= Unordered-pair partition for ordinal data
{title2=$\binom n2=C+D+T_X+T_Y+T_{XY}$}
= Pair-count partition
{synonym}
For $n$ observations of two ordered variables, every unordered pair is concordant, discordant, tied only on the first variable, tied only on the second variable, or tied on both. Consequently
$$
\binom n2=C+D+T_X+T_Y+T_{XY}.
$$
The proof is an exhaustive partition: if neither coordinate is tied the two nonzero comparison signs agree or disagree; otherwise exactly one or both coordinates are tied. For contingency-table counts $n_{ij}$ with row totals $n_{i+}$ and column totals $n_{+j}$,
$$
T_{XY}=\sum_{ij}\binom{n_{ij}}2,\quad T_X=\sum_i\binom{n_{i+}}2-T_{XY},\quad T_Y=\sum_j\binom{n_{+j}}2-T_{XY}.
$$
Two records tied on both variables belong to each marginal tie total, so subtracting them once from each produces the exclusive $T_X,T_Y$. Using incompatible exclusive and inclusive conventions corrupts <Somers' D> and <Kendall's tau-b> even if the untied concordances are correct. The identities also provide a direct arithmetic check on a printed statistical summary.
Back to article page
