The Annals of Applied Statistics

Space–time smoothing of complex survey data: Small area estimation for child mortality

Laina D. Mercer, Jon Wakefield, Athena Pantazis, Angelina M. Lutambi, Honorati Masanja, and Samuel Clark

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Many people living in low- and middle-income countries are not covered by civil registration and vital statistics systems. Consequently, a wide variety of other types of data, including many household sample surveys, are used to estimate health and population indicators. In this paper we combine data from sample surveys and demographic surveillance systems to produce small area estimates of child mortality through time. Small area estimates are necessary to understand geographical heterogeneity in health indicators when full-coverage vital statistics are not available. For this endeavor spatio-temporal smoothing is beneficial to alleviate problems of data sparsity. The use of conventional hierarchical models requires careful thought since the survey weights may need to be considered to alleviate bias due to nonrandom sampling and nonresponse. The application that motivated this work is an estimation of child mortality rates in five-year time intervals in regions of Tanzania. Data come from Demographic and Health Surveys conducted over the period 1991–2010 and two demographic surveillance system sites. We derive a variance estimator of under five years child mortality that accounts for the complex survey weighting. For our application, the hierarchical models we consider include random effects for area, time and survey and we compare models using a variety of measures including the conditional predictive ordinate (CPO). The method we propose is implemented via the fast and accurate integrated nested Laplace approximation (INLA).

Article information

Ann. Appl. Stat. Volume 9, Number 4 (2015), 1889-1905.

Received: November 2014
Revised: September 2015
First available in Project Euclid: 28 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian smoothing infant mortality small area estimation survey sampling


Mercer, Laina D.; Wakefield, Jon; Pantazis, Athena; Lutambi, Angelina M.; Masanja, Honorati; Clark, Samuel. Space–time smoothing of complex survey data: Small area estimation for child mortality. Ann. Appl. Stat. 9 (2015), no. 4, 1889--1905. doi:10.1214/15-AOAS872.

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  • Alkema, L. and New, J. R. (2014). Global estimation of child mortality using a Bayesian B-spline Bias-reduction model. Ann. Appl. Stat. 8 2122–2149.
  • Alkema, L., New, J. R., Pedersen, J., You, D. et al. (2014). Child mortality estimation 2013: An overview of updates in estimation methods by the United Nations inter-agency group for child mortality estimation. PloS ONE 9 e101112.
  • Allison, P. (1984). Event History Analysis: Regression for Longitudinal Event Data. Number 46. Sage, Thousand Oaks, CA.
  • Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics. Ann. Inst. Statist. Math. 43 1–59.
  • Binder, D. A. (1983). On the variances of asymptotically normal estimators from complex surveys. Int. Stat. Rev. 51 279–292.
  • Byass, P., Worku, A., Emmelin, A. and Berhane, Y. (2007). Dss and dhs: Longitudinal and cross-sectional viewpoints on child and adolescent mortality in Ethiopia. Population Health Metrics 5 12.
  • Clark, S. J., Wakefield, J., McCormick, T. and Michelle, R. (2012). Hyak mortality monitoring system innovative sampling and estimation methods: Proof of concept by simulation. Technical Report 118, Center for Statistics and the Social Sciences (CSSS), Univ. Washington.
  • Clark, S. J., Kahn, K., Houle, B., Arteche, A., Collinson, M. A., Tollman, S. M. and Stein, A. (2013). Young children’s probability of dying before and after their mother’s death: A rural South African population-based surveillance study. PLoS Med. 10 e1001409.
  • Demographic and Health Surveys (1992). Demographic Health Survey 1991/1992. Bureau of Statistics Planning Commission.
  • Demographic and Health Surveys (1997). Tanzania Demographic and Health Survey 1996. Bureau of Statistics Tanzania and Macro International Inc.
  • Demographic and Health Surveys (2000). Tanzania Demographic and Health Survey 1999. Bureau of Statistics Tanzania and Macro International Inc.
  • Demographic and Health Surveys (2005). Tanzania Demographic and Health Survey 2004–05. National Bureau of Statistics (NBS) Tanzania and ORC Macro.
  • Demographic and Health Surveys (2010). Tanzania Demographic and Health Survey 2010. National Bureau of Statistics (NBS) Tanzania and ICF Macro.
  • Dwyer-Lindgren, L., Kakungu, F., Hangoma, P., Ng, M., Wang, H., Flaxman, A. D., Masiye, F. and Gakidou, E. (2014). Estimation of district-level under-5 mortality in Zambia using birth history data, 1980–2010. Spat. Spatiotemporal Epidemiol. 11 89–107.
  • Fong, Y., Rue, H. and Wakefield, J. (2010). Bayesian inference for generalized linear mixed models. Biostatistics 11 397–412.
  • Fottrell, E., Enquselassie, F. and Byass, P. (2009). The distribution and effects of child mortality risk factors in Ethiopia: A comparison of estimates from dss and dhs. Ethiopian Journal of Health Development 23 163–168.
  • Gelman, A. (2007). Struggles with survey weighting and regression modeling. Statist. Sci. 22 153–164.
  • Hammer, G. P., Kouyaté, B., Ramroth, H. and Becher, H. (2006). Risk factors for childhood mortality in sub-Saharan Africa. A comparison of data from a demographic and health survey and from a demographic surveillance system. Acta Trop. 98 212–218.
  • Held, L., Schrödle, B. and Rue, H. (2010). Posterior and cross-validatory predictive checks: A comparison of MCMC and INLA. In Statistical Modelling and Regression Structures 91–110. Physica-Verlag/Springer, Heidelberg.
  • Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc. 47 663–685.
  • INDEPTH Network (2014). Health and demographic surveillance systems. Available at
  • Jenkins, S. P. (1995). Easy estimation methods for discrete-time duration models. Oxford Bulletin of Economics and Statistics 57 129–136.
  • Knorr-Held, L. (2000). Bayesian modelling of inseparable space–time variation in disease risk. Stat. Med. 19 2555–2567.
  • Lohr, S. L. (2010). Sampling: Design and Analysis, 2nd ed. Brooks/Cole, Cengage Learning, Boston, MA.
  • Lumley, T. (2004). Analysis of complex survey samples. Journal of Statistical Software 9 1–19.
  • Mercer, L., Wakefield, J., Chen, C. and Lumley, T. (2014). A comparison of spatial smoothing methods for small area estimation with sampling weights. Spat. Stat. 8 69–85.
  • Mercer, L. D., Wakefield, J., Pantazis, A., Lutambi, A., Masanja, H. and Clark, S. (2015). Supplement to “Space–time smoothing of complex survey data: Small area estimation for child mortality.” DOI:10.1214/15-AOAS872SUPP.
  • Paris21 (2014). Paris21: Partnership for statistics in development in the 21st century. Available at
  • Pedersen, J. and Liu, J. (2012). Child mortality estimation: Appropriate time periods for child mortality estimates from full birth histories. PLoS Med. 9 e1001289.
  • Plummer, M. (2008). Penalized loss functions for Bayesian model comparison. Biostatistics 9 523–539.
  • Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability 104. Chapman & Hall/CRC, Boca Raton, FL.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 319–392.
  • Rutstein, S. O. and Rojas, G. (2006). Tanzania Demographic and Health Survey 1996. ORC Macro, Calverton, MD.
  • Schrödle, B. and Held, L. (2011). Spatio-temporal disease mapping using INLA. Environmetrics 22 725–734.
  • Sørbye, S. H. and Rue, H. (2014). Scaling intrinsic Gaussian Markov random field priors in spatial modelling. Spat. Stat. 8 39–51.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 583–639.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2014). The deviance information criterion: 12 years on. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 485–493.
  • UN (2000). Millennium development goals. Available at
  • UN (2014a). Civil registration and vital statistics coverage. Available at
  • UN (2014b). Data revolution for sustainable development. Available at
  • UN (2014c). Millennium development goal number 4: Reduce by two thirds, between 1990 and 2015, the under-five mortality rate. Available at
  • UN (2014d). The post-2015 development agenda. Available at
  • UN (2014e). Sustainable development goals. Available at
  • USAID (2014). Demographic and health surveys. United States Agency for International Development. Available at
  • Wakefield, J. (2009). Multi-level modelling, the ecologic fallacy, and hybrid study designs. Int. J. Epidemiol. 38 330–336.
  • Wang, H., Liddell, C. A., Coates, M. M., Mooney, M. D., Levitz, C. E., Schumacher, A. E., Apfel, H., Iannarone, M., Phillips, B., Lofgren, K. T. et al. (2014). Global, regional, and national levels of neonatal, infant, and under-5 mortality during 1990–2013: A systematic analysis for the global burden of disease study 2013. The Lancet 384 957–979.
  • World Bank and World Health Organization (2014). Global civil registration and vital statistics scaling up investment plan 2015–2024. Available at
  • Ye, Y., Wamukoya, M., Ezeh, A., Emina, J. B. and Sankoh, O. (2012). Health and demographic surveillance systems: A step towards full civil registration and vital statistics system in sub-Sahara Africa? BMC Public Health 12 741.

Supplemental materials

  • Supplement to “Space–time smoothing models for complex survey data: Small area estimation for child mortality”. The organization of the supplementary material is as follows. In Section 1 we provide the details of the discrete survival model. In Section 2 we provide the derivation of the standard error for U5M. Section 3 describes a simulation study aimed to test the coverage performance of the derived standard error against the jackknife standard error used by DHS. In Section 4 we describe the hyperprior specifications for the Bayesian hierarchical model. Section 5 provides a summary of the posterior distribution of the random effects. In Section 6 we provide a comparison of weighted and unweighted direct estimates of U5M. In Section 7 we have included some exploratory analysis looking at the rates and magnitude of regional decreases in U5M and how they relate to the fourth millennium development goal of two thirds reduction in child mortality by 2015. The results of our model validation are presented in Section 8. Lastly, Section 9 includes example R code for the analyses.