An alternative jackknife variance estimator when calibrating weights to adjust for unit nonresponse in a complex survey
Section 1. Introduction
Calibration weighting is a method for adjusting the weights in probability-sampling theory by forcing the weighted sum of each variable in a set of survey variables to equal a specified target. When that happens, the analysis weights are said to satisfy the calibration equation. There are several reasons to calibrate analysis weights. The reason we focus on here is to remove potential selection bias resulting from unit nonresponse.
It is common in the survey-sampling literature to argue that a survey respondent’s calibration-weight adjustment implicitly estimates the inverse of its probability of response (see, for example, Section 5.1 of Fuller, 2009). Kott and Liao (2012) show that using calibration weighting to adjust for unit nonresponse can provide double protection against nonresponse bias when estimating a population total. This means that if either a linear outcome model or an implied selection model holds, then the resulting estimator is asymptotically unbiased in some sense. They go on to describe a linearization-based variance estimator for an estimated total based on a stratified multistage (or single-stage) sample with calibration-adjusted analysis weights.
The brief treatment in Sections 2 and 3 of calibration weighting for nonresponse and of linearization-based variance estimation for a calibrated estimator of a population total are developed in more depth in Kott and Liao. Proofs of the various assertions made in these sections can be found there. Here they set up the theory behind variance estimation with a jackknife.
Given a stratified multistage probability sample, a traditional delete-1 jackknife variance estimator creates sets of replicate weights, one set corresponding to each selected primary sampling unit (PSU). One selected PSU is dropped at a time, and the replicate weights of its subsampled elements are set to zero. To compensate, the replicate probability weights of the remaining elements in the same stratum as the dropped PSU are increased by the factor where is the original number of PSUs selected from the stratum. The replicate probability weights are calibrated in a manner analogous to the original analysis weights. Section 4 describes this jackknife and shows its near equality to the nearly-unbiased linearization-based variance estimator for a population total.
The advantage of a delete-1 jackknife over linearization for variance estimator is that once replicate weights are computed, estimating the variance of smooth function of estimated totals (such as a regression coefficient) is straightforward. Krewski and Rao (1981) provides a rigorous treatment of the delete-1 jackknife and its properties.
Sometimes no solution to a calibration equation exists when starting with a set of replicate probability weights. The main contribution of this paper is contained in the remainder of Section 4, where an alternative method of constructing jackknife replicate weights that can usually overcome this problem is described and justified. This method was introduced in Kott (2006) for another purpose.
Section 6 uses a weight-adjustment function described in Section 5 to illustrate how to implement this method. It then favorably compares the results of the method to those of two popular competitors. Section 7 discusses a variant of the alternative jackknife methodology.
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