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RE: Modeling of two time-to-event outcomes

From: Stephen Duffull <stephen.duffull>
Date: Thu, 23 Jul 2009 16:35:28 +1200

Hi Nick

> I've been hearing about copulas for a couple of years now but haven't
> seen anything which reveals how they can be translated into the real
> world.

This is a good point. I have seen very few applications of copulas outside of statistics or actuary processes in the specific sense of joining two or more parametric distributions together to form a multivariate distribution.

Obviously we (implicitly) use copulas all the time when we model interval data since a multivariate normal is a specific example of a copula of two marginal normal distributions and we do this when modelling bivariate continuous measure responses such as parent-metabolite data.

Explicit use of copulas are considered when joining distributions that either don't have multivariate forms (e.g. a multivariate Poisson) or distributions that aren't of the same form (e.g. logistic-normal).

Part of the complexity is there are many types of copulas and it seems important to match the copula type to the marginal distribution type.

> If we take the example I gave of hospitalization for heart disease and
> death as being two 'correlated' events. Is there something like a
> correlation coefficient that you can get from a copula to describe the
> assocation between the two event time distributions?

Yes. Most copulas seem to be parameterised with an "alpha" parameter that describes the amount of co-dependence between the observations. Note that the values of alpha are not necessarily interchangeable between copulas and are mostly bounded on -inf to +inf or 0 to +inf.

> If one then added
> a
> fixed effect, such as cholesterol in the example I proposed, would you
> then see a fall in this correlation coefficient?

Yes. I would expect that the degree of co-dependence would decrease.

> It would be helpful to me and perhaps to others if you could give some
> specific example of what copulas contribute.

I haven't seen a PKPD estimation application (yet).

Professor Stephen Duffull
Chair of Clinical Pharmacy
School of Pharmacy
University of Otago
PO Box 913 Dunedin
New Zealand
E: stephen.duffull
P: +64 3 479 5044
F: +64 3 479 7034

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Received on Thu Jul 23 2009 - 00:35:28 EDT

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