Unit nonresponse and item nonresponse both occur frequently in surveys. Unit nonresponse is customarily handled by weighting adjustment, whereas item nonresponse is usually treated by some form of imputation. In particular, deterministic or stochastic imputation is often used to assign values for missing item responses. We provide an account of some recent work on jackknife variance estimation based on adjusted imputed values, using only a single imputation and hence a single completed data set. We also present linearized versions of the jackknife variance estimators. We study both stratified simple random sampling and stratified multistage sampling. Existing computer programs for jackknife and linearization variance estimation can be modified to implement the proposed variance estimators without requiring the creation and permanent retention of supplemental data sets. But for secondary analyses, the completed data set must include information on response status for each item as well as on the imputation class.