Variance Estimation With Complex Data and Finite Population Correction—A Paradigm for Comparing Jackknife and Formula‐Based Methods for Variance Estimation

Abstract

The finite population correction (FPC) factor is often used to adjust variance estimators for survey data sampled from a finite population without replacement. As a replicated resampling approach, the jackknife approach is usually implemented without the FPC factor incorporated in its variance estimates. A paradigm is proposed to compare the jackknifed variance estimates with those yielded by the delta method because the delta method has the effect of the FPC factor implicitly integrated. The goal is to examine whether the grouped jackknife approach properly estimates the variance of complex samples without incorporating the FPC effect, in particular for data sampled from a finite population with a high sampling rate. The investigation focuses on the data drawn by two‐stage sampling with probability proportional to size of schools and with simple random sampling of students. Moreover, the Hájek approximation of the joint probabilities is used in the delta method for a Horvitz–Thompson (H–T) estimator. Samples of the National Assessment of Educational Progress (NAEP) state science assessments are used in the analysis.