# Re: Modeling of two time-to-event outcomes

From: Nick Holford <n.holford>
Date: Thu, 23 Jul 2009 22:26:18 +1200

Steve,

Thanks for your comments. I am still struggling with the copula concept.

The ideas in this sentence are not obvious to me:

"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."

Why is a model for interval data an example?

I think of a parent-metabolite model as being a fixed effect model
connecting parent with metabolite in just the same way as my earlier
example of using chol(t) as a fixed effect affecting the hazard
functions for time to hospitalization and time to death. Putting aside
any random effect correlations between the parent and metabolite
structural model parameters in what way does a parent-metabolite model
involve an implicit copula?

Thanks

Nick

Stephen Duffull wrote:
> 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.
>
>
>> 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).
>
> Steve
> --
> 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
>
> Design software: www.winpopt.com
>
>

--
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand
n.holford
mobile: +33 64 271-6369 (Apr 6-Jul 20 2009)
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
Received on Thu Jul 23 2009 - 06:26:18 EDT

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