Abstract
We present an approach to inform decisions about nonresponse follow-up sampling. The basic idea is (i) to create completed samples by imputing nonrespondents’ data under various assumptions about the nonresponse mechanisms, (ii) take hypothetical samples of varying sizes from the completed samples, and (iii) compute and compare measures of accuracy and cost for different proposed sample sizes. As part of the methodology, we present a new approach for generating imputations for multivariate continuous data with nonignorable unit nonresponse. We fit mixtures of multivariate normal distributions to the respondents’ data, and adjust the probabilities of the mixture components to generate nonrespondents’ distributions with desired features. We illustrate the approaches using data from the 2007 U.S. Census of Manufactures.
References
1.
Andridge, R. and K.J. Thompson. 2015. “Assessing Nonresponse Bias in a Business Survey: Proxy Pattern-Mixture Analysis for Skewed Data.” Annals of Applied Statistics 9: 2237–2265. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1214/15-AOAS878.
2.
Andridge, R.R. and R.J. Little. 2011. “Proxy Pattern-Mixture Analysis for Survey Nonresponse.” Journal of Official Statistics 27(2): 153. Available at: http://www.jos.nu/articles/abstract.asp?article=272153.
3.
Daniels, M. and J. Hogan. 2008. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis . Chapman & Hall/CRC Press. Taylor & Francis, Boca Raton.
4.
Diggle, P. and M.G. Kenward. 1994. “Informative Drop-Out in Longitudinal Data Analysis.” Journal of the Royal Statistical Society, 43(1): 49–93. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2986113.
5.
Dunson, D.B. 2010. “Nonparametric Bayes Applications to Biostatistics.” In Bayesian Nonparametrics , edited by N. Hjort, C. Holmes, P. Müller, and S. Walker. Cambridge Series in Statistical and Probabilistic Mathematics, 223–273. Cambridge University Press.
6.
Dunson, D.B. and C. Xing. 2009. “Nonparametric Bayes Modeling of Multivariate Categorical Data.” Journal of the American Statistical Association 104(487): 1042–1051. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1198/jasa.2009.tm08439.
7.
Escobar, M.D. and M. West. 1995. “Bayesian Density Estimation and Inference Using Mixtures.” Journal of the American Statistical Association 90(430): 577–588. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1080/01621459.1995.10476550.
8.
Ferguson, T.S. 1973. “A Bayesian Analysis of Some Nonparametric Problems.” The Annals of Statistics 1(2): 209–230. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1214/aos/1176342360.
9.
Ferguson, T.S. 1983. “Bayesian Density Estimation by Mixtures of Normal Distributions.” In Recent Advances in Statistics 24: 287–302. New York Academic Press.
10.
Finamore, J., B. Reist, and S. Coffey. 2013. 2013 National Survey of College Graduates: A practice-based investigation of adaptive design. In AAPOR (American Association for Public Opinion Research) 68th Annual Conference, Boston, MA, May 16–19, 2013. Available at: https://www.aapor.org/Conference/Recent-Conferences/2013-Conference.aspx.
11.
Fraley, C. and A.E. Raftery. 2007. “Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering.” Journal of Classification 24(2): 155–181. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1007/s00357-007-0004-5.
12.
Ghosal, S. and A. van der Vaart. 2001. “Entropies and Rates of Convergence for Maximum Likelihood and Bayes Estimation for Mixtures of Normal Densities.” The Annals of Statistics 29(5): 1233–1263. Available at: https://www-jstor-org.web.bisu.edu.cn/stable/2699987.
13.
Ghosal, S. and A. van der Vaart. 2007. “Posterior Convergence Rates of Dirichlet Mixtures at Smooth Densities.” The Annals of Statistics 35(2): 697–723. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1214/009053606000001271.
14.
Glynn, R., N. Laird, and D. Rubin. 1986. “Selection Modeling Versus Mixture Modeling with Nonignorable Nonresponse.” In Drawing Inferences from Self-Selected Samples , edited by H. Wainer, 115–142. New York: Springer.
15.
Glynn, R.J., N.M. Laird, and D.B. Rubin. 1993. “Multiple Imputation in Mixture Models for Nonignorable Nonresponse with Follow-Ups.” Journal of the American Statistical Association 88(423): 984–993. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2290790.
16.
Groves, R. 2004. Survey Errors and Survey Costs . Wiley Series in Probability and Statistics. New York: Wiley.
17.
Ishwaran, H. and L.F. James. 2001. “Gibbs Sampling Methods for Stick-Breaking Priors.” Journal of the American Statistical Association 96(453). Doi: https://dx-doi-org.web.bisu.edu.cn/10.1198/016214501750332758.
18.
Kaputa, S.J., L. Bechtel, K.J. Thompson, and D. Whitehead. 2014. “Strategies for Subsampling Nonrespondents for Economic Programs.” In JSM Proceedings, Survey Research Methods Section. American Statistical Association, Alexandria, VA, August 2–7, 2014. Available at: http://ww2.amstat.org/meetings/JSM/2014/onlineprogram/AbstractDetails.cfm?abstractid=311881.
19.
Kim, H.J., L.H. Cox, A.F. Karr, J.P. Reiter, and Q. Wang. 2015. “Simultaneous Edit-imputation for Continuous Microdata.” Journal of the American Statistical Association 110(511): 987–999. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2139/ssrn.2698601.
20.
Kim, H.J., J.P. Reiter, Q. Wang, L.H. Cox, and A.F. Karr. 2014. “Multiple Imputation of Missing or Faulty Values Under Linear Constraints.” Journal of Business and Economic Statistics 32(3): 375–386. Doi: https://dx-doi-org.web.bisu.edu.cn/{10.1080/07350015.2014.885435}.
21.
Little, R. 1993. “Pattern-Mixture Models for Multivariate Incomplete Data.” Journal of the American Statistical Association 88(421): 125–134. Available at: https://www-jstor-org.web.bisu.edu.cn/stable/2290705.
22.
Little, R. 1994. “A Class of Pattern-Mixture Models for Normal Incomplete Data.” Biometrika 81(3): 471–483. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1093/biomet/81.3.471.
23.
Little, R. 1995. “Modeling the Drop-Out Mechanism in Repeated-Measures Studies.” Journal of the American Statistical Association 90(431): 1112–1121. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2291350.
24.
Little, R. 2008. “Selection and Pattern-Mixture Models.” In Longitudinal Data Analysis , edited by G. Fitzmaurice, M. Davidian, G. Verbeke, and G. Molenberghs, 409–431. Chapman and Hall/CRC: Boca Raton.
25.
Little, R. and D. Rubin. 2002. Statistical Analysis with Missing Data . Wiley Series in Probability and Statistics. Wiley: Hoboken, NJ. Second edition.
26.
Miller, P.V. 2013. “Adaptive Design at the Census Bureau – a New Way of Doing Business.” In Proceedings of AAPOR (American Association for Public Opinion Research) 68th Annual Conference, 171–171, Boston, MA, May 16–19, 2013. Available at: https://www.aapor.org/Conference/Recent-Conferences/2013-Conference.aspx.
27.
Molenberghs, G., M.G. Kenward, and E. Lesaffre. 1997. “The Analysis of Longitudinal Ordinal Data with Nonrandom Drop-Out.” Biometrika 84(1): 33–44. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1093/biomet/84.1.33.
28.
Molenberghs, G., B. Michiels, M.G. Kenward, and P.J. Diggle. 1998. “Monotone Missing Data and Pattern-Mixture Models.” Statistica Neerlandica 52(2): 153–161. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1111/1467-9574.00075.
29.
Müller, P. and R. Mitra. 2013. “Bayesian Nonparametric Inference – Why and How.” Bayesian Analysis 8(2): 269–302. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1214/13-BA811.
30.
Müller, P. and F.A. Quintana. 2004. “Nonparametric Bayesian Data Analysis.” Statistical Science 19(1): 95–110. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1214/088342304000000017.
31.
Murray, J.S. and J.P. Reiter. 2016. “Multiple Imputation of Missing Categorical and Continuous Values via Bayesian Mixture Models with Local Dependence.” Journal of the American Statistical Association 111(516): 1466–1479.
32.
R Core Team. 2013. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1080/01621459.2016.1174132.
33.
Rao, R.S., M.E. Glickman, and R.J. Glynn. 2008. “Stopping Rules for Surveys with Multiple Waves of Nonrespondent Follow-up.” Statistics in Medicine 27(12): 2196–2213. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1002/sim.3063.
34.
RStudio and Inc. 2014. “shiny: Web Application Framework for R. R package version 0.9.1”.
35.
Rubin, D. 1987. Multiple Imputation for Nonresponse in Surveys . Wiley: Hoboken, NJ.
36.
Rubin, D.B. 1976. “Inference and Missing Data.” Biometrika 63(3): 581–592. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2335739.
37.
Schouten, B., J. Bethlehem, K. Beullens, Ø. Kleven, G. Loosveldt, A. Luiten, K. Rutar, N. Shlomo, and C. Skinner. 2012. “Evaluating, Comparing, Monitoring, and Improving Representativeness of Survey Response Through r-Indicators and Partial r-Indicators.” International Statistical Review 80(3): 382–399. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1111/j.1751-5823.2012.00189.x.
38.
Schouten, B., F. Cobben, and J. Bethlehem. 2009. “Indicators for the Representativeness of Survey Response.” Survey Methodology 35(1): 101–113.
39.
Schouten, B., N. Shlomo, and C. Skinner. 2011. “Indicators for Monitoring and Improving Representativeness of Response.” Journal of Official Statistics 27(2): 1–24.
40.
Sethuraman, J. 1994. “A Constructive Definition of Dirichlet Priors.” Statistica Sinica 4: 639–650. Available at: https://www-jstor-org.web.bisu.edu.cn/stable/24305538.
41.
Shen, W., S.T. Tokdar, and S. Ghosal. 2013. “Adaptive Bayesian Multivariate Density Estimation with Dirichlet Mixtures.” Biometrika 100(3): 623–640. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1093/biomet/ast015.
42.
Thijs, H., G. Molenberghs, B. Michiels, G. Verbeke, and D. Curran. 2002. “Strategies to Fit Pattern-Mixture Models.” Biostatistics 3(2): 245–265. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1007/978-1-4757-3625-0_27.
43.
U. S. Census Bureau. 2015. Annual Survey of Manufactures (ASM). Available at: https://www.census.gov/programs-surveys/asm.html (accessed June 30, 2017).
44.
Wagner, J. 2008. “Adaptive Survey Design to Reduce Nonresponse Bias.” PhD thesis, University of Michigan. Available at: http://hdl.handle.net/2027.42/60831.
45.
Wagner, J. and T.E. Raghunathan. 2010. “New Stopping Rule for Surveys.” Statistics in Medicine 29(9): 1014–1024. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1002/sim.3834.
46.
West, M. and G. Cao. 1993. “Assessing Mechanisms of Neural Synaptic Activity.” In Case Studies in Bayesian Statistics , edited by Gatsonis, Constantine and Hodges, James S. and Kass, Robert E. and Singpurwalla, Nozer D. 416–428. Springer-Verlag: New York.
47.
West, M., P. Müller, and M.D. Escobar. 1994. “Hierarchical Priors and Mixture Models, with Application in Regression and Density Estimation.” In Aspects of Uncertainty: A Tribute to D.V. Lindley , edited by A. Smith and P. Freeman, 363–386. London: Wiley.
48.
Woo, M.-J., J.P. Reiter, and A.F. Karr. 2008. “Estimation of Propensity Scores Using Generalized Additive Models.” Statistics in Medicine 27(19): 3805–3816. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1002/sim.3278.
49.
Woo, M.-J., J.P. Reiter, A. Oganian, and A.F. Karr. 2009. “Global Measures of Data Utility for Microdata Masked for Disclosure Limitation.” Journal of Privacy and Confidentiality 1(1): 7. Available at: http://repository.cmu.edu/jpc/vol1/iss1/7.