[R] Problem when creating matrix of values based on covariance matrix

[R. Michael Weylandt]

On Sun, Aug 12, 2012 at 7:52 PM, R. Michael Weylandt
<michael.weylandt at gmail.com> wrote:
> I am not sure if this is a general/fixed bias in the spearman
> estimator or if it's just a function of the covMat I randomly chose.
> Prof. Dalgaard and many others on this list must know.

To somewhat answer myself, I restricted my attention to 2x2 random
normals and played around with variations of this script, using
standard normals for the marginals:

######################

library(MASS)

x <- seq(-1, 1, length.out = 1000)
bias <- numeric(length(x))
n <- 100

pb <- txtProgressBar(0, length(x))

for(i in seq_along(x)) {
  rands <- mvrnorm(n, c(0,0), matrix(c(1, x[i],x[i],1),2), empirical = TRUE)
  bias[i] <- cor(rands, method = "s")[2] - cor(rands, method = "p")[2]
  # Change "s" -> "k" for kendall's tau below
  if(!(i %% 50)) setTxtProgressBar(pb, i)
}

plot(x, bias, type = "l", xlab = "True Correlation", ylab = "Est.
Bias: Spearman - Pearson")

#######################

# For n <- 100

> summary(bias)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
-0.1112000 -0.0186000 -0.0007219 -0.0007062  0.0172600  0.1177000

> summary(abs(bias))
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
0.000000 0.008928 0.017900 0.024240 0.035650 0.117700

# For n <- 10000

> summary(bias)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
-2.642e-02 -1.272e-02 -5.491e-04 -8.178e-05  1.277e-02  2.372e-02

> summary(abs(bias))
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
0.000000 0.006904 0.012750 0.011840 0.016490 0.026420

where the general trend is sinusoidal with zeros at x = -1, 0, 1. The
sinusoid is more definite for large n, but has less amplitude.

I then moved to kendall's tau[0] which gives a very different shape
(sort of like nike swoosh, but also an odd function). Notably the
estimated difference is of a different order of magnitude here, even
for a  large sample:

# n <- 50

> summary(bias)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
-0.2966000 -0.1490000  0.0047720  0.0004882  0.1479000  0.3003000

> summary(abs(bias))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.00000 0.08273 0.14850 0.13690 0.19210 0.30030

# n <- 5000

> summary(bias)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
-2.131e-01 -1.525e-01  0.000e+00  8.379e-05  1.531e-01  2.137e-01
> summary(abs(bias))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.00000 0.08542 0.15290 0.13640 0.19550 0.21370

which seems to put me at odds with Prof Therneau's pronouncement [1]
that spearman and kendall are basically the same for large samples
(though I'm not sure if n in their discussion is observation dimension
or sample size)

Hope this is of interest to someone,

Michael

[1] http://tolstoy.newcastle.edu.au/R/e17/devel/12/06/1434.html

[0] Much slower so I'll give a dput here:

n <- 5000
dput(bias)

c(0, 0.0379451874358857, 0.052342945385874, 0.0640785709093772,
0.072964986590912, 0.0792379195759072, 0.0858528409585821, 0.0934189925873063,
0.0987845440960066, 0.103788103206227, 0.106233070598104, 0.112044791340651,
0.115695599900761, 0.120068472873754, 0.123673832344046, 0.128009737923561,
0.127985731520679, 0.13220481373552, 0.135410253221816, 0.138467183006171,
0.13961445085814, 0.14122325101657, 0.142922789322629, 0.146233529869137,
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