Bayesian predictive inference of a proportion under a two-fold small area model with heterogeneous correlations
Section 4. Concluding remarks

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We have extended a homogeneous two-fold model, Nandram (2015) to a heterogeneous two-fold model which adds a degree of flexibility to our data analysis. Weakly identified parameters in these models posed serious computational problems. Therefore, we have done two additional things. First, we have introduced unimodal constraints on the parameters of the beta prior distributions. Second, we have used a blocked Gibbs sampler to perform the computations. To compare these models, we have performed a Bayesian predictive inference. As an illustrative example, we have used data from TIMSS, a study of the performance of US students at the third grade in mathematics. Also, we have performed a simulation study to compare these two two-fold models even further.

It is important to model the two-fold sample design using the heterogeneous model because for many applications the intracluster correlations may vary from area to area, making the heterogeneous two-fold model more appropriate than the homogeneous two-fold model. Indeed, using an illustrative example and the simulation study with several diagnostics, we have demonstrated that the heterogeneous two-fold model is to be preferred over the homogeneous two-fold model when the correlations vary significantly.

It is possible to extend our work to accommodate multivariate binary data. This can be viewed as a problem of pooling data from multinomial distributions in order to infer about the finite population proportions. For example, in TIMSS we can use both mathematics and science scores as bivariate binary responses (correlation). Then it is possible to develop a hierarchical Bayesian model for multinomial responses and a Dirichlet prior to the model of cell probabilities. In this study we can work with two issues. First, we can investigate how much the prediction will be improved when using the multivariate data. Second, we can also study how much the precision of inference will be improved when considering a model with heterogeneous intracluster correlations over one with homogeneous correlation with respect to the multivariate data.

Acknowledgements

The authors are grateful to the two reviewers for their careful reading of the manuscript and their suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2014R1A1A2058954). Also this work was supported by a grant from the Simons Foundation (#353953, Balgobin Nandram).

Appendix A

Proofs of formulas (2.12) and (2.13)

It is easy to show that

$$\begin{array}{ll}\text{Cov}\left(y_{ijk}\mathrm{,}{y}_{ij{k}^{\prime }}|{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\hfill & \mathrm{<mo>=</mo>}\text{Var}\left(p_{ij}|{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)={\mu }_i\left(1-{\mu }_i\right){\rho }_i\mathrm{,}\hfill \\ \text{Var}\left(y_{ijk}|{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\hfill & =E\left[\text{Var}\left(y_{ijk}|p_{ij}\mathrm{,}{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\right]+\text{Var}\left[E\left(y_{ijk}|p_{ij}\mathrm{,}{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\right]\mathrm{,}\hfill \\ \hfill & =E\left(p_{ij}|{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\left[1-E\left(p_{ij}|{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\right]={\mu }_i\left(1-{\mu }_i\right).\hfill \end{array}$$

Thus, $\text{Cor}\left(y_{ijk}\mathrm{,}{y}_{ij{k}^{\prime }}|{\mu }_i\mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\mathrm{<mo>=</mo>}{\rho }_i\left(k\ne k^{\prime }\right)\mathrm{,}$ thereby proving (2.12).

Similarly, it is easy to show that

$$\begin{array}{ll}\text{Cov}\left(y_{ijk},y_{i{j}^{\prime }{k}^{\prime }}|\theta \mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\hfill & \mathrm{<mo>=</mo>}E\left[\text{Cov}\left(p_{ij},p_{i{j}^{\prime }}|{\mu }_i\mathrm{,}\theta \mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\right]\hfill \\ \hfill & +\text{Cov}\left[E\left(p_{ij}|{\mu }_i\mathrm{,}\theta \mathrm{,}\gamma \mathrm{,}{\rho }_i\right),E\left(p_{i{j}^{\prime }}|{\mu }_i\mathrm{,}\theta \mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}\text{Var}\left({\mu }_i|\theta \mathrm{,}\gamma \right)\mathrm{<mo>=</mo>}\theta \left(1-\theta \right)\gamma \mathrm{,}\hfill \\ \text{Var}\left(y_{ijk}|\theta \mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\hfill & \mathrm{<mo>=</mo>}E\left({\mu }_i|\theta \mathrm{,}\gamma \right)-{\left[E\left({\mu }_i|\theta \mathrm{,}\gamma \right)\right]}^2\mathrm{<mo>=</mo>}\theta \left(1-\theta \right)\mathrm{.}\hfill \end{array}$$

Thus, $\text{Cor}\left(y_{ijk},y_{i{j}^{\prime }{k}^{\prime }}|\theta \mathrm{,}\gamma \mathrm{,}{\rho }_i\right)\mathrm{<mo>=</mo>}\gamma \left(j\ne j^{\prime }\mathrm{,}k\ne k^{\prime }\right),$ thereby proving (2.13).

Appendix B

Computation with unimodality constraints

It is well known that a beta pdf with parameters $\alpha$ and $\beta$ is unimodal if $\alpha \mathrm{<mo>></mo>\; 1}$ and $\beta \mathrm{<mo>></mo>\; 1.}$ This can be established easily using calculus. In our case, $\mu |\theta \mathrm{,}\gamma \sim \text{Beta}\left\{\theta \frac{\left(1-\gamma \right)}{\gamma }\mathrm{,}\left(1-\theta \right)\frac{\left(1-\gamma \right)}{\gamma }\right\}.$ Therefore, we have two inequalities,

and simple algebra gives

$$\frac{\gamma }{1-\gamma }\mathrm{<mo><</mo>}\theta \mathrm{<mo><</mo>}\frac{1-2\gamma }{1-\gamma }\mathrm{,}\mathrm{0\; <mo><</mo>}\gamma \mathrm{<mo><</mo>}\frac{1}{3}\mathrm{.}$$

Next, we describe briefly how to apply these constraints to the computation in the two-fold model with heterogeneous correlations. Recall the conditional marginal posterior distribution of $\gamma$ ,

$$p\left(\gamma |\rho ,\varphi ,\delta ,y\right)\approx {\displaystyle \sum _{g\mathrm{<mo>=</mo>\; 1}}^G{\omega }_{g}p}\left(x_g,\gamma |\rho ,\varphi ,\delta ,y\right),$$

where $\left\{{\omega }_g\right\}$ are the weights and $\left\{x_g\right\}$ are roots of the Legendre polynomial. Here, we use the univariate grid method to sample $\gamma .$ So, we divide the interval $\left(0,\frac{1}{3}\right),$ the first constraint, into G1 subintervals $\left[{\gamma }_0\mathrm{,}{\gamma }_1\right),\left[{\gamma }_1\mathrm{,}{\gamma }_2\right),\dots ,\left[{\gamma }_{G1-1}\mathrm{,}{\gamma }_{G1}\right].$ For a uniform random number, $u^{*},$ from any grid, say, $\left[{\gamma }_{\nu -1}\mathrm{,}{\gamma }_{\nu }\right),$ we compute the height, i.e., the value of the conditional marginal posterior density function of $\gamma$ as

$$\frac{1-3u^{*}}{1-u^{*}}{\displaystyle \sum _{g\mathrm{<mo>=</mo>\; 1}}^{G^{*}}{\omega }_g^{*}p\left(x_g^{*},u^{*}|\rho ,\varphi ,\delta ,y\right),}$$

where $\left\{{\omega }_g^{*}\right\}$ are the weights and $\left\{x_g^{*}\right\}$ are roots of the Legendre polynomial with the interval $\left[\frac{u^{*}}{1-u^{*}},\frac{1-2u^{*}}{1-u^{*}}\right],$ the second constraint. Similarly, we can apply unimodality criterion to sample $\left(\varphi \mathrm{,}\delta \right).$

References

Brier, S.S. (1980). Analysis of contingency tables under cluster sampling. Biometrika, 67, 591-595.

Caslyn, C., Gonzales, P. and Frase, M. (1999). Highlights from TIMSS. National Center for Education Statistics, Washington, DC.

Damien, P., Laud, P.W. and Smith, A.F.M. (1997). Bayesian estimation of unimodal distributions. Communications in Statistics, 26(2), 429-440.

Foy, P., Rust, K. and Schleicher, A. (1996). Sample design. In TIMSS Technical Report, Volume I: Design and Development, (Eds., M.O. Martin and D.L. Kelly), Chestnut Hill, MA: Boston College.

Fuller, W.A., and Battese, G.E. (1973). Transformations for estimation of linear models with nested-error structure. Journal of the American Statistical Association, 68, 626-632.

Ghosh, M., and Lahiri, P. (1988). Bayes and empirical Bayes analysis in multistage sampling. In Statistical Decision Theory and Related Topics IV, (Eds., S.S. Gupta and J.O. Berger), New York: Springer, Vol. 1, 195-212.

Molina, I., Nandram, B. and Rao, J.N.K. (2014). Small area estimation of general parameters with application to poverty indicators: A hierarchical Bayes approach. Annals of Applied Statistics, 8(2), 852-885.

Nandram, B. (1998). A Bayesian analysis of the three-stage hierarchical multinomial model. Journal of Statistical Computation and Simulation, 61, 97-126.

Nandram, B. (2015). Bayesian predictive inference of a proportion under a two-fold small area model. Journal of Official Statistics (accepted).

Nandram, B., and Sedransk, J. (1993). Bayesian predictive inference for a finite population proportion: Two-stage cluster sampling. Journal of the Royal Statistical Society, Series B, 55, 399-408.

Nandram, B., Bhatta, D., Sedransk, J. and Bhadra, B. (2013). A Bayesian test of independence in a two-way contingency table using surrogate sampling. Journal of Statistical Planning and Inference, 143, 1392-1408.

Ntzoufras, I. (2009). Bayesian Modeling Using WinBUGS. New Jersey: Wiley, Hoboken.

Rao, J.N.K., and Molina, I. (2015). Small Area Estimation. New York: John Wiley & Sons, Inc.

Rao, J.N.K., and Scott, A.J. (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.

Rao, J.N.K., and Scott, A.J. (1984). On chi-squared tests for multi-way tables with cell proportions estimated from survey data. Annals of Statistics, 12, 46-60.

Ritter, C., and Tanner, M.A. (1992). Facilitating the Gibbs sampler: The Gibbs stopper and the Griddy Gibbs sampler. Journal of the American Statistical Association, 87, 861-868.

Scott, A., and Smith, T.M.F. (1969). Estimation in multi-stage surveys. Journal of the American Statistical Association, 101, 1387-1397.

Stukel, D.M., and Rao, J.N.K. (1997). Estimation of regression models with nested error regression structure and unequal variances under two and three stage cluster sampling. Statistics & Probability Letters, 35, 401-407.

Stukel, D.M., and Rao, J.N.K. (1999). On small-area estimation under two-fold nested error regression models. Journal of Statistical Planning and Inference, 78, 131-147.

Toto, M.C.S., and Nandram, B. (2010). A Bayesian predictive inference for small area means incorporating covariates and sampling weights. Journal of Statistical Planning and Inference, 140, 2963-2979.

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