cspr() is Spearman's rho: the Pearson correlation of
the ranks, so it measures monotone association rather than linear
association and is unaffected by any order-preserving transformation of
either variable. cmidrk() exposes the ranks themselves;
tied values share the average of the ranks they span, which is what
makes the result agree with stats::cor() on
tied data. Use Spearman's rho when two variables rise and fall together
but not along a straight line (counts against rates, incomes against
ranks), or when a few extreme values would dominate a Pearson
correlation.
Value
cspr(): a length-1 numeric in \([-1, 1]\)
(NaN as described above). cmidrk(): a numeric vector the
length of x, the rank of each value (1 for the smallest; tied values
share the mean of their ranks; NA kept as NA).
Details
With \(r_i\) and \(s_i\) the midranks of \(x_i\) and \(y_i\), $$\rho = \frac{\sum_i (r_i - \bar r)(s_i - \bar s)} {\sqrt{\sum_i (r_i - \bar r)^2 \sum_i (s_i - \bar s)^2}},$$ the Pearson correlation of the ranks (Spearman, 1904). Because ties get midranks, the shortcut \(1 - 6 \sum d_i^2 / (n(n^2 - 1))\) is NOT what is computed; the full formula above is, and it stays correct with ties.
cspr() does not remove missing values: an NA in either
vector gives NaN, and so does a constant vector (its ranks have no
variance). cmidrk() leaves an NA in place and ranks the rest,
as rank(x, na.last = "keep") does. Refusals: vectors of different
lengths ("must have the same length") and a character, factor or list
argument ("must be numeric"). Ranking sorts, so the cost is
\(O(n \log n)\).
References
Spearman C (1904). The proof and measurement of association between two things. The American Journal of Psychology 15(1), 72–101. doi:10.2307/1412159
See also
ccorr() for the Pearson correlation.
Examples
x <- c(1, 2, 3, 4, 5)
y <- c(2, 4, 9, 16, 25)
# Perfectly monotone but not linear: rho is 1 where Pearson is not.
cspr(x, y)
#> [1] 1
ccorr(x, y)
#> [1] 0.9737247
# Agrees with stats::cor(), ties included.
xt <- c(1, 2, 2, 2, 5, 5, 7)
yt <- c(3, 1, 1, 4, 4, 9, 2)
all.equal(cspr(xt, yt), stats::cor(xt, yt, method = "spearman"))
#> [1] TRUE
# Tied values share the average of the ranks they cover.
cmidrk(xt)
#> [1] 1.0 3.0 3.0 3.0 5.5 5.5 7.0
all.equal(cmidrk(xt), rank(xt))
#> [1] TRUE
# Invariant to any monotone rescaling.
all.equal(cspr(x, y), cspr(exp(x), log(y)))
#> [1] TRUE
# Missing values are kept in place by cmidrk().
cmidrk(c(3, NA, 1, 3))
#> [1] 2.5 NA 1.0 2.5
try(cspr(1:3, 1:4))
#> Error : cspr: `x` and `y` must have the same length