5.5 Tied Ranks

\begin {align*} r_s &=\frac {\sum S_i^2-\Big [\dfrac {n^3-n}{6}\Big ]\Big [\dfrac {7n+5}{n-1}\Big ]-\sum t_X-\sum t_Y}{\Bigg \{\Big [\dfrac {n^3-n}{6}-2\sum t_X\Big ]\Big [\dfrac {n^3-n}{6}-2\sum t_Y\Big ]\Bigg \}^{\dfrac {1}{2}}}\\ \end {align*}

where the corrections are computed separately for each variable, \[\sum t_X=\dfrac {\sum _{p}\left (t_p^{3}-t_p\right )}{12} \hspace {0.6cm}\text {over the tied groups in } X, \hspace {0.6cm} \sum t_Y=\dfrac {\sum _{q}\left (u_q^{3}-u_q\right )}{12} \hspace {0.6cm}\text {over the tied groups in } Y,\] with \(t_p\) the size of the \(p^{th}\) group of tied \(X\) values and \(u_q\) the size of the \(q^{th}\) group of tied \(Y\) values. A group of size \(1\) contributes nothing, since \(1^{3}-1=0\), so untied observations may be ignored.

\(r_s\) computed with and without the tie correction differs noticeably only when ties are numerous.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.