Bayesian predictive inference of small area proportions under selection bias
Section 5. Concluding remarks
We have extended the homogeneous nonignorable selection model of CNK to accommodate selection probabilities that have different distributions in different areas. This is an improvement to handle the relationship between the binary variables and the selection probabilities. As there are numerous additional parameters, the computation has become a lot more difficult and we have used the Metropolis-Hastings sampler to overcome this difficulty.
We have used an example on severe activity limitation in the National Health Interview Survey in which the small areas are formed by crossing education (pre-college, college, post-college), sex (male, female) and race (white, nonwhite). The heterogeneous nonignorable selection model appears to perform better than the homogeneous selection model and the baseline ignorable selection model.
We have used a simulation study to assess the performance of our heterogeneous nonignorable selection model. We have drawn data from the homogeneous nonignorable selection model and fitted the ignorable selection model and the heterogeneous nonignorable selection model. We found little difference between the two nonignorable selection models but substantial difference from the ignorable selection model. However, when we have drawn data from the heterogeneous nonignorable selection model and fitted the ignorable selection model and the homogeneous nonignorable selection model, we found that the heterogeneous nonignorable selection model is a lot better especially when there is medium to strong selection bias using bias, mean square error and coverage. This is evident in Tables 4.9-4.12 and Figures 4.3-4.8.
Within the framework of our heterogeneous nonignorable selection model, we can think about several additional problems. First, we can accommodate polychotomous data that are numerous in survey problems. Second, although a bit different from our approach, we can accommodate covariates (e.g., age, race, sex in our application on activity limitation). Third, as there are nonrespondents in numerous surveys, we can attempt to accommodate nonignorable nonresponse and selection simultaneously (e.g., Nandram and Choi, 2010). This can be done within our framework. Fourth, as in a two-fold model, we can accommodate clustering or stratification within each area (e.g., Nandram, 2016; Lee, Nandram and Kim, 2017). Our work is potentially useful to solve problems in nonprobability samples as well (e.g., Elliot and Valliant, 2017).
Acknowledgements
This research was supported by a grant from the Simons Foundation (#353953, Balgobin Nandram).
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