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I am trying to evaluate whether there is any bivariate correlation between several non-normally distributed variables.

Is it appropriate, or not, to claim positive and negative correlations in these cases based on a Spearman rank-order test statistic?

These variables have skewed distributions and some outliers and I was hoping that a Spearman rank correlation would fit the bill; however, I think my data are failing the monotonic assumption.

I am a rank amateur, so to speak, and would appreciate some informed perspectives on these scatter plots.

Edit: It is possible, and sometimes likely, for values to be either 0% or 100%, as indicated in the scatter plots.

Each data point represents a case (inquiry), in which different types of sources are consulted to a lesser or greater degree. All sources may be of type a, in which case 100% of sources consulted are type a, etc.

Sources can be simultaneously of type a & b, or a & c, but not c & b. Alternatively, they may exclusively type a, or exclusively type b, or exclusively c. The question is whether there are evident relationships in consultation choices.

2 scatter plots

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Spearman correlation is to be thought of as measuring monotonicity and such correlations will achieve absolute value of 1 if and only if relationships are perfectly monotonic. There is no more an assumption of monotonicity than there is an assumption in grading an examination that everyone will achieve 100%. Rather, (perfect) monotonicity is a reference standard. (Similarly Pearson correlation is not based on an assumption of linearity, but is designed to measure how well a linear relationship would summarize bivariate data.)

You seem also to be conflating the problem of testing correlation, usually in the form of establishing whether a correlation is definitely (significantly) not zero, with the problem of measuring it. Although many texts and courses give space to significance tests, it is arguable that they are of little real use scientifically or practically, as usually it is known that there should be some relationship between variables, the only real doubt being over how strong it is precisely and what form it takes.

In your case, the variables are anonymous but evidently measured on a percent scale. We would need more information to give more detailed advice. What can be done with percent variables depends on detail on whether exact zeros and exact 100% values are possible.

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  • $\begingroup$ Thanks for your response. I have edited my question to try to clarify. My confusion about meeting assumptions comes from trying to teach myself stats from online sources like: bit.ly/1NIncKc. My impression was that if too many of my data points do not increase/decrease in tandem (along x&y) that this indicates a Spearman rank correlation test would be an inappropriate way to establish whether a statistically notable relationship exists or not. $\endgroup$ – Nominally Innumerate Jul 6 '15 at 12:08
  • $\begingroup$ Not so; that's like saying that you need to know how to drive a car before you can take a driving test. $\endgroup$ – Nick Cox Jul 6 '15 at 12:42
  • $\begingroup$ More importantly, it's not clear that correlation is going to answer your questions. it's mushing together A, B, C, AB, AC, when you may want to keep those separate. $\endgroup$ – Nick Cox Jul 6 '15 at 12:46
  • $\begingroup$ I thought one does need to know how to drive before taking a driving test. But I guess I'm missing your point. $\endgroup$ – Nominally Innumerate Jul 6 '15 at 15:35
  • $\begingroup$ OK, we need a better analogy. The point is that monotonicity is not a prerequisite; it is what Spearman correlation is assessing. This isn't always explained especially well; Harold Jeffreys, Theory of probability is especially lucid on the topic. $\endgroup$ – Nick Cox Jul 6 '15 at 15:51

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