Came up in a conversation, so I just wanted to store it: the stratified mean and inverse-probability weighted mean are algebraically equivalent:<\/p>\n
\n$latex \\underbrace{N^{-1}\\sum_{s=1}^S \\sum_{i \\in s}y_{is}\\frac{R_{is}N_s}{n_s}}_{\\text{IPW mean}} = \\sum_{s=1}^S\\frac{N_s}{N}\\frac{1}{n_s}\\sum_{i \\in s}y_{is}R_{is} = \\underbrace{\\sum_{s=1}^S\\frac{N_s}{N}\\bar{y}_s}_{\\text{stratified mean}}$,<\/p>\n
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 $latex s$.<\/p>\n","protected":false},"excerpt":{"rendered":"
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; … <\/p>\n