Which regression: Model I or Model II?

For Excel®, MATLAB® and most other commercial programs the inherent line fitting method is the model-I regression.

To determine whether you are using a model-I or a model-II regression

• first find the slope of Y vs X where Y is plotted on the vertical axis and X is plotted on the horizontal axis — this is the “normal” way of doing things. It is also known as the regression of Y-on-X. Call this slope m(y).
• Now reverse X and Y and fit another line; call this slope m(x)’.
• Since X and Y are reversed, we need to find the inverse of m(x)’ to properly compare against m(y), so let m(x) = 1 / m(x)’.
• Now if m(x) = m(y) exactly and r is not equal to 1, then you are using a model-II regression.
• If m(x) is not equal to m(y), then you are using a model-1 regression.

Note that for either model, r^2 = m(y) / m(x). This is known as the Pearson product-moment correlation coefficient. It is a measure of the linearity of the data, not the fit of the line to the data.

To quickly calculate the model-II geometric mean regression slope, m(gm), first determine the model-I regression slope, m(y), and the correlation coefficient, r. The geometric mean slope is then calculated as: m(gm) = m(y) / r. Or, you can use the MATLAB® script file lsqfitgm.

Also note that for datasets where r = 1, m(y) = m(x) = m(gm). In those cases, this test will not tell you which method you are using.