Fractional or beta regression, with weights that reflect the inverse variance of the proportion

[Nicholas Syn]

Dear Statalisters,

I am thinking of fitting a fractional response or beta regression for Y on X, for a dataset which looks something like:

Identifier	Numerator	Denominator	Y	X1	X2
1	1000	2000	0.5	1	1
2	2000	3000	0.666667	2	3
3	3000	4000	0.75	3	5
4	4000	5000	0.8	4	7
5	5000	6000	0.833333	5	9
6	6000	7000	0.857143	6	11
7	7000	8000	0.875	7	13
8	8000	9000	0.888889	8	15
9	9000	10000	0.9	9	17

However, to me, it seems intuitive to assign more "weight" to observations where the sample size and/or number of events is larger. For example, it just seems to make sense that Observation 9 should contribute more weight to the analysis as it has 10000 observations and therefore should have the greatest precision / narrowest confidence intervals (if we were to compute the 95% CI of the proportion). Meaning to say that the confidence intervals around the proportion 9000/10000 is smaller than the confidence intervals around a proportion of say, 1000/2000.

In line with the above intuition, I was thinking of assigning inverse variance or inverse-SE weights to the analysis, by computing the SE of the logit of Y.

But I am also conscientious of the fact that I have not found any references to support such an approach..

I was hoping to get some expert advice if you are aware of any statistical theory / simulation papers which might support my above methodology...

Thanks in advance!
Best regards,
Nicholas Syn