MCMC variance covariance

[Lise]

Hello
We are using the MCMC variance covariance option to estimate process correlation between survival and dead recovery rates.
How does one properly convert the estimated correlation coefficients (rhos) for the beta parameters to correlation coefficients for the real parameters?
We tried univariate transformation with the link function (we used sin link), but some simple calculation checks indicate this might not be correct.
Is the appropriate transformation more complex?
Cheers!

[cooch]

Salut Lise...

Hello
We are using the MCMC variance covariance option to estimate process correlation between survival and dead recovery rates.

They're geese -- mortality is additive.

How does one properly convert the estimated correlation coefficients (rhos) for the beta parameters to correlation coefficients for the real parameters?
We tried univariate transformation with the link function (we used sin link), but some simple calculation checks indicate this might not be correct.
Is the appropriate transformation more complex?
Cheers!

Think of it this way -- you have a correlation between two sets of parameters estimates on a particular scale, and are interested in the correlation among these parameters on a different scale. What complicates things here is that logit is -/+ infinite, whereas the real probability scale is [0,1] bounded. But, more to the point -- whether or not you need to back-transform at all is a function of whether or not the transformation is linear (or nearly so) over the range of the data. For example, if you take the following values for phi (0.45, 0.475, 0.5, 0.525, 0.5, 0.575, 0.6, 0.625), and 'correlate' them with p (0.6, 0.58, 0.58, 0.52, 0.46, 0.55, 0.45, 0.41), you'll get -0.877. If you do the same correlation on logit transformed values (which correspond to beta under an identity matrix), you'll get a correlation of -0.879 (close enough for government work). However, if you have a bigger range of values (say, by including some near either the 0-1 boundaries), then the assumption of 'linearity' over the range of the transformation is violated, and the concordance of the two correlation values blows up.

The formal 'back-transform' from the logit scale (which we often don't care much about) to the real probability scale (which we invariably are interested in) -- well, c'est compliqué (as you might say). I'm working on this as we speak.