Dear Carsten,

Here is why the model II slope is based on the 
assumption that the correlation coefficient, in 
the absence of measurement error of Y and X, is 
1. I'm going to use the example of the RMA, or 
reduced major axis, but the same would be true 
for other methods that attempt to estimate the 
effect of error in X, without having any measure of that error*.

The easiest demonstration is mathematical. The 
slope of an OLS regression can be written as r · 
Sy / Sx. The correlation coefficient provides the 
regression aspect of the relationship: the 
predictions of Y are a restricted range of Y 
because some of the variation in Y is not due to 
variation in X. The slope of the RMA is (sign) Sy 
/ Sx. The covariance between Y and X is still 
present, through the assumption that r = 1 or -1.

The next argument is more difficult, but more 
crucial, as I see it. Model II regression is an 
attempt to get something for nothing. It is an 
attempt to get "truth" (this concept is almost 
always invoked) by discarding random noise. The 
idea is that you will recover a line linking Y 
and X, by minimizing "error" along both 
horizontal and vertical axes. In OLS we know what 
the "error" in Y is - it's equation error, 
including all sources of variation in Y that are 
not correlated with X. In model II, what is error 
in X? It includes variation in X that is not 
correlated with Y. The only way the model II 
slope makes sense in this case is if X contains 
two parts, a part that is wholly correlated with 
Y (deterministic, r=1) and another part that we 
can ignore, throw away, because it is not real 
information. It is not other variables. There are 
no other variables affecting Y and there are no 
other variables affecting X, and the relation 
between Y and X is linear, and there is only 
measurement error. By assuming equal error (or 
proportional error) in X and Y you are saying 
that error is random and other variables play no 
role. In almost all cases, this assumption is 
patently wrong and therefore the Model II slope 
that derives from that assumption is a 
never-never land slope that doesn't correspond with any "truth".

Think about this: what if the hidden relationship 
between Y and X is a cloud of points? And then 
random error is added. Now apply any model II 
protocol you want **. You will derive a 
meaningless and misleading slope, because the 
true relationship between X and Y is not perfect***.

* In other words, if you have measured the 
measurement error in X, I have no objection to 
using model II corrections. If you haven't 
measured this error, and in particular, if you 
think that "random error" in X can be "corrected 
for" without any measurement of it, you are 
living in a dream world. Literally. I know, I 
used to live in that world too. I "drank the 
koolaid" of model II regression right from the 
beginning, having used the RMS model II slope in 
a paper in 1984. In fact, I confess it, I thought 
that model II methods allowed us to cheat God or 
the Flying Spaghetti Monster - to see the "true" 
mechanisms in the world by a mathematical trick. I should have known better.

** Except a true model II protocol that is based 
on a knowledge of measurement error in X, derived 
by measuring it, by repeated measurements of individual X's.

*** Unless the true relationship between X and Y 
is indeed perfect, in which case model II should work just fine.

Best regards
David





>as one of the ecologists "inflamed by your comments", I would like to learn:
>Why is "the model II slope based on the assumption that the correlation
>coefficient, in the absence of measurement error of Y and X, is 1.", as
>you wrote? This may be a simple answer ...
>(I assumed that the error of X and Y are distributed similarly, since I
>minimise equally in X and Y direction. But why do I assume that they are
>correlated?)
>
>I like to think of a model II as a regression through data points with
>error bars both vertically and horizontally and that I find more
>realistic, given that 15% cover rarely ever are indeed 15% cover.
>
>If this "picture" is correct, it does not imply a correlation of 1
>between error-free X and Y, or does it?
>
>Cheers,
>
>Carsten
>
>
>On 02.11.10 20:57, David Bird wrote:
> > Karen,
> >
> >> Carsten: did you imply that beta regression is necesarily model I
> >> regression (no variance in predictor variable)?? I'd be interested to hear
> >> anyone's thoughts on how much of a limitation this is for situations where
> >> both y and x are random variables. Is it the same as for OLS regression,
> >> where OLS is acceptable if the error variance in x is less than a third of
> >> that in y?
> >
> > In most cases, model I is the proper model to employ. The possible
> > impact of error in X in prediction is an interesting and important
> > question which many in ecology get wrong, from all evidence. To put
> > it simply, the calculations of regression models do not change when
> > both X and Y are random variables, when you are modeling the
> > distribution of observed Y conditional on observed X, that is to say,
> > when you are making predictions, and I would argue that this is
> > generally what should be sought. This would be just as true for beta
> > regression as for OLS. The slope of this relation is an unbiased
> > estimate of the true slope linking observed Y and observed X.
> >
> > There are many different kinds of Model II regression, and since you
> > are interested in measuring the correlation between two variables and
> > the form of the relation, I doubt than any of them will be of use to
> > you.  The most common kinds, that are based on the major axis, the
> > reduced major axis, the ranged major axis, or the slope bisector,
> > will give you biased predictions if based on observed values of X
> > (and those are the only kind of X values you have).
> >
> > Here is something to make you think about what this sort of Model II
> > regression means. You've apparently estimated the correlation
> > coefficient, and then you have estimated a model II slope. However,
> > the model II slope is based on the assumption that the correlation
> > coefficient, in the absence of measurement error of Y and X, is 1. So
> > if you believe that model II is the right way to go, I can tell you
> > already what your corrected, model II, true 
> correlations are. They are rho=1.
> >
> > The calculation of the correlation coefficient is exactly as biased
> > an estimate of rho, in the presence of measurement error in X, as the
> > estimated slope is of the true slope. This little fact means that the
> > test of the null hypothesis, that the slope is zero, is unbiased,
> > even in the presence of measurement error in X. Biases in the
> > numerator and denominator of the t-test for the slope exactly cancel out.
> >
> > This inflammatory comment is designed to take the temperature of
> > ecologists on this issue. If I'm right, not many people will agree
> > with the above. I'd like to think that I'm wrong, however, and that
> > this is old hat to everyone.
> >
> > Best wishes
> > David Bird
> >
> >       [[alternative HTML version deleted]]
> >
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>
>--
>Dr. Carsten F. Dormann
>Department of Computational Landscape Ecology
>Helmholtz Centre for Environmental Research-UFZ
>(Department Landschafts?kologie)
>(Helmholtz Zentrum f?r Umweltforschung - UFZ)
>Permoserstr. 15
>04318 Leipzig
>Germany
>
>Tel: ++49(0)341 2351946
>Fax: ++49(0)341 2351939
>Email: carsten.dormann@ufz.de
>internet: http://www.ufz.de/index.php?de=4205
>
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>Wilfried Kraus
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>Dr. Georg Teutsch
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>Andreas Schmidt
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>
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>End of R-sig-ecology Digest, Vol 32, Issue 7
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