# Justifying a linear model

I 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?

• 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. Commented Jul 30 at 7:30
• 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? Commented Jul 30 at 10:00
• 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. Commented Jul 30 at 10:03
• 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). Commented Jul 30 at 10:15
• @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. Commented Jul 30 at 10:18

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 C Hennig, regression might make some sense, but have a look at the residuals plot!