Meeting in the middle; or fudging model II regression with nls
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My colleague Karen needed an equation to predict trunk diameter given tree height, which she hoped to base on measurements of trees in semi-arid Australian woodlands. This is the dark art of allometry and a quick google found a large number of formulae that have been used in different studies of tree dimensions. No problem: I started to play with a few of them and eventually settled on this one:Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
dbh = exp( b0 + b1 / (b2 + h) )
where: dbh is trunk diameter at breast height; h is tree height.
Karen also needed to do reverse predictions, ie. predict a tree’s height given its trunk diameter. Again no problem, the inverse equation is simply:
height = b1 / (log( dbh ) – b0) – b2
But then, the pièce de résistance: the forward predictions had to agree with the reverse predictions, ie. if plugging height h into the forward equation gave trunk diameter d, then plugging d into the reverse equation should get you back to h. Karen pointed out that this seemed to be a Model II regression problem because as well as needing symmetric predictions, both variables were subject to measurement error.
R has at least two packages with Model II regression functions: lmodel2 and smatr, but both seemed to be restricted to a single predictor (I’d be interested to hear if I’m wrong about that). Meanwhile, the trusty nls function seemed to do a good job of fitting the forward and reverse equations while dealing with the pattern of variance displayed by the tree data (mu^3).
A solution arrived in the form of this post in the r-help list archive explaining how the coefficients from the forward and reverse fits could be combined by taking their geometric mean to arrive at a form of Model II regression. We tried this and it appeared to work nicely.
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The plot above shows the separate fits, plus the combined fit which does indeed give symmetric predictions, for one of the tree species.
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