Abstract
The inclusion of geographically-based information in many epidemiological studies has led to the development of statistical and estimation methods that account for spatially dependent risks of health outcomes across geographical areas. Many of these developments have been concerned with spatial modelling of aggregated count data, for example incidence rates of cancer at the small–area level. In this chapter, we consider individual timetoevent data, where the individual subjects are hierarchically nested in natural or administrative areas. The individual failure time data are modelled using proportional hazards models, which are modified to include both spatially uncorrelated and correlated area frailty random effects; the latter accounting for local spatial dependence in the data. This model is expanded to accommodate multiple failure events, where the set of within and between failure-event spatial frailty random effects are assumed to have a multivariate normal distribution. We illustrate the proposed methodology with an analysis of timing of first childbirth and timing of first marriage across health districts in South Africa for women aged between 15 and 49 years. For each failure event, the spatial dependence is modelled using a multiple membership multiple classification (MMMC) model. A multivariate version of the MMMC model is then used to obtain estimates of covariance parameters between various failure-event spatial random effects.
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References
Abrahantes, J. C., Legrand, C., Burzykowski, T., Janssen, P., Ducrocq, V., & Duchateau, L. (2007). Comparison of different estimation procedures for proportional hazards model with random effects. Computational Statistics and Data Analysis, 51, 3913–3930.
Andersen, P. K., & Gill, R. D. (1982). Cox’s regression models for counting processes. The Annals of Statistics, 10, 1100–1120.
Banerjee, S., & Carlin, B. P. (2003). Semiparametric spatio-temporal frailty modeling. Environmetrics, 14, 523–535.
Banerjee, S., Wall, M. M., & Carlin, B. P. (2003). Frailty modeling for spatially correlated survival data, with application to infant mortality in Minnesota. Biostatistics, 4, 123–142.
Banerjee, S., Carlin, B. P., Alan, E., & Gelfand, A. E. (2004). Hierarchical modeling and analysis for spatial data. Boca Raton: Chapman & Hall.
Besag, J., York, J., & Mollie, A. (1991). Bayesian image restoration, with two applications in spatial statistics (with discussion). Annals of the Institute of Statistical Mathematics, 43, 1–59.
Browne, W. J., Goldstein, H., & Rasbash, J. (2001). Multiple membership multiple classification (MMMC). Statistical Modelling, 1, 103–124.
Cai, B., & Meyer, R. (2011). Bayesian semiparametric modeling of survival data based on mixtures of B-spline distributions. Computational Statistics and Data Analysis, 55, 1260–1272.
Carlin, B. P., & Hodges, J. S. (1999). Hierarchical proportional hazards regression models for highly stratified data. Biometrics, 55, 1162–1170.
Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–152.
Clayton, D. G. (1991). A Monte Carlo method for Bayesian inference in frailty models. Biometrics, 47, 467–485.
Clayton, D. G., & Cuzick, J. (1985). Multivariate generalizations of the proportional hazards model (with discussion). Journal of the Royal Statistical Society A, 148, 82–117.
Department of Health, Medical Research Council, Measure DHS+. (2002). South Africa Demographic and Health Survey 1998. Pretoria: Department of Health.
Feltbower, R. G., Manda, S. O. M., Gilthorpe, M. S., Greaves, M. F., Parslow, R. C., Kinsey, S. E., Bodansky, H. J., Patricia, A., & McKinney, P. A. (2005). Detecting small area similarities in the epidemiology of childhood acute lymphoblastic leukaemia and type 1 diabetes: A Bayesian approach. American Journal of Epidemiology, 161, 1168–1180.
Gupta, N., & Mahy, M. (2003). Adolescent childbearing in sub-Saharan Africa: Can increased schooling alone raise ages at first birth? Demographic Research, 8, 93–106.
Hougaard, P. (2000). Analysis of multivariate survival data. New York: Springer.
Human Sciences Research Council. (2009) Teeange pregancy in South Africa-with specific focus on school-going children (Full report). Pretoria: Human Sciences Research Council.
Jewkes, R., Morrell, R., & Christofides, N. (2009). Empowering teen ages to prevent pregnancy: Lessons from South Africa. Culture, Health and Sexuality, 11, 675–688.
Kalule-Sabiti, I., Palamuleni, M., Makiwane, M., & Amoateng, A. Y. (2007). Family formation and dissolution patterns. In A. Y. Amoateng & T. B. Heaton (Eds.), Families and households in post-apartheid South Africa: Socio-demographic perspectives (pp. 89–112). Cape Town: HSRC Press.
Leyland, A. H., Langford, I. H., Rasbash, J., & Goldstein, H. (2000). Multivariate spatial models for event data. Statistics in Medicine, 19, 2469–2478.
Lloyd, C. B., & Mensch, B. S. (2006). Marriage and childbirth as factors in school exit: An analysis of DHS data from sub-Saharan Africa. New York: Population Council.
Lunn, D. J., Thomas, A., Best, N., & Spiegelhalter, D. (2000). WinBUGS: A Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.
Magadi, M. (2004). Poor pregnancy outcomes among adolescents in South Nyansa region of Kenya. Working paper: A04/04 Statistical Sciences Research Institute. University of Southampton.
Manda, S. O. M. (2011). A nonparametric frailty model for clustered survival data. Communications in Statistics – Theory and Methods, 40(5), 863–875.
Manda, S. O. M., & Leyland, A. (2007). An empirical comparison of maximum likelihood and Bayesian estimation methods for multivariate spatial disease model. South African Statistical Journal, 41, 1–21.
Manda, S. O. M., & Meyer, R. (2005). Age at first marriage in Malawi: A Bayesian multilevel analysis using a discrete time-to-event model. Journal of the Royal Statistical Society A, 168, 439–455.
Manda, S. O. M., Feltbower, R. G., & Gilthorpe, M. S. (2009). Investigating spatio-temporal similarities in the epidemiology of childhood leukaemia and diabetes. European Journal of Epidemiology, 24, 743–752.
Oakes, D. (1982). A concordance test for independence in the presence of censoring. Biometrics, 38, 451–455.
Palamuleni, M. E. (2011). Socioeconomic determinant of age at first marriage in Malawi. International Journal of Sociology and Anthropology, 3, 224–235.
Palamuleni, M. E., Kalule-Sabiti, I., Makiwane, M. (2007). Fertility and child bearing in South Africa. In A. Y. Amoateng & T. B. Heaton (Eds.), Families and households in post-apartheid South Africa: Sociol-demographics perspectives (pp. 113–134). Cape Town: HSRC Press.
Sargent, D. J. (1998). A general framework for random effects survival analysis in the Cox proportional hazards setting. Biometrics, 54, 1486–1497.
Sastry, N. (1997). A nested frailty model for survival data, with an application to the study of child survival in northeast Brazil. Journal of the American Statistical Association, 92, 426–435.
Sharma, A. K., Verma, K., & KhatriS, K. A. T. (2003). Determinants of pregnancy in adolescents in Nepal. Indian Journal of Paediatrics, 69, 19–22.
South, S. J. (1993). Racial and ethnic differences in the desire to marry. Journal of Marriage and Family, 55, 357–370.
Statistics South Africa. (2010). Estimation of fertility from the 2007 Community Survey of South Africa/Statistics South Africa. Pretoria: Statistics South Africa.
Upchurch, D. M., Levy-Storms, L., Sucoff, C. A., & Aneshensel, C. S. (1998). Gender and ethnic differences in the timing of first sexual intercourse. Family Planning Perspectives, 30, 121–127.
Vaupel, J. W., Manton, K. G., & Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16, 439–454.
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Manda, S.O.M., Feltbower, R.G., Gilthorpe, M.S. (2012). A Multivariate Random Frailty Effects Model for Multiple Spatially Dependent Survival Data. In: Tu, YK., Greenwood, D. (eds) Modern Methods for Epidemiology. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-3024-3_9
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