[R] competing risks survival analysis

gb gb at stat.umu.se
Tue Oct 31 15:46:19 CET 2000

On Tue, 31 Oct 2000, Bill Simpson wrote:

> > 
> > 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.

Now, as I see it, this simplifies matters enormously! Since you only are 
interested in the ratio between the two hazards, you can (a) throw away
all censored observations, be it right or left, (b) just count the number 
of events of each type, i.e., disregard exposure time, which will be the
same for both types of events. 

Two (non-parametric) approaches are possible:

I. Divide the time axis into suitable intervals, count the number of
   events of each type in each interval, and calculate the appropriate
   ratios. This is what Cox & Oakes did in Figure 9.1.

II. For each  t, t > 0, count the events of type  a  and  b, n_a(t)  and
    n_b(t), respectively, that occurs before  t. Then calculate
    n_a(t) / (n_a(t) + n_b(t)). This is not exactly what you asked for,
    but a "cumulative" version of it. The advantage could be that you
    don't have to do an (arbitrary) interval splitting of the time axis.

> 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.

So, your censored observations are not truly censored in that you know 
the type of event, a  or  b? In that case, since the right and left
censoring time points are constant over individuals (fixed by "design"), 
I guess it would make sense to count events in (0, 100ms) and in 
(1500ms, \infty) separately, by type, and continue as above. 
> 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. 

If censored observations carry no information of type you can throw them
out, since you only bother about the ratio between the intensities.

 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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