Skip to main content
SpringerLink
Log in

Robust small area estimation in generalized linear mixed models

  • Published:
METRON Aims and scope Submit manuscript

Abstract

Small area estimation with categorical outcomes often requires intensive computation, as the marginal likelihood does not have a closed form in general. The likelihood analysis is further complicated by deviations in distributional assumptions often arise through outliers in the data. In this paper, the author proposes a robust method for estimating the small area parameters. Finite-sample properties of the estimators are investigated using Monte Carlo simulations. The empirical study shows that the proposed robust method is very useful for bounding the influence of outliers on the small area estimators. To approximate the mean squared errors of the estimators, a parametric bootstrap method is adopted. An application is also provided using actual data from a public health survey.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Battese, G.E., Harter, R.M., Fuller, W.A.: An error-components model for prediction of county crop areas using survey and satellite data. J. Am. Stat. Assoc. 83, 28–36 (1988)

    Article  Google Scholar 

  2. Cantoni, E., Ronchetti, E.: Robust inference for generalized linear models. J. Am. Stat. Assoc. 96, 1022–1030 (2001)

    Article  MathSciNet  Google Scholar 

  3. Chambers, R.L., Tzavidis, N.: M-quantile models for small area estimation. Biometrika 93, 255–268 (2006)

    Article  MathSciNet  Google Scholar 

  4. Datta, G.S., Rao, J.N.K., Smith, D.D.: On measuring the variability of small area estimators under a basic area level model. Biometrika 92, 183–196 (2005)

    Article  MathSciNet  Google Scholar 

  5. Franco, C., Bell, W.R.: Borrowing information over time in binomial/logit normal models for small area estimation. Joint Issue Stat. Transit. Survey Methodol. 16, 563–584 (2015)

    Google Scholar 

  6. Ghosh, M., Lahiri, P.: A hierarchical Bayes approach to small area estimation with auxiliary information (with discussion). Bayesian Analysis in Statistics and Econometrics, pp. 107–125. Springer, Berlin (1992)

    Chapter  Google Scholar 

  7. Ghosh, M., Natarajan, K., Stroud, T.W.F., Carlin, B.P.: Generalized linear models for small-area estimation. J. Am. Stat. Assoc. 93, 273–282 (1998)

    Article  MathSciNet  Google Scholar 

  8. Hall, P., Maiti, T.: On parametric bootstrap methods for small area prediction. J. Roy. Stat. Soc. B 68, 221–238 (2006)

    Article  MathSciNet  Google Scholar 

  9. Jiang, J., Lahiri, P.: Empirical best prediction for small area inference with binary data. Ann. Inst. Stat. Math. 53, 217–243 (2001)

    Article  MathSciNet  Google Scholar 

  10. Jiang, J., Lahiri, P.: Mixed model prediction and small area estimation. Test 15, 1–96 (2006)

    Article  MathSciNet  Google Scholar 

  11. Jiang, J., Lahiri, P., Wan, S.-M.: A unified jackknife theory for empirical best prediction with M-estimation. Ann. Stat. 30, 1782–1810 (2002)

    Article  MathSciNet  Google Scholar 

  12. Maiti, T.: Robust generalized linear mixed models for small area estimation. J. Stat. Plan. Inf. 98, 225–238 (2001)

    Article  MathSciNet  Google Scholar 

  13. Malec, D., Sedransk, J., Moriarity, C.L., LeClere, F.B.: Small area inference for binary variables in the national health interview survey. J. Am. Stat. Assoc. 92, 815–826 (1997)

    Article  Google Scholar 

  14. McCulloch, C.E.: Maximum likelihood algorithms for generalized linear mixed models. J. Am. Stat. Assoc. 92, 162–170 (1997)

    Article  MathSciNet  Google Scholar 

  15. Prasad, N.G.N., Rao, J.N.K.: The estimation of the mean squared error of small area estimators. J. Am. Stat. Assoc. 85, 163–171 (1990)

    Article  MathSciNet  Google Scholar 

  16. Preisser, J.S., Qaqish, B.F.: Robust regression to clustered data with application to binary responses. Biometrics 55, 574–579 (1999)

    Article  Google Scholar 

  17. Rao, J.N.K.: Small area estimation. Wiley, New Jersey (2003)

    Book  Google Scholar 

  18. Rao, J.N.K.: Interplay between sample survey theory and practice: an appraisal. Survey Methodol. 31, 117–138 (2005)

    Google Scholar 

  19. Rao, J.N.K., Sinha, S.K., Dumitrescu, L.: Robust small area estimation under semi-parametric mixed models. Can. J. Stat. 42, 126–141 (2014)

    Article  MathSciNet  Google Scholar 

  20. Rousseeuw, P.J., van Zomeren, B.C.: Unmasking multivariate outliers and leverage points. J. Am. Stat. Assoc. 85, 633–639 (1990)

    Article  Google Scholar 

  21. Salibian-Barrera, M., Van Aelst, S.: Robust model selection using fast and robust bootstrap. Comput. Stat. Data Anal. 52, 5121–5135 (2008)

    Article  MathSciNet  Google Scholar 

  22. Sinha, S.K.: Robust analysis of generalized linear mixed models. J. Am. Stat. Assoc. 99, 451–460 (2004)

    Article  MathSciNet  Google Scholar 

  23. Sinha, S.K., Rao, J.N.K.: Robust small area estimation. Can. J. Stat. 37, 381–399 (2009)

    Article  MathSciNet  Google Scholar 

  24. Torabi, M., Shokoohi, F.: Non-parametric generalized linear mixed models in small area estimation. Can. J. Stat. 43, 82–96 (2015)

    Article  MathSciNet  Google Scholar 

  25. Tzavidis, N., Ranalli, M.G., Salvati, N., Dreassi, E., Chambers, R.L.: Robust small area prediction for counts. Stat. Methods Med. Res. 24, 373–395 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was partially supported by a Grant from the Natural Sciences and Engineering Research Council (NSERC Grant no. RGPIN-2016-06258) of Canada.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Sanjoy K. Sinha

Authors
  1. Sanjoy K. Sinha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanjoy K. Sinha.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

1.1 A1: Variance of the RML Estimator

The variance-covariance matrix of the robust estimators \(\hat{\varvec{\gamma }} = ({\hat{\beta }}, {\hat{\sigma }}_u)^t\) in Sect. 4 may be approximated by

$$\begin{aligned} \text {var} (\hat{\varvec{\gamma }}) \approx {\mathbf {M}}_k^{-1} (\hat{\varvec{\gamma }}) \, {\mathbf {Q}}_k (\hat{\varvec{\gamma }}) \, {\mathbf {M}}_k^{-1} (\hat{\varvec{\gamma }}), \end{aligned}$$

where \({\mathbf {M}}_k (\varvec{\gamma })\) and \({\mathbf {Q}}_k (\varvec{\gamma })\) may be partitioned as

$$\begin{aligned} {\mathbf {M}}_k (\varvec{\gamma })= & {} \left( \begin{matrix} {M}_{\beta \beta } &{} {M}_{\beta \sigma }\\ {M}_{\sigma \beta } &{} {M}_{\sigma \sigma }\\ \end{matrix} \right) , \end{aligned}$$

and

$$\begin{aligned} {\mathbf {Q}}_k (\varvec{\gamma })= & {} \left( \begin{matrix} {Q}_{\beta \beta } &{} {Q}_{\beta \sigma }\\ {Q}_{\sigma \beta } &{} {Q}_{\sigma \sigma }\\ \end{matrix} \right) , \end{aligned}$$

with components

$$\begin{aligned} {M}_{\beta \beta }= & {} - \sum _{i=1}^k \sum _{j=1}^n \left. E \left\{ w_{ij} (\varvec{\gamma }, {v}_i) D_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) \, \right| \, {\mathbf {y}}_i \right\} {\mathbf {x}}_{ij} \, {\mathbf {x}}_{ij}^t\\&+ \sum _{i=1}^k E \left. \left[ \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij} \right\} \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij}^t \right\} \right| \, {\mathbf {y}}_i \right] \\&- \sum _{i=1}^k \left. E \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij} \right| \, {\mathbf {y}}_i \right\} E \left. \left\{ \sum _{t=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij}^t \right| \, {\mathbf {y}}_i \right\} , \\ {M}_{\beta \sigma }= & {} - \sum _{i=1}^k \sum _{j=1}^n \left. E \left\{ w_{ij} (\varvec{\gamma }, {v}_{i}) D_{ij} (\varvec{\gamma }, y_{ij}, {v}_{i}) \, {\mathbf {x}}_{ij} {\mathbf {h}}_{ij}^{t} \, \right| \, {\mathbf {y}}_i \right\} \\&+ \sum _{i=1}^k E \left. \left[ \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij} \right\} \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) \, {\mathbf {h}}_{ij}^{t} \, \right\} \right| \, {\mathbf {y}}_i \right] \\&- \sum _{i=1}^k \left. E \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij} \, \right| \, {\mathbf {y}}_i \right\} E \left. \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) \, {\mathbf {h}}_{ij}^t \, \right| \, {\mathbf {y}}_i \right\} , \\ {M}_{\sigma \beta }= & {} - \sum _{i=1}^k \sum _{j=1}^n \left. E \left\{ w_{ij0} (\varvec{\gamma }, {v}_{i}) D_{ij} (\varvec{\gamma }, y_{ij}, {v}_{i}) \, {\mathbf {h}}_{ij} {\mathbf {x}}_{ij}^t \, \right| \, {\mathbf {y}}_i \right\} \\&+ \sum _{i=1}^k E \left. \left[ \left\{ \sum _{j=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij} \right\} \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) \, {\mathbf {x}}_{ij}^t \, \right\} \right| \, {\mathbf {y}}_i \right] \\&- \sum _{i=1}^k \left. E \left\{ \sum _{j=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij} \, \right| \, {\mathbf {y}}_i \right\} E \left. \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) \, {\mathbf {x}}_{ij}^t \, \right| \, {\mathbf {y}}_i \right\} , \\ {M}_{\sigma \sigma }= & {} - \sum _{i=1}^k \sum _{j=1}^n \left. E \left\{ w_{ij0} (\varvec{\gamma }, {v}_i) D_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij} \, {\mathbf {h}}_{ij}^{t} \, \right| \, {\mathbf {y}}_i \right\} \\&+ \sum _{i=1}^k E \left. \left[ \left\{ \sum _{j=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij} \right\} \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij}^{t} \right\} \right| \, {\mathbf {y}}_i \right] \\&- \sum _{i=1}^k \left. E \left\{ \sum _{j=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij} \right| \, {\mathbf {y}}_i \right\} E \left. \left\{ \sum _{j=1}^n g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij}^{t} \right| \, {\mathbf {y}}_i \right\} , \end{aligned}$$

with \(g_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) = y_{ij} - \mu _{ij} (\varvec{\gamma }, {v}_i)\), and

$$\begin{aligned}&{Q}_{\beta \beta }\\&\quad = \sum _{i=1}^k E \left. \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij} \right| {\mathbf {y}}_i \right\} E\\&\qquad \left. \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij}^t \right| {\mathbf {y}}_i \right\} , \\&{Q}_{\beta \sigma }\\&\quad = \sum _{i=1}^k E \left. \left\{ \sum _{j=1}^n w_{ij} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {x}}_{ij} \right| {\mathbf {y}}_i \right\} E\\&\qquad \left. \left\{ \sum _{t=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij}^{t} \right| {\mathbf {y}}_i \right\} , \\&{Q}_{\sigma \sigma }\\&\quad = \sum _{i=1}^k E \left. \left\{ \sum _{j=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij} \right| {\mathbf {y}}_i \right\} E\\&\qquad \left. \left\{ \sum _{j=1}^n w_{ij0} (\varvec{\gamma }, {v}_i) d_{ij} (\varvec{\gamma }, y_{ij}, {v}_i) {\mathbf {h}}_{ij}^{t} \right| {\mathbf {y}}_i \right\} . \end{aligned}$$

Also, we have \({Q}_{\sigma \beta } = {Q}_{\beta \sigma }^t\).

1.2 A2: Predicted area means and root MSPEs for CCHS data

See Table 6.

Table 6 Predicted area means and root MSPEs
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sinha, S.K. Robust small area estimation in generalized linear mixed models. METRON 77, 201–225 (2019). https://doi.org/10.1007/s40300-019-00161-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40300-019-00161-6

Keywords

Search

Navigation