Spearman's Rank-Order Correlation Coefficient is a nonparametric
measure of statistical dependence between two variables. It assesses
how well the relationship between two variables can be described using
a monotonic function.
So says Wikipedia.
My question: is it also possible extend this to higher dimensions?
Say I have scattered data of the form $(x_i,y_i,z_i)$, is it possible to compute Spearman's Rank-Order Correlation Coefficient to shed light on the question whether the relation between $x$ and $z$ can be described by a monotonic function?
I highlighted the word scattered, since I cannot simply fix $y$ and look upon it as data of the form $(x_i,z_i)$, because I only have a limited number of data, which is scattered.
Best Answer
Yes, you could in principle extend the idea of a rank correlation to higher dimensions as long as you have a way of ordering the points. For instance, consider two vectors $x_i = (x_{i1}, x_{i2}, \ldots , x_{ip})$ and $x_j = (x_{j1}, x_{j2}, \ldots, x_{jp})$. We could start by comparing the first two coordinates and say that $x_i < x_j$ if $x_{i1} < x_{j1}$ (or vice versa), and if $x_{i1} = x_{j1}$ then go to the next coordinate and repeat.
Now suppose we have a data set $\{ (x_1, y_1), (x_2, y_2), \ldots , (x_n, y_n) \}$ where each element is a pair of vectors. It's straightforward to apply an ordering such as the one above to the set of $x$ and $y$ vectors separately and convert each point to a pair of ranks and then calculate Spearman's rank correlation.
This shows that it's possible, but it's not clear when you would actually want to do this. In practice it would depend on whether or not some ordering makes sense and is interesting given the problem.