Your data show a relationship between the two variables. However, it is apparently not a linear relationship for the entire region of X. It looks that Y has a transition around X=17. I would fit the data with a linear regression for the range of X=0 to 17, and fit the rest of the data with a constant , say 18, for the range of X >17. I would then calculate the residuals and do residual analysis for the fitting.

Try to use ranks instead of actuals: https://en.wikipedia.org/wiki/Rank_correlation

Additional comments:

In one of my old studies, I dealt with the data that had the similar pattern as your data. The data were fitted well with two asymptotic solutions based on the dimensional analysis. It was expected that the data were scattering in a transition region. I also developed an empirical equation that fits the entire range of the data.

My objective is just to see if there is a relationship (association) between the two variables and to quantify it, not to fit a model for predictions (I edited my post for more clarity).

You then need the Spearman's rho coef.

Use the cor.test() function in R.

Your data tells a relationship between the two variables for the range of X=0 to 17. However, there is no correlation between them for the range of X >17.

Hi,

-- or you try to get rid off the extreme values of y - with the corresponding loss of information, and perform a simple linear regression on what is left;

-- or, if the extreme values (2 and 20) are known a priori, you may convert the y to values between 0 and 1 and perform a logistic regression;

-- or the extreme values may/shall be adjusted to the data, then fit a 4 parameters logistic (cf stats::SSasymp, or various functions of the drc pack.)

Best regards.

Pauline -

It looks like you have a nice ratio relationship (one regressor with a straight line through the origin) with expected heteroscedasticity, which is more useful than a correlation coefficient, but you have neither with the restrictions you have on y. In regression you could say it fits between certain values of x, but since your restriction is on y, and y is a random variable, it could come up anywhere.

Correlation does not interest me so much. It is of far less practical importance. But perhaps you might look at it as Hening suggested for this particular data set. ??? Maybe?

Regarding the regression line shown on the scatterplot, you should ignore that. It is artificial.

Cheers - Jim

Via ranks.

send some data

as W. E. Deming (1900-1993) said "in God we trust, the others bring the data"

Suggestion:

If you know the y-values that you restricted to less than 20 or more than 2, then place those values, for now, back into your working data.  If you don't know those values, then remove any corresponding x-values associated with the restricted y-values, as well as those artificial/restricted y-values.  Next take your data and find a regression, likely, it appears, regression through the origin.  (If there were no restrictions, would you expect y to be zero, if x is zero?  If so, do not include an intercept term.)  There appears to be substantial heteroscedasticity.  (Never fear, heteroscedasticity in regression is a feature, not a bug.)  You can use the first example in https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

as a guide, and the following tool:

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx.

At this point you will have a regression, likely of the form

y = bx + (e_0)(x^gamma).

You can use that to "predicted" (estimate) y from x.  Then and only then, you can apply the restrictions you have on y. 

Not knowing why you have such restrictions, there may be some reason not to do this, but it occurs to me that you have a natural relationship between x and y, one that you can graph, and model, and for which you have a correlation, but then you apply these artificial restrictions - well, artificial as far as we know.  Your specific application should be considered to see if this solution is appropriate. 

Review:  Establish the natural relationship between x and y.  Then you can predict y.  If it is above 20 or below 2, "restrict" it as required. 

(Whatever your application, you need to consider the practical ramifications of such restrictions.) 

Be safe. Cheers!