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RE: Modeling of two time-to-event outcomes

From: Stephen Duffull <stephen.duffull>
Date: Fri, 24 Jul 2009 07:15:18 +1200

Nick

> "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?

Only because for continuous normally distributed data we have an automatic solution in the use of a multivariate normal and we implement this in NONMEM wit $SIGMA BLOCK and the L1 L2 data item flags.

> 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?

This is an important point. There are two scenarios here:
1) The fixed effect doesn't fully explain the correlation structure in the data (i.e. all models are wrong)
2) There is correlation in the residual random effects, e.g. both assay: parent and metabolite are assayed in the same run and both are affected by process errors: recording errors with wrong dose & wrong time

I think most agree that we would account for correlated observations when considering parent-metabolite and concentration-effect modelling and that a fixed effect alone may not be sufficient to handle this correlation.

For empirical models, e.g. logistic regression and survival, there is even more reason to believe our model is wrong and we cannot hope to account for co-dependence structure based on fixed effects only.

Steve
--

>
> 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.
> >
> >
> >> 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).
> >
> > Steve
> > --
> > Professor Stephen Duffull
> > Chair of Clinical Pharmacy
> > School of Pharmacy
> > University of Otago
> > PO Box 913 Dunedin
> > New Zealand
> > E: stephen.duffull go.ac.nz
> > 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 fax:+64(9)373-7090
> 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 - 15:15:18 EDT

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