Spearman's Rank Correlation Coefficient

#1
Since Spearman’s rho is a measure of general dependence between two variables, does this mean a correlation of zero means zero relationship between variables?
 

Dason

Ambassador to the humans
#2
Spearman's tests for a monotone relationship. So we could have a perfect relationship between two variables and still have spearmans give us a correlation of 0.

Here's some R code to illustrate an example (even if you don't know R hopefully this should be relatively simple to follow along with):
Code:
> x <- seq(-10, 10)
> y <- x^2
> cbind(x, y)
        x   y
 [1,] -10 100
 [2,]  -9  81
 [3,]  -8  64
 [4,]  -7  49
 [5,]  -6  36
 [6,]  -5  25
 [7,]  -4  16
 [8,]  -3   9
 [9,]  -2   4
[10,]  -1   1
[11,]   0   0
[12,]   1   1
[13,]   2   4
[14,]   3   9
[15,]   4  16
[16,]   5  25
[17,]   6  36
[18,]   7  49
[19,]   8  64
[20,]   9  81
[21,]  10 100
> cor(x, y, method = "spearman")
[1] 0
 
#3
How could this be true, because the reason a correlation of zero does not mean zero dependence between variables for a Pearson product moment correlation coefficient is that Pearson correlation coefficient measures linear relationship. Now Spearman’s rho measures general dependence at ordinal scale. Why can’t it be zero dependence when we have zero correlation?
 

Dason

Ambassador to the humans
#4
"0 dependence" might imply a correlation of 0. But all I was showing was that just because you have a correlation of 0 doesn't mean that you don't have any dependence at all. I think my example illustrates pretty clearly that you can have dependence between two variables and still get a spearman correlation of 0.

In the example I gave you would get pearson's correlation giving you 0 because the relationship isn't linear (and because of the x values I chose to make it symmetric). You get a spearman's correlation of 0 because the relationship isn't monotone - which is what spearman's is looking for.
 
#5
I guess the phrase “general dependence” is misleading because Spearman’s rho is looking for something specific and that is monotone relationship between two variables. I presume this leads us to the conclusion that a Spearman’s rank correlation coefficient of zero does not mean zero relationship but zero monotone relationship.