# Use of ordinary least squares line in correlation analyses

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.

1. 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.
2. 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.
• You might enjoy reading relevant chapters from Tukey's book EDA showing how robust regressions of $X$ on $Y$ and $Y$ on $X$ reveal a great deal about co-variation among those variables. You will acquire useful (but not very well known) exploratory tools for any kind of correlation analysis.
– whuber
Oct 27, 2020 at 15:57

First, if you can data center (subtract the mean) from your explanatory variable(s), then upon further dividing the centered data by their standard deviation(s), a regression-based on the standardized data produces so-called beta regression coefficient(s) or beta weights. To quote Wikipedia:

In statistics, standardized (regression) coefficients, also called beta coefficients or beta weights, are the estimates resulting from a regression analysis where the underlying data have been standardized so that the variances of dependent and independent variables are equal to 1.[1]...Some statistical software packages like PSPP, SPSS and SYSTAT label the standardized regression coefficients as "Beta" while the unstandardized coefficients are labeled "B".

More over per another source, to quote:

When you have only one predictor variable in your model, then beta is equivalent to the correlation coefficient (r) between the predictor and the criterion variable.

Further:

When you have more than one predictor variable, you cannot compare the contribution of each predictor variable by simply comparing the correlation coefficients. The beta (B) regression coefficient is computed to allow you to make such comparisons and to assess the strength of the relationship between each predictor variable to the criterion variable.

In the case of a simple Least-Squares (LS) regression with one standardized variable, the beta coefficient is mathematically equal to the Pearson Correlation Coefficient. An important implication, at least in my opinion, is that one can consider, say, something other than LS, including, for example, robust regression (including, for example, Least-Absolute Deviations with data centered on the median) as options. With robust methods, the beta (or correlation) coefficient is expected to be correspondingly robust as well.

My personal advice in general, do a simulation exercise generating random deviates (in accord with the data you are examining) and test the various outputs of correlation measures with respect to point accuracy and over repeated runs, a measure of deviation from known underlying value. I advise you to incorporate in the simulation, your knowledge of, for example, the relative sample size (relating again to your situation) and particular measures of appropriateness. These could include bias, accuracy, and robustness with respect to the select error distribution of interest.

This approach should give you more confidence in the particular measure you select and verification as to its likely validity for others.