I'm not an expert in complex survey design, but I took a class from a world-class expert a few years ago. Since you are doing complex survey design work, using -if- to subset your data is risky because you are only using a subset of the weights for your calculations, which can be very problematic. This is why Stata's svy command prefix has a subpop option. The subpop option allows you to use all of the weighting information for a calculation where you want to make inferences to a subpopulation. I'm not comfortable promising that the traditional standard error calculation for the prtest is correct here. I'm not completely convinced #9 is without problems related to the standard error calculation (and Clyde is one of the best posters here). If you look back over the thread, you should notice George and Mike also express some uncertainty as to the validity of what you are doing, and these are both very knowledgeable posters.
I am not saying your method definitely doesn't work, but I do think you should proceed with a great deal of caution. If your question is "can we use a subset of weights on a subset of data to make inferences" the general answer is "no", and even if all of your tests all fail to reject the null it's still not obvious to me that you can safely make any old inference on a subset of the data. Depending on the design, I think you would be much better off making use of the svy prefix with the subpop option to make inferences about subsets of the data whenever possible. If you do that, you should not need to calculate any other weights.
Crossed with #15, but that's a great question. Do city parts define clusters or are they a different kind of division? Are there other levels of strata or clusters in the design?
I am not saying your method definitely doesn't work, but I do think you should proceed with a great deal of caution. If your question is "can we use a subset of weights on a subset of data to make inferences" the general answer is "no", and even if all of your tests all fail to reject the null it's still not obvious to me that you can safely make any old inference on a subset of the data. Depending on the design, I think you would be much better off making use of the svy prefix with the subpop option to make inferences about subsets of the data whenever possible. If you do that, you should not need to calculate any other weights.
Crossed with #15, but that's a great question. Do city parts define clusters or are they a different kind of division? Are there other levels of strata or clusters in the design?

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