand index is presented. Because the splitting and
merging of clusters have to be considered differently in some situations, an
asymmetric modified Wallace Index is also proposed. For all presented indices,
the correction for chance is described, which allows different values of a
selected index to be compared.
