Sampling Variances for Surveys With Weighting, Poststrati cation, and Raking

It is common practice to use weighting, poststratiication, and raking to correct for sampling and nonsampling biases and to improve eeciency of estimation in sample surveys. However, there is no standard method for computing sampling variances of estimates that use these adjustments in combination. In this paper we develop such a method, using three ideas: (1) a general notation that uniies the diierent forms of weighting adjustment, (2) a variance decomposition to estimate sampling variances conditional and unconditional on sample sizes within poststratiication categories, and (3) numerical computation using a delta method. We apply our approach to the problem that motivated this research, the New York City Social Indicators Survey, a telephone survey that uses inverse-probability weighting, poststratiication, and raking to correct for sampling design and nonresponse. Our variance estimates systematically diier from those obtained using methods that do not account for the design of the weighting scheme. Assuming simple random sampling leads to underestimating the sampling variance, and treating all weights as inverse-probability causes variances to be overestimated.