Mundane algebra: stratified mean and IPW mean

Came up in a conversation, so I just wanted to store it: the stratified mean and inverse-probability weighted mean are algebraically equivalent:

$latex \underbrace{N^{-1}\sum_{s=1}^S \sum_{i \in s}y_{is}\frac{R_{is}N_s}{n_s}}_{\text{IPW mean}} = \sum_{s=1}^S\frac{N_s}{N}\frac{1}{n_s}\sum_{i \in s}y_{is}R_{is} = \underbrace{\sum_{s=1}^S\frac{N_s}{N}\bar{y}_s}_{\text{stratified mean}}$,

where $latex N$ is population size; $latex N_s$ and $latex n_s$ are stratum $latex s$ population and sample size, respectively; and $latex R_{is}$ is the response indicator for unit $latex i$ in stratum $latex s$.


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