Do I need a GMR?
Recently, we had a paper revision come back with the suggestion that we use RMA versus ordinary least squares regression. My first thought was: what the heck is RMA? RMA (reduced major axis) regression is a re-branding of GMR (geometric mean regression). Whereas in a "traditional" OLS regression (y=a+bx), our goal is to minimize errors in the y -direction, in GMA, we minimize the product of the distances in the y- and x-directions. I have since gone on to find it called least products regression, diagonal regression, line of organic correlation, least areas line, Deming regression, Model II regression, and an errors in variables model.
(... Seriously? Can we just call a cod fish a cod fish? It does not taste any better if you call it a Sable Fish.)
In any event, I had not seen a GMR since my consulting days back in Vancouver in 2000-2001. I remember my very smart and talented supervisor using it to make adjustments to site index assessments made by experts using field data of top height. These two variables are both subject to a lot of error, so GMR makes sense. GMR is appropriate when the dependent and independent variables in the regression equation are "random"., i.e. not controlled by the researcher. OLS regression can underestimate the slope when the independent variable also contains error. Pierre Legendre gives a great table outlining rules for using GMR, OLS, and some variants here. In particular, OLS is a valid method when the error in the dependent variable is >> error in the independent variable:
"If the magnitude of the random variation (i.e. the error variance) on the response variable y is much larger (i.e. more than three times) than that on the explanatory variable x, use OLS. Otherwise, proceed as follows...."
So... was it appropriate for this paper? Well, we developed predictive equations for tree height and crown width based on diameter as a predictor. Even the most experienced forester would be hard-pressed to reduce the errors in height measurements to <3 times that of diameter. So, my opinion? no way. Let's hope I can convince the reviewers...
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