From: Mats Karlsson <*mats.karlsson*>

Date: Fri, 24 Jul 2009 13:36:22 +0200

Dear all,

Awaiting implementation of copulas, I would try a divide and conquer =

approach. Data for event A are analysed first, using observed events B =

as time varying covariate. Thereafter analyse data for event B, using =

event A as a time varying covariate.

This would have the drawback that what with copulas can be captured in a =

single correlation parameter, now requires two parameters - one for each =

covariate relation. When one event is always occurring before the other, =

if it occurs at all, only one covariate relation is necessary and from a =

parsimony principle I don't see why it would be worse than a copula =

function. Nick's example is such a situation, like all examples where =

death is one event...

Best regards,

Mats

Mats Karlsson, PhD

Professor of Pharmacometrics

Dept of Pharmaceutical Biosciences

Uppsala University

Box 591

751 24 Uppsala Sweden

phone: +46 18 4714105

fax: +46 18 471 4003

-----Original Message-----

From: owner-nmusers

On Behalf Of Nick Holford

Sent: Friday, July 24, 2009 8:09 AM

To: nmusers

Subject: Re: [NMusers] Modeling of two time-to-event outcomes

Steve,

Stephen Duffull wrote:

*> 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 still don't understand what this has to do with 'interval data'.

*>> 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.

*>
*

The correlation between parent and metabolite collected at the same time =

is indeed a theoretical possibility but I've not had much success in

showing it has any importance in improving the description of real data

sets. The fixed effect part of the model seems to dominate this kind of

process.

*> 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.

*>
*

Here I agree with you. Time to event models are very tough to diagnose

and typically use very empirical fixed effect structures. Finding some

way to account for some kind of random effect would be very nice but I

still have no idea how to it.

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.
*

*>>
*

*>>>
*

*>>>> 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 *

*>>> 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
*

*>>
*

*>
*

*>
*

--

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 Fri Jul 24 2009 - 07:36:22 EDT

Date: Fri, 24 Jul 2009 13:36:22 +0200

Dear all,

Awaiting implementation of copulas, I would try a divide and conquer =

approach. Data for event A are analysed first, using observed events B =

as time varying covariate. Thereafter analyse data for event B, using =

event A as a time varying covariate.

This would have the drawback that what with copulas can be captured in a =

single correlation parameter, now requires two parameters - one for each =

covariate relation. When one event is always occurring before the other, =

if it occurs at all, only one covariate relation is necessary and from a =

parsimony principle I don't see why it would be worse than a copula =

function. Nick's example is such a situation, like all examples where =

death is one event...

Best regards,

Mats

Mats Karlsson, PhD

Professor of Pharmacometrics

Dept of Pharmaceutical Biosciences

Uppsala University

Box 591

751 24 Uppsala Sweden

phone: +46 18 4714105

fax: +46 18 471 4003

-----Original Message-----

From: owner-nmusers

On Behalf Of Nick Holford

Sent: Friday, July 24, 2009 8:09 AM

To: nmusers

Subject: Re: [NMusers] Modeling of two time-to-event outcomes

Steve,

Stephen Duffull wrote:

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 still don't understand what this has to do with 'interval data'.

aside

model

in the data (i.e. all models are wrong)

assay: parent and metabolite are assayed in the same run and both are =

affected by process errors: recording errors with wrong dose & wrong =

time

when considering parent-metabolite and concentration-effect modelling =

and that a fixed effect alone may not be sufficient to handle this =

correlation.

The correlation between parent and metabolite collected at the same time =

is indeed a theoretical possibility but I've not had much success in

showing it has any importance in improving the description of real data

sets. The fixed effect part of the model seems to dominate this kind of

process.

even more reason to believe our model is wrong and we cannot hope to =

account for co-dependence structure based on fixed effects only.

Here I agree with you. Time to event models are very tough to diagnose

and typically use very empirical fixed effect structures. Finding some

way to account for some kind of random effect would be very nice but I

still have no idea how to it.

Nick

real

Poisson)

parameter

--

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 Fri Jul 24 2009 - 07:36:22 EDT