Why one should incorporate the design weights when adjusting for unit nonresponse using response homogeneity groups
When there is unit (whole-element) nonresponse in a survey sample drawn using probability-sampling principles, a common practice is to divide the sample into mutually exclusive groups in such a way that it is reasonable to assume that each sampled element in a group were equally likely to be a survey nonrespondent. In this way, unit response can be treated as an additional phase of probability sampling with the inverse of the estimated probability of unit response within a group serving as an adjustment factor when computing the final weights for the group's respondents. If the goal is to estimate the population mean of a survey variable that roughly behaves as if it were a random variable with a constant mean within each group regardless of the original design weights, then incorporating the design weights into the adjustment factors will usually be more efficient than not incorporating them. In fact, if the survey variable behaved exactly like such a random variable, then the estimated population mean computed with the design-weighted adjustment factors would be nearly unbiased in some sense (i.e., under the combination of the original probability-sampling mechanism and a prediction model) even when the sampled elements within a group are not equally likely to respond.