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scatterplotI am interested in the association between two variables, cognitive distortion and overall suffering (on scales from 0 to 40 and 0 to 10 respectively). A Spearman rank test suggests positive correlation and hence to study the strength of the association I would have intuitively reached for linear regression to quantify how much suffering varies with each added unit score of cognitive distortion.

I wonder whether based on a dataset like the one attached, linear regression is a reasonable model. Independence, normal distribution and equality of variance assumptions are met (based on Breusch-Pagan testing). Or would other models be more appropriate?

A related but slightly separate question: is normality testing with Kolmogorov-Smirnov or Shapiro tests definitely needed? I noticed they suggest non-normality on data which "appear" normal on inspection.

I would characterize the model as explanatory rather than predictive. If I would consider a linear model appropriate on inspection of the scatter plot how can I formally justify this?

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    $\begingroup$ As an aside: both scales are discrete. Are any of your points actually multiple overplotted points? If so, you could add a little jitter for display purposes, or look into sunflowerplots. $\endgroup$
    – Stephan Kolassa
    Commented Jul 30 at 7:30
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    $\begingroup$ You say the minimum of the suffering score is 0, but the plot shows values smaller than 0. Also larger than 10 (supposedly the maximum). How come? $\endgroup$
    – Christian Hennig
    Commented Jul 30 at 10:00
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    $\begingroup$ Note that discrete data with limited value range can never be normally distributed, but then nothing is ever normally distributed ("all models are wrong but some are useful"), and we apply linear regression in many situations anyway. See stats.stackexchange.com/questions/538561/… and stats.stackexchange.com/questions/538561/… for some guidance regarding how to deal with model assumptions. $\endgroup$
    – Christian Hennig
    Commented Jul 30 at 10:03
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    $\begingroup$ In addition to the good comments above, note that correlation (Spearman or Peasron or other) is a measure of the strength of the relationship. Also, no form of regression that I know of requires normal data, some require normal errors). $\endgroup$
    – Peter Flom
    Commented Jul 30 at 10:15
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    $\begingroup$ @PeterFlom: OP has changed the plot, including jittering. The original plot was quite evidently discrete, and based on the change, did have some overplotting. So the change absolutely improved matters. $\endgroup$
    – Stephan Kolassa
    Commented Jul 30 at 10:18

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I think a linear regression addresses the question "quantify how much suffering varies with each added unit score of cognitive distortion" straight away, so I'd probably run it. Transformation may give you a slightly better model, but my gut feeling is that this may not be worth the hassle. Note in particular that the more data dependent decisions you make, the more you invalidate the inference on the final model. This by the way also is the case if you choose a model conditionally on misspecification testing, although this may be seen as lesser evil compared to running a strongly misspecified model, see https://jdssv.org/index.php/jdssv/%20issue/view/15 for detailed discussion.

Note that discrete data with limited value range can never be normally distributed, but then nothing is ever normally distributed ("all models are wrong but some are useful"), and we apply linear regression in many situations anyway. See Relevance of assumption of normality, ways to check and reading recommendations for non-statisticians for some guidance regarding how to deal with model assumptions (maybe also consider Is normality testing 'essentially useless'? if you like to be confused). It is generally the wrong question whether model assumptions are fulfilled or not, the correct question is, are they violated in such a way that results will be misleading, which is unfortunately a rather subtle and hard question, and often not addressed well by standard misspecification tests.

So by and large I think exploratory use is fine and in fact cross-validation may even tell you something about predictive strength, but results should be interpreted with a pinch of salt due to considerable model uncertainty (on top of the uncertainty within the assumed model that you should quantify by any means).

There is no formal justification for this, and this is actually a good thing, because it will hopefully stop you and your readers from reading too much into the results.

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I added a straight line on your plot

Original plot by asker with straight line added

from $(0,0)$ to $(40, 10)$. Note that the plot shows some (nonperfect) "triangular correlation" (not an official term ...) with more than two-thirds of the points above the line, and the points below somewhat closer ... it is like, to some degree , the suffering score is lower-bounded by total_distortion. This is not perfect, but I would ask myself what is the meaning of this.

As in the answer by @Christian Hennig, regression might make some sense, but have a look at the residuals plot! Looking at the plot, a quadratic regression maybe could make sense.

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    $\begingroup$ Quantile regression could make sense if the OP wanted to pursue this scenario. $\endgroup$
    – Ben Bolker
    Commented Aug 28 at 23:08
  • $\begingroup$ I cannot fathom what you refer to by "triangular correlation." In your example it appears to be something like a bad fit, a nonlinear relationship, a quantile regression, or perhaps something else altogether. Could you describe it or explain it further? $\endgroup$
    – whuber ♦
    Commented Sep 10 at 16:47
  • $\begingroup$ @whuber: In this plot it is quite approximate, but a majority of the points is in the upper triangle above the red line $\endgroup$
    – kjetil b halvorsen ♦
    Commented Sep 10 at 17:14
  • $\begingroup$ That merely characterizes a bad fit! How exactly is the red line determined? You have only told us you added it without explaining what procedure you used. $\endgroup$
    – whuber ♦
    Commented Sep 10 at 17:37

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