[R] competing risks survival analysis

gb gb at stat.umu.se
Mon Oct 30 23:14:23 CET 2000

On Mon, 30 Oct 2000, Thomas Lumley wrote:

> On Mon, 30 Oct 2000, gb wrote:
> > > 
> > > I don't have a copy of Cox & Oakes, but if you just want to know the
> > > probability of a failure of type a before time t you can easily handle the
> > > competing risks issue. If someone has a failure of type b then you know
> > > that they don't have a failure of type a before time t, so you can set
> > > their failure time to a very large number (effective infinity).  If it
> > 
> > Hold it! I think you should _right censor_ the observation at t. This
> > doesn't matter, of course, if you only are interested of _one_ particular
> > t, but usually one wants to do the estimation for a range of  t  values.
> It does matter, even for one particular t (unless it's the first one), and
> I meant what I said.  If someone has a failure of one type you know for
> certain that they will not have a failure of another type: for 
> any t, P(failure of other type before t)=0.  Censoring the
> observation would imply that their chance of having a failure of another
> type was the same as for someone who hadn't failed.
> You would censor failures from other causes if you wanted to estimate the
> cause-specific hazard function, but it seems that Bill wants to estimate
> the crude incidence of failure of one type.  

I always thought of estimating the "cumulative incidence" P_a(t) = 
P(T < t, type = a) by first estimating the cause-specific hazard
function and the overall survival function, and then express the estimate
of P_a(t) in terms of these. See e.g. Kalbfleisch & Prentice (1980), p. 169.
However, your method seems to be simpler (and to give the same answer)!


Back to Bill's plot. I checked Cox & Oakes, figure 9.1, and in my copy of
the book, the figure is a graph of h_a(t) / (h_a(t) + h_b(t)) vs t.
Bill, is that what you want?

 Göran Broström                      tel: +46 90 786 5223
 Professor                           fax: +46 90 786 6614
 Department of Statistics            http://www.stat.umu.se/egna/gb/
 Umeå University
 SE-90187 Umeå, Sweden             e-mail: gb at stat.umu.se

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