RE: Modeling of two time-to-event outcomes

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: [email protected] P: +64 3 479 5044 F: +64 3 479 7034 Design software: www.winpopt.com