An alternative jackknife variance estimator when calibrating weights to adjust for unit nonresponse in a complex survey
Section 6. A simulation example

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The MU281 population of municipalities in Särndal, Swensson and Wretman (1992; data from the slightly revised version is contained in http://lib.stat.cmu.edu/datasets/mu284; one of the municipalities was accidentally dropped in this analysis) has been augmented with an indicator (RESP) for whether an element (municipality) would respond if sampled. Probabilities of element response were generated using a logistic function of one of the data set’s covariates (the log of the element’s 1975 population in thousands). The average probability of response was roughly 70%.

A stratified simple random sample of 10 elements per each of 8 strata was simulated 1,716 times. In each simulated sample, the elements with RESP = 1 were treated as respondents, and the respondent sample was calibrated to the full sample using the weight-adjustment function in equation (5.2) with a lower bound of 1 and an upper bound of 5. In the calibration model, the two components of $x_k$ were 1 and the log of the element’s 1975 population in thousands; $z_k$ was set equal to $x_k.$ 1,225 out of the 1,716 simulations had their respondent samples successfully calibrated on both components (i.e., satisfied the calibration equation in 2.2) and produced linearization-based standard-errors.

Estimated means (ratios of two estimated totals) and standard errors (square roots of estimated variances) were computed for four variables:

Although the SUDAAN procedure WTADJUST can compute standard errors when using a delete-1 jackknife, it will fail when one or more replicates fail to calibrate. Therefore, two versions of the conventional delete-1 jackknife standard errors were computed using a macro the authors created. In one, the set of the imperfect “calibrated” weights from the last iteration for the failed replicates were used. In the other, the replicates that failed to calibrate were dropped and this following modified jackknife variance estimator was computed:

$${\mathrm{var}}_J^{*}(t)={\displaystyle \sum _{h=1}^H\frac{n_h-1}{n_h^{*}}}{\displaystyle \sum _{j=1}^{n_h^{*}}{(t^{(hj)}-t)}^2},(6.1)$$

where $n_h^{*}$ is the number of replicates in stratum $h$ that successfully calibrated. This revised jackknife variance estimator is suggested by Rust (1985) when replicates are dropped at random, which is not what happens here. The SAS-callable (SAS Institute Inc., 2015) SUDAAN code used in the analysis for a single simulation is available from the authors upon request.

Among the 1,225 analyzable samples, 867 simulations had all the replicates using conventional delete-1 jackknife calibrate, while the remaining 358 simulations had at least one replicate that failed to calibrate after 50 iterations (the default is 10). Table 6.1 averages the results for both situations. When no conventional replicate failed, the alternative and conventional jackknife standard errors are close (on average) and slightly higher than those produced by linearization as theory predicts (note that the two versions of the conventional delete-1 jackknife are identical).

When at least one replicate failed to calibrate for the conventional delete-1 jackknife, the alternative jackknife’s standard errors are again close to linearization-based ones, even though it failed to calibrate in 114 out of these 358 simulations due to a (near) singularity in at least one of the replicates. However, including the failed replicates clearly overestimates standard error and dropping them clearly underestimates relative to linearization. It appears that the alternative jackknife variance estimator produces the more useful set of replicate weights in this situation.

Table 6.1 compares standard-errors from competing jackknifes to linearization-based standard errors rather than empirical standard errors because finite-population correction has been ignored. Moreover, the bounded logistic response model fit in the simulations was not the unbounded response model used to generate responses.

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