# [R-meta] Meta-Analysis: Proportion in overall survival rate

Dr. Gerta Rücker ruecker @end|ng |rom |mb|@un|-|re|burg@de
Wed May 27 22:05:03 CEST 2020

```Dear Nelly, dear all,

I have another problem with this approach, and this is the denominator,
"n". You insert n_t for n, but n_t = n_0*S(t) estimates the number of
patients surviving at time t which is the numerator of the ratio. The
denominator is n_0, not n_t. And n_0 should be used as n.

(Suppose a study without any censoring. Then the Kaplan-Meier estimator
in t gives exactly p = n_t/n_0 which is the same as if the study reports
simply the 2-year survival proportion relative to the sample size, n_0.)

Thus the n in your calculation should be n_0, irrespectively of the
transformation method (I agree with Wolfgang that the logit
transformation is preferable).

I don't see a principle problem with this approach, because n_t is an
unbiased estimate of the numerator and sample size n_0 is - for sure -
the denominator.

Best,

Gerta

Am 27.05.2020 um 21:07 schrieb ne gic:
> Many thanks for the insights, WoIfgang!
>
> I concur, the proportion is likely not from a binomial distribution, so I
>
> Sincerely,
> nelly
>
>
> On Wed, May 27, 2020 at 8:17 PM Viechtbauer, Wolfgang (SP) <
> wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>
>> Dear Nelly,
>>
>> Your equation for the SE assumes that the p behaves like a 'regular'
>> proportion computed from a binomial distribution. I am not sure if this is
>> correct when using the Kaplan-Meier estimator to derive such a proportion.
>>
>> As far as your input to rma() is concerned - that is correct. However, I
>> would consider not meta-analyzing the proportions directly, but doing a
>> logit transformation on p, so using qlogis(p) for yi and sqrt(1/(p*n) +
>> 1/((1-p)*n)) for the SE.
>>
>> Best,
>> Wolfgang
>>
>>> -----Original Message-----
>>> From: R-sig-meta-analysis [mailto:
>> r-sig-meta-analysis-bounces using r-project.org]
>>> On Behalf Of ne gic
>>> Sent: Wednesday, 27 May, 2020 20:02
>>> To: Dr. Gerta Rücker
>>> Cc: r-sig-meta-analysis using r-project.org
>>> Subject: Re: [R-meta] Meta-Analysis: Proportion in overall survival rate
>>>
>>> Dear Michael, Gerta and List,
>>>
>>> I would like to cross-check with you what I have done.
>>>
>>> I have restricted myself to Kaplan-Meier studies which gave the number at
>>> risk at 2 years, and also n_0 at baseline.
>>>
>>> I then estimated the absolute number of those surviving as *n_t *=
>> n_0*S(t)
>>> following Gerta's idea. I took the reported proportions at 2 years to
>>> represent the S(t).
>>>
>>> I calculated the standard error (SE) using the formula: *se *= square
>> root (
>>> *p*(1-*p*)/n). Where *p* = proportion at 2 years i.e. S(t)
>>> , n = *n_t*, the estimated number of of those surviving.
>>>
>>> I then used the random effects model in metafor as follows:
>>> rma(yi = *p*, sei = *se*, data=mydata, method="REML")
>>>
>>> The resulting estimate seems reasonable to me. But I want to confirm with
>>> you if this is the way one would input SE and the proportion to the
>>> function.
>>>
>>>
>>> Sincerely,
>>> nelly
> 	[[alternative HTML version deleted]]
>
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```