# Comparing calculated regression line to identity line

I have two large data sets that I do several calculations on. In turn, I end up with a regression line for each data set. Now, I am not interested in comparing them to each other, but I am interested in comparing them to the identity line, since that is my "gold standard". If there are close to the identity line, my data fits perfectly. I am, however, not sure about which test I should use in order to analyse this, and see if one or the other is significantly different from the identity line.

## 3 Answers

A linear regression with one independent variable $x$ fits a line $y=\beta_1 x +\beta_0$ to the data $y$. The identity line between these two variables is given by $y=x$. Thus, to obtain exactly the identity line, the estimated regression coefficients must be $\hat{\beta_1}=1$ & $\hat{\beta_0}=0$, i.e. a slope of 1 and an intercept of 0. So to test whether your regression lines are significantly different from the identity line, I would use t-tests on your regression coefficients, to test whether they are significantly different from 1 (slope) and 0 (intercept).

If your original model is $y = \alpha_0 + \alpha_1 x$ then you could do the regression over $y' = y - x$ instead of $y$. In this way your model is $y' = \alpha_0 + (\alpha_1 - 1) x = \alpha_0 + \beta_1 x$, and you just have to check the significance test results for the fitted coefficient of $\beta_1$ to see if $\alpha_1$ is different from 1 (with some degree of confidence) or not.

The concordance correlation coefficient, "" also called Lin's concordance correlation, "Like Bland-Altman or Deming Regression" is a "method for comparing two measurements of the same variable. This is especially important if you are trying to introduce a new measurement capability...over an existing measurement technique." The two links above should be enough to get you started, but also see interclass correlation, which can be quite similar.