Bayesian inference for a variance component model using pairwise composite likelihood with survey data
Section 5. Conclusion

• Previous

There are well-known philosophical and foundational reasons for considering Bayesian approaches to survey sampling, and there is a long tradition of research in this area. See for example Sedransk (2008). There are also practical advantages. Using a Bayesian approach rather than a frequentist one relies much less on approximations, substituting computation for asymptotic expressions. In the context of random effects models, an important advantage is the ability to constrain the variance components to be non-negative in the prior distribution, without masking deficiencies in the data.

One example where Bayesian methods are used extensively is at the National Agricultural Statistical Service (NASS) of the US Department of Agriculture. At NASS, Bayesian methods are used to produce official statistics at the county and state levels for variables such as planted crop acreage and crop yield. Commonly, these inferences use several data sources. There is special attention to consistent estimation across the hierarchy of geographical areas of interest for inference. See Nandram, Berg and Barboza (2014); Erciulescu, Cruze and Nandram (2020, 2019, 2018); and Cruze, Erciulescu, Nandram, Barboza and Young (2019) for additional details.

We have investigated a use of pairwise composite likelihood in Bayesian inference for survey data, in the sense of developing a posterior distribution for mean $\theta$ and standard deviation parameter ${\sigma }_u$ of a simple random effects model. We have evaluated the posterior distribution in terms of the frequentist coverage properties of credible intervals for the parameters, and found them to work well for $\theta$ but not to be fully satisfactory for inference about ${\sigma }_u$ for the settings considered. There would be corresponding implications for frequentist inference from the pairwise composite likelihood, treated as an approximate likelihood function. It is possible that better results might be obtainable through applying a suitable transformation to ${\sigma }_u,$ and this is a subject of future research.

An ideal situation for the use of composite likelihood in Bayesian inference is one where (a) a model for generation of the data is fully specified, so that a true likelihood function exists, and (b) the true likelihood can be reasonably approximated by the composite likelihood, so that the corresponding posterior distributions agree well. For example, for Stoehr and Friel (2018) the motivation is the use for Bayesian inference of a pseudo-likelihood for data from a Gibbs random field. They establish identities that link the gradient and the Hessian of the log posterior for a parameter to moments of sufficient statistics of the random field, and use these to improve the ability of the log pairwise posterior density to approximate the log posterior density function. The curvature adjustment of RCD, upon which we have based our approach, instead adjusts the log pairwise composite likelihood so that its gradient (which we might call the “pairwise score vector”) has the information-unbiasedness property that leads to credible intervals with frequentist coverage probabilities approximating nominal values. Intuitively, with the increase of the number $n$ of clusters, $m$ remaining fixed, this approximation should improve, and its computation does not require the use of properties of the likelihood itself. In this paper, we have used the availability of the full likelihood in the simple case to evaluate how closely Bayesian inference based on the adjusted pairwise composite likelihood resembles full Bayesian inference.

Acknowledgements

The research is partially supported by grants to Thompson and Yi from the Natural Sciences and Engineering Research Council of Canada (NSERC). Yi is Canada Research Chair in Data Science (Tier 1). Her research was undertaken, in part, thanks to funding from the Canada Research Chairs Program.

References

Cox, D.R., and Reid, N. (2004). A note on pseudolikelihood constructed from marginal densities. Biometrika, 91, 729-737.

Cruze, N., Erciulescu, A., Nandram, B., Barboza, W. and Young, L. (2019). Producing official county-level agricultural estimates in the United States: Needs and challenges. Statistical Science, 34, 301-316.

Erciulescu, A., Cruze, N. and Nandram, B. (2018). Benchmarking a triplet of official estimates. Environmental and Ecological Statistics, 23, 523-547.

Erciulescu, A., Cruze, N. and Nandram, B. (2019). Model-based county level crop estimates incorporating auxiliary sources of information. Journal of the Royal Statistical Society, Series A, 182, 283-303.

Erciulescu, A., Cruze, N. and Nandram, B. (2020). Statistical challenges in combining survey statistics and auxiliary data to produce official statistics. Journal of Official Statistics, 36, 63-88.

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 3, 515-533.

Heyde, C.C. (1997). Quasi-Likelihood and its Application: A General Approach to Optimal Parameter Estimation. New York: Springer-Verlag.

Jørgensen, B., and Knudsen, S.J. (2004). Parameter orthogonality and bias adjustment for estimating functions. Scandinavian Journal of Statistics, 31, 93-114.

Lehmann, E.L. (1999). Elements of Large-Sample Theory. New York: Springer-Verlag.

Léon-Novelo, L., and Savitsky, T. (2019). Fully Bayesian estimation under informative sampling. Electronic Journal of Statistics, 13, 1608-1645.

Lindsay, B.G. (1982). Conditional score functions: Some optimality results. Biometrika, 69, 505-512.

Lindsay, B.G. (1988). Composite likelihood methods. Contemporary Mathematics, 80, 220-239.

Lindsay, B.G., Yi, G.Y. and Sun, J. (2011). Issues and strategies in the selection of composite likelihoods. Statistica Sinica, 21, 71-105.

Nandram, B., Berg, E. and Barboza, W. (2014). A hierarchical Bayesian model for forecasting state-level corn yield. Journal of Environmental and Ecological Statistics, 21, 507-530.

Pfeffermann, D., Skinner, C.J., Holmes, D.J., Goldstein, H. and Rasbash, J. (1998). Weighting for unequal selection probabilities in multi-level models. Journal of the Royal Statistical Society, Series B, 60, 23-56.

Rao, J.N.K., Verret, F. and Hidiroglou, M.A. (2013). A weighted composite likelihood approach to inference for two-level models from survey data. Survey Methodology, 39, 2, 263-282. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/2013002/article/11887-eng.pdf.

Rabe-Hesketh, S., and Skrondal, A. (2006). Multilevel modeling of complex survey data. Journal of the Royal Statistical Society, Series A, 169, 805-827.

Ribatet, M., Cooley, D. and Davison, A.C.D. (2012). Bayesian inference from composite likelihoods, with an application to spatial extremes. Statistica Sinica, 22, 813-845.

Sedransk, J. (2008). Assessing the value of Bayesian methods for inference about finite population quantities. Journal of Official Statistics, 24, 495-506.

Skinner, C.J., Holt, D. and Smith, T.F.M. (1989). Analysis of Complex Surveys. Wiley.

Stoehr, J., and Friel, N. (2018). Calibration of conditional composite likelihood for Bayesian inference on Gibbs random fields. Artificial Intelligence and Statistics, 921-929. arXiv:150201997v2.

Varin, C. (2008). On composite marginal likelihoods. AStA Advances in Statistical Analysis, 92, 1. https://doi.org/10.1007/s10182-008-0060-7 (accessed July 1, 2020).

Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 5-24.

Yi, G.Y. (2017). Composite likelihood/pseudolikelihood. Wiley StatsRef: Statistics Reference Online. 1-14.

Yi, G.Y., Rao, J.N.K. and Li, H. (2016). A weighted composite likelihood approach for analysis of survey data under two-level models. Statistica Sinica, 26, 569-587.

• Previous
• Article main page