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I have some data measured on a continuous scale. The data have been collected from the same subjects and I want to look for correlation between two dependent variables. The data are not normally distributed.

A cursory look at the raw data suggests correlation, but when I generate a scatter plot I get the following:

enter image description here

That looks pretty random to me or is it?

However when I run a Spearman's correlation I get the following result:

enter image description here

So can someone confirm if there is actually any correlation or not?

Edit-Graph with cross to create four quadrants here...

enter image description here

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  • $\begingroup$ R^2 shown on the 1st pic is r=.645. Quite high correlation. The shape of the cloud is somewhat elliptic. Plus to it, note the two clots of merging points at the opposite poles of the pic. $\endgroup$
    – ttnphns
    Nov 20, 2016 at 15:14
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    $\begingroup$ Why did the line change position when you added the cross? $\endgroup$
    – Yuval Spiegler
    Nov 20, 2016 at 15:45
  • $\begingroup$ Looks at least somewhat correlated to me: if you draw lines parallel to your regression line with one intersecting the y-axis at about 0.5 and another intersecting the x-axis at about 0.2, the vast majority of the data points are between those lines. Of course, it's easier to see that once you've been told where the regression line is. But even without that, there are very few data points towards the top left and bottom right corners. $\endgroup$
    – David Richerby
    Nov 20, 2016 at 19:45
  • $\begingroup$ Not the question but I guess that the central statistical issue here is whether or how well the methods agree, which is not the same as whether or how strongly they are correlated. It seems that the measure concerned varies between 0 and 1, which would have implications for modelling, but doesn't bite hard here. $\endgroup$
    – Nick Cox
    Dec 11, 2019 at 19:38

4 Answers 4

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One simple approach here would be to impose a cross on your graph which divided it into four quadrants, perhaps at the 0.5 position on each axis. Then ask yourself: "Are there more points in the North-East and South-West quadrants of the graph compared to the other two?". If the answer is yes then you have a positive correlation. A more precise version could be obtained by imposing a 3 by 3 grid perhaps at 0.33 and 0.67, and so on.

Incidentally if you are comparing measurements from two laboratories as your labelling suggests then there are better ways of doing this than correlation.

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    $\begingroup$ I have put a cross in the graph and it does indeed look like there is correlation. I am not expert though so i woudl be grateful if you could confirm this looks to be the case. I have addeed this to the original post. $\endgroup$
    – pat18
    Nov 20, 2016 at 14:26
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    $\begingroup$ Certainly looks that way to me. $\endgroup$
    – mdewey
    Nov 20, 2016 at 14:34
  • $\begingroup$ Could you elaborate on the better ways of comparing labs please? I regularly see labs comparing results in this way, so any pointers to better approaches would be invaluable! $\endgroup$
    – Mooks
    Nov 20, 2016 at 15:02
  • $\begingroup$ @Mooks this is really a separate issue but the problem is that correlation does not help if one lab (rater, measurer, ...) rates higher or lower than the other since correlation is unaffected by adding a constant. $\endgroup$
    – mdewey
    Nov 20, 2016 at 15:08
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Scatterplots containing even modest numbers of data points always look like a blur. One approach to uncovering the "signal" summarized in a dependence metric like a Spearman correlation is to create ranked buckets of information based on the feature (or independent) variables. Then, based on those groupings, average both the feature and the target (or dependent) variable. A scatterplot can be formed with the x-axis as the feature buckets and the y-axis as the average of the target variable across those buckets.

Here's an example based on 1,643 observations:

Spearman correlation between X and Y, rho=0.261, p-value=<.0001

Here's a scatterplot of the raw data:

enter image description here

Here's a scatterplot of the same information after grouping X into 20 buckets:

enter image description here

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    $\begingroup$ Another suggestion to combat over-printing is hexagonal binning (at least in R and matlab). $\endgroup$
    – mdewey
    Nov 20, 2016 at 15:20
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I think this is just a visual effect. The points which are not on the regression line are very striking because they lay far apart from each other. However, there are many points close to the regression line that are on top of each other -- appearing as one.

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    $\begingroup$ Yes, thats correct $\endgroup$
    – mRcSchwering
    Nov 20, 2016 at 21:43
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Scatterplots sometimes "hide" points if they have (almost) the same coordinates as others.

Consider using a "heatmap" density plot instead, which may better convey point density.

But also beware: the data shows that strong positive correlation may exist even when humans would consider it to be weak. Here, you simply have several points in the top-right (at 1,1) and bottom-left (0,0) corners, but next to no points in the other corners. By definition, this is a positive correlation. In reality, it may be useless.

If you intend to check if the lab results are correlated, consider splitting the data set into "obvious" cases (where both results are almost exactly 0 or 1) and "difficult" cases. Look at the difficult cases only.

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