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I'm looking for a correlation between 2 parameters, neither of which is normally distributed. Hence from my (very limited) understanding of statistics, I should be running a Spearman correlation or log-transforming before running Pearson's. Both of these tests are not significant. However, the Pearson's correlation on untransformed data is strongly significant and the scatterplot looks like a trend does exist:

Pearson's correlation

After some googling I know that the scatterplot resembles this scenario of significant outliers: enter image description here so the Pearson's correlation might not be valid, but I am confident that the stray points are not random noise because using alternative surrogate parameters for the y-axis (fib4) that are measured in completely different ways gives the exact same result. It looks like something is going on but I don't even know how to articulate it- like there is a positive linear? correlation that only holds at the positive extreme values. I would really appreciate it if someone can set me on the right path. Thanks kindly in advance.

The opposite situation (significant Spearman, non-significant Pearson's) has been asked before and the answer was that Spearman is more robust to scattering by outliers as it only uses rank vs actual values, which makes sense. I also know that using values makes Pearson's more powerful which could explain the situation, but it doesn't seem a valid test anyway.

Addit: log-transformed Pearson's correlation is also significant, but isn't robust to removal of just 1 point. For anyone with the same question I believe the conclusion is that these inconsistencies reflect a weak (at best) relationship.

scatterplots with axes log-transformed

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    $\begingroup$ To understand why the correlations differ, you could calculate the ranks and then plot those, remembering that Spearman correlation is just Pearson correlation on the ranks. $\endgroup$
    – Nick Cox
    Feb 14 at 13:41
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    $\begingroup$ If I had access to these data I would plot APRI score against log of the predictor (which I think is strongly indicated as likely to be helpful) -- and I would also try log APRI score (although I doubt that would be as useful as the first). I suspect many other statistically minded people would be happy if the first plot produced a simple linear pattern and would then consider Pearson correlation appropriate. @Frank Harrell has excellent reasons for his approach, which is nevertheless likely to be harder to explain to clinicians not strong specialists in statistical methods. $\endgroup$
    – Nick Cox
    Feb 15 at 12:46
  • $\begingroup$ Thank you for your advice Nick and I'm very sorry for the late reply. I am confused by the scatterplots with log transformation and would really appreciate your advice- imgur.com/a/nrXYG8c $\endgroup$
    – houston
    Mar 4 at 9:32
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    $\begingroup$ I would post the graphs here. You have weak or insubstantial relationships period, and necessarily I have no idea how far that is surprising clinically or scientifically. Omitting outliers because they are awkward I regard as poor practice statistically, $\endgroup$
    – Nick Cox
    Mar 4 at 9:40
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    $\begingroup$ You're pushing me into regions where I can only guess wildly. There is no oracle here to tell you the right thing to do. On this evidence I would work with log of APRI score or use a generalized linear model with log link if I were told to analyse the data independently. Your present thinking needs a new question with some evidence on what happens. From a little Googling on APRI score it may be that much hinges on whether (a) you have or (b) you want to include data on really sick patients, but that's utterly beyond my expertise. $\endgroup$
    – Nick Cox
    Mar 4 at 10:35

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To stir the pot a little I suggest that it primarily means that one too many correlation coefficients was estimated. It is better to choose a measure based on statistical principles and stick with it. Unless one has prior evidence strongly suggesting linearity and some confidence that extreme values that would distort the result have a very small chance of being sampled, the default position would be to use Spearman's $\rho$. It is resistant to extreme values and is efficient under non-linearity as long as the relationship is monotonic (doesn't go up then back down or down then back up). $\rho$ quantifies the degree to which Y goes up (or down) as X goes up. To top it off were normality to actually hold, $\rho$ is $\frac{3}{\pi}$ as efficient as Pearson's $r$. A loss of 0.05 efficiency under ideal conditions for $r$ is a small price to pay for $\rho$ having a much higher efficiency than $r$ under non-normality in many cases.

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  • $\begingroup$ Thank you very much for the explanation. Can you please kindly stir the pot on a few follow-up questions: $\endgroup$
    – houston
    Feb 13 at 5:24
  • $\begingroup$ 1. Does simply stating no correlation throw anything away or am I deluding myself? I ask because from pathophysiology I expect a monotonic (though not necessarily linear) positive relationship between the parameters, though other influences no doubt dominate at low values. I am confident the datapoints in the right upper quadrant are real. 2. Do you know why log-transformation -> Pearson's correlation seems to be the default in medical studies rather than Spearman's? Is it simply because the data needs to be log-transformed for multiple linear regression anyway? $\endgroup$
    – houston
    Feb 13 at 5:25
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    $\begingroup$ Spearman's is used frequently in medicine. For your goal it is the appropriate one and the other should not be used. You seem to be hung up on the meaningless concept of "statistical significance". Quote the p-value for $\rho$ and perhaps provide a confidence interval for $\rho$ and don't use dichotomous thinking about significance. And note that on the untransformed scale you have high leverage points that the make results misleading. $\endgroup$ Feb 13 at 13:15
  • $\begingroup$ Thanks again for wrangling with my understanding, I really appreciate it. This is almost a separate question but the Spearman's correlation coefficient ρ is only 0.08 (p-value 0.4) which is basically no correlation, while Pearson's r is 0.43. I now know this is because of the influential high leverage points and it's a false linear correlation. My question is whether some other type of relationship describes this data? Words like a threshold effect come to mind. I just feel that even if I had never performed the Pearson's correlation I would look at that scatterplot and see something! $\endgroup$
    – houston
    Feb 14 at 13:02
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    $\begingroup$ Why not stop with the degree of monotonic association as quantified by $\rho$? You may also want to use a spline function to fit a nonlinear relationship. $\endgroup$ Feb 14 at 14:15

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