Suppose we want to compare two classifications summarized by the
contingency table \(T=[t_{ij}]\) where \(i,j=1,\ldots,K\) and \(t_{ij}\)
denotes the number of data points which are in class \(i\) in the
first partition and in class \(j\) in the second partition. If both
classifications use the same labels, then obviously the two
classification agree completely if only elements in the main diagonal
of the table are non-zero. On the other hand, large off-diagonal
elements correspond to smaller agreement between the two
classifications. If match.names is TRUE, the class labels
as given by the row and column names are matched, i.e. only columns and
rows with the same dimnames are used for the computation.
If the two classification do not use the same set of labels, or if
identical labels can have different meaning (e.g., two outcomes of
cluster analysis on the same data set), then the situation is a little
bit more complicated. Let \(A\) denote the number of all pairs of data
points which are either put into the same cluster by both partitions or
put into different clusters by both partitions. Conversely, let \(D\)
denote the number of all pairs of data points that are put into one
cluster in one partition, but into different clusters by the other
partition. Hence, the partitions disagree for all pairs \(D\) and
agree for all pairs \(A\). We can measure the agreement by the Rand
index \(A/(A+D)\) which is invariant with respect to permutations of
the columns or rows of \(T\).
Both indices have to be corrected for agreement by chance if the sizes
of the classes are not uniform.