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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