# [R] competing risks survival analysis

Bill Simpson wsi at gcal.ac.uk
Tue Oct 31 10:18:05 CET 2000

```Thanks very much Thomas and Goran for your help on my problem.

On Mon, 30 Oct 2000, gb wrote:

> 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?

Yes.

If it would help matters, could we consider the simpler cases
- competing risk with no censoring
- competing risk with right-censoring only

I haven't used survival5 at all, and am not sure what the R code would
look like. Some code snippets would be a great help.

In answer to your question Thomas, the times are probably going to have a
density function that this skewed to the right, so I was thinking of
something simple like log-logistic or Gumbel.

Final point:
I did not present the full picture in my original post. I distorted it to
make it simpler. In reality the short and long times are problematic but
not really censored. In the experiment, on each trial either stimulus A or
B is presented with equal probability. The subject must respond a or b,
and the time of the respose is recorded. If the time is short, say <100
ms, this is an "anticipation": just like when the sprinter jumps off the
blocks prematurely in a race. It is not a true reaction time, and
therefore we don't want to treat it the same as the other responses. If
the resp time is too long, say >1500 ms, this is also not a real reaction
time--it means the subject missed the button or blinked or something like
that. We really will have right-censoring: we won't record times >1.5 s,
but that is not really the issue.

Traditionally these short and long times are just thrown out in the
analysis. I am not sure what the right thing to do is. Censoring doesn't
really capture the problem at all. It is more of an "outlier" or "multiple
mechanism" problem: one mechanism produces normal reaction times, another
produces anticipations, and another produces the really long ones. I
suppose it could be handled as a mixture of three distributions (although
the short and long responses are quite rare).

There's the full story, ugly I know. I was going to try some simple stuff
for a start and slowly try to work my way up to a sensible approach.

Thanks again for the help!

Bill

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```