Equivalent to Spearman correlation for non-monotonic data I have several datasets of independent variables that have a monotonic (but non-linear) relationship. If I want to assess if they're correlated, the test of choice is Spearman's (rho) or Kendall's (tau) rank correlation coefficients.
Yet, sometimes I've observed a slight U-shape distribution in scatter plots, in what I suspect to be non-monotonic datasets.
I have a number of questions:


*

*Is there a way to test if my data is monotonic prior to Spearman's rho / Kendall's tau correlation calculations?

*Is it possible to decompose my dataset into monotonic sections, to analyse them separately?

*Is there any equivalent to Spearman's rho test (or Kendall's tau) that accounts for multiple monotonic components?


I'm not sure if the last question makes sense. 
Thanks a lot.
 A: 
  
*
  
*Is there a way to test if my data is monotonic prior to Spearman's rho / Kendall's tau correlation calculations?
  

You could plot the data and look for a non-monotone shape. 
Also, you could fit a generalized additive model (GAM) which estimates nonparametric functions of the predictor variables. This can be done in the mgcv package in R. 
For example:
require(mgcv)
set.seed(123)
n <- 100

x <- runif(n,-5,5)

y <- x^2 + rnorm(n,0,4) 
plot(x,y, col="red")

which produces:

Note that
> cor.test(x, y, method = "kendall")

sample estimates:
        tau 
-0.01454545 

> cor.test(x, y, method = "spearman")

sample estimates:
         rho 
-0.005664566 

So, both Spearman's rho and Kendall's tau are not helpful.
Now, if we run a GAM, we get
> summary(m0 <- gam(y~s(x)))

.
.
.
Approximate significance of smooth terms:
       edf Ref.df     F p-value    
s(x) 8.277  8.861 46.72  <2e-16 ***
.
.
.

With edf>1 there is evidence of non-linearity in the data, which doesn't prove that the association is non-monotonic, but nevertheless suggests that it might be.

Is it possible to decompose my dataset into monotonic sections, to analyse them separately?

Yes ! Sticking with the same dataset, we can do:
x1 <- x[x<0]
y1 <- y[x<0]

x2 <- x[x>=0]
y2 <- y[x>=0]

cor.test(x1, y1, method = "kendall")
cor.test(x1, y1, method = "spearman")

which gives:
sample estimates:
       tau 
-0.5878084 

sample estimates:
       rho 
-0.7905983 

and this handles the first segment of the data, then:
cor.test(x2, y2, method = "kendall")
cor.test(x2, y2, method = "spearman")

which gives:
sample estimates:
      tau 
0.7446809 

sample estimates:
      rho 
0.9155874 

So here we can see a strong negative association in the first segment and a strong positive association in the second.


  
*Is there any equivalent to Spearman's rho test (or Kendall's tau) that accounts for multiple monotonic components?
  

Not that I am aware of.
