Re: st: Dropping right-censored spells in the Cox model

[Nicole Boyle <nicboyle@gmail.com>]

Hi Kai,

I'll take a stab at this.

I'm going to assume that those "censored spells" you're considering
dropping do not include those that are censored due to the competing
risk, but only include noninformative censorship: For subjects
censored at time "c," probability of survival past some future time
t>c is the same as probability of survival past t for subjects who
were merely known to have survived past c. I'm also assuming those
"censored spells" you're referring to don't include administratively
censored subjects.

Given the assumption of noninformative censorship in conjunction with
a central property of Cox regression, I do NOT think dropping censored
observations is a good idea.

Important to Cox regression is the maximum partial-likelihood
estimator property, which dictates that hazard estimates are computed
only at times of failure, using the risk pools available at each of
these failure times for these computations. All those contributing to
the "at risk" pool at time=t are not weighted any differently at
time=t with regards to future failure or future censorship.

Since subjects censored at time=t are considered just as likely to
fail in the future as subjects NOT censored at time=t, and because
censored subjects by definition NEVER fail in the data, dropping
censored subjects from the study reduces the "# at risk" denominator
from each failure time (where censored observations would have
normally contributed to this denominator prior to censorship) while
keeping the "# failed" numerator intact. This will very likely bias
results.

I suggest that instead of thinking about the data in terms of [total #
failed] / [total # at risk], which is more of a risk ratio or odds
ratio mentality, consider thinking of each failure time as an
independently conducted prevalence estimate.

_Here's an example to clarify_
Say you're conducting small study, and the ONLY data involves one
instantaneous prevalence calculation:
[# dead at this particular moment] / [# at risk for death at this
particular moment]

Now, imagine one month has gone by since you performed the
calculation. A colleague (involved in a completely unrelated study)
informs you that 50% of your original "at risk" sample has now just
declined participation in one HIS/HER studies.

If YOUR study only concerns the original prevalence calculation, would
it make sense to drop those subjects unavailable to your colleague's
study from your original prevalence calculation? Since their
declination from your colleague's study has no bearing on whether they
were dead or alive in YOUR study one month ago, it makes no sense to
drop them. They contributed valuable information.

But furthermore, removing them might be harmful. Only those subjects
who were alive (not in the "# dead" numerator) in your prevalence
calculation are the ONLY subjects in your study capable of declining
study involvement one month later. DeclinationTherefore, eliminating
these subjects BY DEFINITION biases your prevalence calculation.


So, in conclusion:
The likelihood of censorship at time=t is independent of the
likelihood of failure at time>t. However, the likelihood of failure at
time=t is NOT independent of the likelihood of censorship at time>t.

If you discount subjects' past survival contributions based on future
censorship, you will be at risk of artifactually "powering" your
resulting HRs. Given the 50% censorship you've mentioned, this
increased "power" will surely result when dropping this many subjects,
and may result in more "powerful" HRs or even a change in HR
direction.

Nicole

On Sat, Sep 14, 2013 at 7:37 AM, Kai Huang <demonsecret@hotmail.com.hk> wrote:
> Dear all,
>
> I have estimated competing-risks Cox models on unemployment spells. The estimates are not very significant probably due to the small number of spells. I wonder whether the high proportion of censored spells in the model (50%) matter. Does it make sense if I drop all the right-censored spells and estimate the models with all spells being completed? Thank you very much in advance.
>
> Best regards,
> Kai Huang
>
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