[Rd] cor.test(method = spearman, exact = TRUE) not exact (PR#14095)

[Petr Savicky]

On Mon, Nov 30, 2009 at 04:00:12AM +0100, dsimcha at gmail.com wrote:
> > a <- c(1:10)
> > b <- c(1:10)
> > cor.test(a, b, method = "spearman", alternative = "greater", exact = TRUE)
> 
>         Spearman's rank correlation rho
> 
> data:  a and b 
> S = 0, p-value < 2.2e-16
> alternative hypothesis: true rho is greater than 0 
> sample estimates:
> rho 
>   1 
> 
> > 1 / factorial(10)
> [1] 2.755732e-07
> 
> Since we have perfect rank correlation and only one permutation out of 10! could
> give this for N = 10, the p-value should be 1/10!.  Reading the code in prho.c,
> it appears that the "exact" calculation uses the Edgeworth approximation for N >
> 9.  This makes sense because, for similar examples with N <= 9, the results are
> as expected (1 / N!).

The Edgeworth approximation is not exact, although the error is quite small.
Computing the exact values has time complexity roughly 2^n, if Ryser's formula
for permanent is used. In cases, where one needs the exact values for small n,
it is possible to use the package pspearman at CRAN, which includes precomuted
values up to n=22. See also the paper
  M.A. van de Wiel and A. Di Bucchianico,
  Fast computation of the exact null distribution of Spearman's rho and Page's L statistic
  for samples with and without ties, J. Stat. Plann. Inf. 92 (2001), pp. 133-145.
which deals with this question.

  library(pspearman)
  a <- c(1:10)
  b <- c(1:10)
  out <- spearman.test(a, b, alternative = "greater", approximation="exact")
  out
  #         Spearman's rank correlation rho
  # data:  a and b 
  # S = 0, p-value = 2.756e-07
  # alternative hypothesis: true rho is greater than 0 
  # sample estimates:
  # rho 
  #   1 
  out$p.value - 1/factorial(10) # [1] 0

PS.