Spearman

1

• Spearmans correlation coefficient , rs, can be
  computed as Pearsons r on the ranks i.e., rank
  the Xs (among the Xs) and the Ys (among the
  Ys) and then compute the correlation of the
  ranks
• See Table 5.2.1 and lets do it in R
• We may test the null hypothesis of no association
  between X and Y by doing a permutation test on
  the ranks all possible assignments of the ranks
  of the Ys to the ranks of the Xs if our
  correspondence yields an unusually high (or low)
  value of rs, then we should reject the hypothesis
  of no association between X and Y.
• We may also test the above hypothesis with the
  same normal approximation used for Pearsons r
  Z rs(sqrt(n-1)) i.e. rs is approx.
  N(0,1/(sqrt(n-1))
• What about ties?? There are two methods mentioned
  on p.155ff
• compute adjusted ranks (midranks) and apply the
  same formulas weve just mentioned
• use the tie-adjusted formulae given on page 156
• the author (and I too!) recommend the former.

2

• Another measure of association is Kendalls Tau,
  t, which looks at the distribution of concordant
  and discordant pairs of the (X,Y)s
• (Xi,Yi) and (Xj,Yj) are concordant if Xi lt Xj
  implies Yi lt Yj and discordant if Xi lt Xj
  implies Yi gt Yj (or equivalently, concordant if
  (Xi Xj)( Yi - Yj ) gt 0 discordant if (Xi
  Xj)( Yi - Yj ) lt 0). X and Y are positively
  associated if pairs are more likely to be
  concordant than discordant and negatively
  associated if pairs are more likely to be
  discordant than concordant.
• Note that tau is just rescaled to be between -1
  and 1 if there is no association, then the
  probability of a concordant pair is the same as
  the probability of a discordant pair, .5, so t
  0.
• We estimate tau by counting the fraction of
  concordant pairs in the data, doubling it and
  subtracting 1

3

• Here,
• Ranks may also be used to compute tau, since
  pairs of ranks are concordant or discordant
  according to whether the original pairs are
  concordant or discordant.
• R computes Kendalls tau in cor.test and SAS
  computes it in PROC CORR
• Exact p-values for testing the hypothesis of no
  association between X and Y may be obtained by a
  permutation test approximate p-values may be
  obtained from the large sample properties of
  Kendalls tau statistic
• HW Read Chapter 5 through page 163 we will
  complete this topic (association between two
  continuous variables) next time have your
  questions ready by then. Do problems 3 and 4 on
  page 189