A method to correct for frame membership error in dual frame estimators
Section 5. Inference for ${\widehat{\widehat{Y}}}_{\text{BC}}$

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Lohr (2011, Section 4) noted that inference for dual frame estimators with non-random constructed weights is straightforward using standard survey software. This is true for both ${\widehat{Y}}_L$ and ${\widehat{Y}}_{\text{BC}}.$ For ${\widehat{Y}}_{\text{BC}},$ the constructed weights for units in the four domains are ${\tilde{w}}_i^A\mathrm{<mo>=</mo>}{w}_i^A\left(p_{a|a^{*}}+\theta \left(1-p_{a|a^{*}}\right)\right)$ and ${\tilde{w}}_i^A\mathrm{<mo>=</mo>}{w}_i^A\left(p_{a|a{b}^{*}\left(A\right)}+\theta \left(1-p_{a|a{b}^{*}\left(A\right)}\right)\right)$ for units in domains $a^{*}$ and $a{b}^{*}$ sampled from frame $A,$ and ${\tilde{w}}_i^B\mathrm{<mo>=</mo>}{w}_i^B\left(\left(1-\theta \right)\left(1-p_{b|b^{*}}\right)+p_{b|b^{*}}\right)$ and ${\tilde{w}}_i^B\mathrm{<mo>=</mo>}{w}_i^B\left(\left(1-\theta \right)\left(1-p_{b|a{b}^{*}\left(B\right)}\right)+p_{b|a{b}^{*}\left(B\right)}\right)$ for units in domains $b^{*}$ and $a{b}^{*}$ sampled from frame $B.$ Then ${\widehat{Y}}_{\text{BC}}$ and its standard error can be calculated by providing to the software files containing data and weights from both frames. When the misclassification probabilities must be estimated, as they are for ${\widehat{\widehat{Y}}}_{\text{BC}},$ however, the variances are inflated, as illustrated in Figure 4.2, panel (a). In this case, linearization, jackknife, or bootstrap methods could be used to accommodate this increased variance.

Lin (2014, Section 5.2.2) produced an approximate variance expression for ${\widehat{\widehat{Y}}}_{\text{BC}}$ for the case of a SRS at both phase 1 and phase 2, based on the linearization method. However, implementing the method requires special purpose coding, and would have to be adapted for complex designs at the two phases. Thus, we chose to investigate the accuracy of an approximate method that ignores the additional variability introduced by estimation of the misclassification probabilities and produces confidence intervals using standard survey software. The only pre-processing required is that the estimated misclassification probabilities must be computed from the phase 2 sample and used to replace the known values in the weight expressions above.

To test this method, we simulated a population with the characteristics of that of example 2 in the previous section. We examined a subset of the misclassification patterns considered there: $db,$ $cb$ and $ca.$ The sample sizes for the phase 1 samples in this case were set to $n_A=$ 400 and $n_B=$ 200, and three phase 2 sampling rates were chosen, ranging from 5% to 20% (i.e., from $m_A\mathrm{<mo>=</mo>}$ 20 and $m_B\mathrm{<mo>=</mo>}$ 10 to $m_A\mathrm{<mo>=</mo>}$ 80 and $m_B\mathrm{<mo>=</mo>}$ 40), chosen as SRS and stratified designs. 10,000 replicates were generated under each setting. From each sample, misclassification probabilities were estimated and substituted in the weight expressions above. For comparison purposes, we also produced weights using the known misclassification probabilities to test the performance of confidence intervals based on ${\widehat{Y}}_{\text{BC}}.$ All computation was done using R’s survey package. The software-produced estimates of standard error and a 95% confidence interval for the population total (based on survey’s standard jackknife procedure) were obtained from each replicate.

The results are summarized in Table 5.1. The columns labeled Sim.Var. displays the variance of each estimator as computed from the 10,000 replicates, which is our best assesment of the true variances. The columns labeled Est.Var. show the average over the 10,000 replicates of the software-produced variance estimates of ${\widehat{Y}}_{\text{BC}}$ and ${\widehat{\widehat{Y}}}_{\text{BC}}.$ The columns labeled Suc.Rate shows the proportion of the replicates for which the confidence interval includes the true total. Panels (a) and (b) display results for the phase 2 SRS and stratified sample designs, respectively.

By comparing the Sim.Var. and Est.Var. columns, we show that the variance of ${\widehat{\widehat{Y}}}_{\text{BC}}$ is underestimated, on average for all settings. As would be expected, the underestimation is worse for the smallest sample size and the inefficient (SRS) design. As a result, the confidence interval coverage is less than its nominal value for most settings. However the undercoverage is small (less than 5%) for all cases except the smallest sample size for a phase 2 SRS. Since the coverage of the confidence intervals based on ${\widehat{Y}}_{\text{BC}}$ held their nominal values, we can conclude that the undercoverage was due completely to the estimation of misclassification probabilities. We suggest that the additional variation added from estimation of misclassification probabilities can be safely ignored in inference if an efficient phase 2 design is used, unless the sample sizes are very small. Based on this simulation, if misclassification probabilities are estimated from at least 10 units in the each perceived domain, the coverage probabilities were no more than a few percent off. This can be accomplished with a smaller total sample size when the phase 2 sample is stratified, than when a SRS is used, so a stratified phase 2 design is recommended.

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