[R] Cox model approximaions (was "comparing SAS and R survival....)

[Mike Marchywka]

----------------------------------------
> From: therneau at mayo.edu
> To: aboueslati at gmail.com
> Date: Fri, 22 Jul 2011 07:04:15 -0500
> CC: r-help at r-project.org
> Subject: Re: [R] Cox model approximaions (was "comparing SAS and R survival....)
>
> For time scale that are truly discrete Cox proposed the "exact partial
> likelihood". I call that the "exact" method and SAS calls it the
> "discrete" method. What we compute is precisely the same, however they
> use a clever algorithm which is faster. To make things even more
> confusing, Prentice introduced an "exact marginal likelihood" which is
> not implemented in R, but which SAS calls the "exact" method.
>
> Data is usually not truly discrete, however. More often ties are the
> result of imprecise measurement or grouping. The Efron approximation
> assumes that the data are actually continuous but we see ties because of
> this; it also introduces an approximation at one point in the
> calculation which greatly speeds up the computation; numerically the
> approximation is very good.
> In spite of the irrational love that our profession has for anything
> branded with the word "exact", I currently see no reason to ever use
> that particular computation in a Cox model. I'm not quite ready to
> remove the option from coxph, but certainly am not going to devote any
> effort toward improving that part of the code.
>
> The Breslow approximation is less accurate, but is the easiest to
> program and therefore was the only method in early Cox model programs;
> it persists as the default in many software packages because of history.
> Truth be told, unless the number of tied deaths is quite large the
> difference in results between it and the Efron approx will be trivial.
>
> The worst approximation, and the one that can sometimes give seriously
> strange results, is to artificially remove ties from the data set by
> adding a random value to each subject's time.

Care to elaborate on this at all? First of course I would agree that doing
anything to the data, or making up data, and then handing it to an analysis tool that doesn't
know you maniplated it can be a problem ( often called interpolation or something 
with a legitimate name LOL).  However, it is not unreasonable to do a sensitivity
analysis by adding noise and checking the results.  Presumaably adding 
noise to remove things the algorighm doesn't happen to like would
work but you would need to take many samples and examine stats
of how your broke the ties. Now if the model is bad to begin with or
the data is so coarsely binned that you can't get much out of it then ok.

I guess in this case, having not thought about it too much, ties would
be most common either with lots of data or if hazards spiked over time scales simlar to your
measurement precision or if the measurement resolution is not comparable to hazard
rate. In the latter 2 cases of course the approach is probably quite  limited . Consider turning
exponential curves into step functions for example. 

>
> Terry T
>
>
> --- begin quote --
> I didn't know precisely the specifities of each approximation method.
> I thus came back to section 3.3 of Therneau and Grambsch, Extending the
> Cox
> Model. I think I now see things more clearly. If I have understood
> correctly, both "discrete" option and "exact" functions assume "true"
> discrete event times in a model approximating the Cox model. Cox partial
> likelihood cannot be exactly maximized, or even written, when there are
> some
> ties, am I right ?
>
> In my sample, many of the ties (those whithin a single observation of
> the
> process) are due to the fact that continuous event times are grouped
> into
> intervals.
>
> So I think the logistic approximation may not be the best for my problem
> despite the estimate on my real data set (shown on my previous post) do
> give
[[elided Hotmail spam]]
> I was thinking about distributing the events uniformly in each interval.
> What do you think about this option ? Can I expect a better
> approximation
> than directly applying Breslow or Efron method directly with the grouped
> event data ? Finally, it becomes a model problem more than a
> computationnal
> or algorithmic one I guess.
>
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