I want to study and plot the correlation between two variables, X,Y. Both are measured, so they have comparable noise. Correlation (and not regression) is the correct analysis here.
In a paper, I followed the approach I (and my colleagues) have always followed: calculate the Pearson correlation coefficient and plot the data together with a least squares line. A comment by a reviewer pointed out that this approach is wrong, because the regression line assumes no (or negligible) error on the x axis.
My question: Why is it standard procedure to plot the least squares line (as implemented in e.g. matlab by
lsline or in ggplot by
geom_smooth(method='lm', formula= y~x) in correlation analyses, instead of total least squares?
I understand that the slope of the least squares regression line corresponds to the Pearson correlation coefficient of the standardized data, so the two are related. But they are not the same. (How) Is this nonetheless valid?
I have two motivations to ask this question.
- I'm a bit surprised that neither I nor anybody around me I asked ever realized that we in our research field have been plotting the wrong thing. Which makes me think it can't be so wrong. Also, we've been using the Robust Correlation toolbox code and paper, where they use matlab's
lsline, which again makes me think that the approach can't be wrong.
- A reviewer of our paper argues that total least squares is what we should be using. But estimating confidence intervals for TLS seems complicated, controversial, and (in our case) doesn't work. I'd appreciate any insight that helps me understand the reasons to use OLS as a valid approximation for TLS (if visual only, not for statistical inference) clearly enough to argue for it.