[R-sig-eco] R-sig-ecology Digest, Vol 32, Issue 1

[Carsten Dormann]

Dear 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
>
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>
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-- 
Dr. Carsten F. Dormann
Department of Computational Landscape Ecology
Helmholtz Centre for Environmental Research-UFZ	
(Department Landschaftsökologie)
(Helmholtz Zentrum für Umweltforschung - UFZ)
Permoserstr. 15	
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Germany

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