We extend the twofold small-area model of Stukel and Rao (1997; 1999) to accommodate binary data. An example is the Third International Mathematics and Science Study (TIMSS), in which pass-fail data for mathematics of students from US schools (clusters) are available at the third grade by regions and communities (small areas). We compare the finite population proportions of these small areas. We present a hierarchical Bayesian model in which the firststage binary responses have independent Bernoulli distributions, and each subsequent stage is modeled using a beta distribution, which is parameterized by its mean and a correlation coefficient. This twofold small-area model has an intracluster correlation at the first stage and an intercluster correlation at the second stage. The final-stage mean and all correlations are assumed to be noninformative independent random variables. We show how to infer the finite population proportion of each area. We have applied our models to synthetic TIMSS data to show that the twofold model is preferred over a onefold small-area model that ignores the clustering within areas. We further compare these models using a simulation study, which shows that the intracluster correlation is particularly important.
Ando, T. 2007. “Bayesian Predictive Information Criterion for the Evaluation of Hierarchical Bayesian and Empirical Bayes Models.” Biometrika 94: 443-458. Doi: http://dx.doi.org/10.1093/biomet/asm017.
Bedrick, E.J. 1983. “Adjusted Chi-Squared Tests for Cross-Classified Tables of Survey Data.” Biometrika 70: 591-595. Doi: http://dx.doi.org/10.1093/biomet/70.3.591.
Brier, S.S. 1980. “Analysis of Contingency Tables Under Cluster Sampling.” Biometrika 67: 591-596. Doi: http://dx.doi.org/10.1093/biomet/67.3.591.
Calsyn, C., P. Gonzales, and M. Frase. 1999. “Highlights from TIMSS.” National Center for Education Statistics, Washington, DC. Doi: http://mces.ed.gov/timss.
Datta, G.S. and M. Ghosh. 1991. “Bayesian Prediction in Linear Models: Applications to Small Area Estimation.” Annals of Statistics 19: 1748-1770.
Foy, P., K. Rust, and A. Schleicher. 1996. “Sample Design.” In TIMMS Technical Report, Volume I: Design and Development, edited by M.O. Martin and D.L. Kelly, pagenumber. Chestnut Hill, MA: Boston College.
Fuller, W.A. and G.E. Battese. 1973. “Transformations for Estimation of Linear Models with Nested-Error Structure.” Journal of the American Statistical Association 68: 626-632. Doi: http://dx.doi.org/10.1080/01621459.1973.10481396.
Gelfand, A., D. Dey, and H. Chang. 1992. “Model Determination using Predictive Distributions with Implementation via Sampling-based Methods.” In Bayesian Statistics 4, 147-168. New York: Oxford University Press.
Geisser, S. and W. Eddy. 1979. “A Predictive Approach to Model Selection.” Journal of the American Statistical Association 74: 153-160. Doi: http://dx.doi.org/10.1080/01621459.1979.10481632.
Gelman, A., J.B. Carlin, H.S. Stern, D.B. Dunson, A. Vehtari, and D.B. Rubin. 2013. Bayesian Data Analysis, 3rd ed. New York: Chapman & Hall/CRC.
Ghosh, M. and P. Lahiri. 1988. “Bayes and Empirical Bayes Analysis in Multistage Sampling.” In Statistical Decision Theory and Related Topics IV, Vol. 1, edited by S.S. Gupta and J.O. Berger. 195-212. New York: Springer.
Hamilton, J. 2009. President Obama, U.S. Secretary of Education Duncan Announce National Competition to Advance School Reform. U.S. Department of Education: Available at: http://www.ed.gov/news/pressreleases/2009/07/07242009.html.
Holt, D., A.J. Scott, and P.D. Ewings. 1980. “Chi-Squared Tests with Survey Data.” Journal of the Royal Statistical Society, Series A 143: 303-320. Doi: http://dx.doi.org/10.2307/2982131.
Malec, D. and J. Sedransk. 1985. “Bayesian Inference for Finite Population Parameters in Multistage Cluster Sampling.” Journal of the American Statistical Association 80: 897-902. Doi: http://dx.doi.org/10.1080/01621459.1985.10478200.
Molina, I., B. Nandram, and J.N.K. Rao. 2014. “Small Area Estimation of General Parameters with Application to Poverty Indicators: A Hierarchical Bayes Approach.” Annals of Applied Statistics 8: 852-885. Doi: http://dx.doi.org/10.1214/13-AOAS702.
Nandram, B. 2014. Bayesian Predictive Inference for a Proportion Under a Two-Fold Small Area Model. Technical Report, Department of Mathematical Sciences, Worcester Polytechnic Institute, 1-43. (Available on request.) Nandram, B., D.R. Bhatta, J. Sedransk, and D. Bhadra. 2013. “A Bayesian Test of Independence in a Two-Way Contingency Table Using Surrogate Sampling.” Journal of Statistical Planning and Inference 143: 1392-1408. Doi: http://dx.doi.org/10.1016/j.jspi.2013.03.011.
Nandram, B. 1998. “A Bayesian Analysis of the Three-Stage Hierarchical Multinomial Model.” Journal of Statistical Computation and Simulation 61: 97-126. Doi: http://dx.doi.org/10.1080/00949659808811904.
Nandram, B. and J. Sedransk. 1993. “Bayesian Predictive Inference for a Finite Population Proportion: Two-Stage Cluster Sampling.” Journal of the Royal Statistical Society, Series B 55: 399-408.
Natarajan, R. and R.E. Kass. 2000. “Reference Bayesian Methods for Generalized Linear Mixed Models.” Journal of the American Statistical Association 95: 227-237. Doi: http://dx.doi.org/10.1080/01621459.2000.10473916.
Rao, J.N.K. 2003. Small Area Estimation. New York: Wiley.
Rao, J.N.K. and A.J. Scott. 1981. “The Analysis of Categorical Data from Complex Sample Surveys: Chi-squared Tests for Goodness of Fit and Independence in Two-Way Tables.” Journal of the American Statistical Association 76: 221-230. Doi: http://dx.doi.org/10.1080/01621459.1981.10477633.
Rao, J.N.K. and A.J. Scott. 1984. “On Chi-Squared Tests for Multi-way Tables with Cell Proportions Estimated from Survey Data.” Annals of Statistics 12: 46-60.
Scott, A.J. and D. Holt. 1982. “The Effect of Two-Stage Sampling on Ordinary Least Squares Methods.” Journal of the American Statistical Association 77: 848-854. Doi: http://dx.doi.org/10.1080/01621459.1982.10477897.
Scott, A. and T.M.F. Smith. 1969. “Estimation in Multi-Stage Surveys.” Journal of the American Statistical Association 101: 1387-1397. Doi: http://dx.doi.org/10.1080/01621459.1969.10501015.
Silverman, B.W. 1986. Density Estimation for Statistics and Data Analysis. New York: Chapman & Hall.
Stukel, D.M. and J.N.K. Rao. 1997. “Estimation of Regression Models with Nested Error Regression Structure and Unequal Error Variances Under Two and Three Stage Cluster Sampling.” Statistics & Probability Letters 35: 401-407. Doi: http://dx.doi.org/10.1016/S0167-7152(97)86602-3.
Stukel, D.M. and J.N.K. Rao. 1999. “On Small-Area Estimation Under Two-Fold Nested Error Regression Models.” Journal of Statistical Planning and Inference 78: 131-147. Doi: http://dx.doi.org/10.1016/S0378-3758(98)00211-0.
Toto, M.C.S. and B. Nandram. 2010. “A Bayesian Predictive Inference for Small Area Means Incorporating Covariates and Sampling Weights.” Journal of Statistical Planning and Inference 140: 2963-2979. Doi: http://dx.doi.org/10.1016/j.jspi.2010.03.043.
Yan, G. and J. Sedransk. 2007. “Bayesian Diagnostic Techniques for Detecting Hierarchical Structure.” Bayesian Analysis 2: 735-760. Doi: http://dx.doi.org/10.1214/07-BA230.
Yan, G. and J. Sedransk. 2010. “A Note on Bayesian Residuals as a Hierarchical Model Diagnostic Technique.” Statistical Papers 51: 1-10. Doi: http://dx.doi.org/10.1007/s00362-007-0111-2.