Abstract
In hazard models, it is assumed that all heterogeneity is captured by a set of theoretically relevant covariates. In many applications however, there are ample reasons for unobserved heterogeneity due to omitted or unmeasured factors. If there is unmeasured frailty, the hazard will not only be a function of the covariates but also of the unmeasured frailty. This paper discusses the implications of unobserved heterogeneity on parameter estimates with application to the analysis of infant death on subsequent birth timing in Ghana and Kenya using DHS data. Using Lognormal Accelerated Failure Time models with and without frailty, we found that standard models that do not control for unobserved heterogeneity produced biased estimates by overstating the degree of positive dependence and underestimating the degree of negative dependence. The implications of the findings are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aalen O.O.: Effects of frailty in survival analysis. Stat. Methods Med. Res. 3, 227–243 (1994)
Addai I.: Ethnicity and contraceptive use in sub-Saharan Africa: the case of Ghana. J. Biosoc. Sci. 31, 105–120 (1999)
Bachieri A., Hinde A.: The proximate determinants of fertility and birth intervals in Egypt: an application of calendar data. Demogr. Res. 16(3), 59–96 (2007)
Baker M., Melino A.: Duration dependence and nonparametric heterogeneity: a Monte Carlo study. J. Econom. 96(2), 357–393 (1999)
Blossfeld H.-P., Hamerle A.: Unobserved heterogeneity in event history models. Qual. Quant. 26(2), 157–168 (1992)
Bongaarts J.: The causes of stalling fertility transitions. Stud. Family Plan. 37(1), 1–16 (2006)
Brass, W., Jolly, C.L. (eds.): Population dynamics of Kenya. National Research Council Working Group on Kenya (1993)
Clayton D.: A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151 (1978)
Cochrane S.: Effects of education and urbanization on fertility. In: Bulatao, R.A., Lee, R.D. (eds) Determinants of Fertility in Developing Countries, A Summary of Knowledge, Part B, Academic Press, New York (1983)
Congdon, P.: Occupation mobility in the Irish Republic. In: Uncles, M.D. (eds.) Longitudinal Data Analysis: Methods and Applications. Pion Limited, London (1988)
Defo K.B.: Fertility response to infant and child mortality in Africa with special reference to Cameroon. In: Montgomery, M.R., Cohen, B. (eds) From Death to Birth: Mortality Decline and Reproductive Change, National Academy Press, Washington D.C. (1998)
Flinn C.J., Heckman J.J.: Models for the analysis of labor force dynamics. Adv. Econom. 1, 35–95 (1981)
Ghana Statistical Service (GSS), Macro International (MI): Ghana demographic and health survey 1998. Gss and MI, Calverton, Maryland (1999)
Guiterrez R.G.: Parametric frailty and shared frailty survival models. Stata J. 2(1), 22–44 (2002)
Guo G., Rodriguez G.: Estimating a multivariate proportional hazards model for clustered data using EM Algorithm, with an application to child survival in Guatemala. J. Am. Stat. Assoc. 87(420), 969–976 (1992)
Gyimah S.O.: A cohort analysis of timing of the first birth and cumulative fertility in Ghana. Popul. Res. Policy Rev. 22(3), 251–266 (2003)
Gyimah S.O.: The dynamics of timing and spacing of births in Ghana. J. Comp. Family Stud. 36(1), 41–60 (2005)
Gyimah S.O., Rajulton F.: The physiological effects of infant death on fertility: A comparative analysis of DHS data from Ghana and Kenya. Soc. Biol. 49(1-2), 44–57 (2003)
Gyimah S.O., Rajulton F.: Intentional replacement of dead children in sub-Saharan Africa: evidence from Ghana and Kenya. Can. Stud. Popul. 34(1), 33–53 (2004)
Gyimah S.O., Maxim P., White J.: Cohort perspectives on educational attainment and timing of parenthood in Ghana. J. Family Econ. Issues 26(1), 123–142 (2005)
Heckman J., Singer B.: Population heterogeneity in demographic models. In: Land, K., Rogers, A. (eds) Multidimensional Mathematical Demography, Academic Press, New York (1982)
Heckman J., Singer B.: A method of minimizing the impact of distributional assumptions in econometric models for duration data. Econom. 52, 271–320 (1984)
Heckman J., Hotz J., Walker J.: New evidence on the timing and spacing of births. Am. Econ. Rev. Papers Proc. 75, 179–184 (1985)
Jenkins, S.P.: Survival Analysis (2005). http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/pdfs/ec968lnotesv6.pdf
Lancaster T.: The Econometric Analysis of Transition Data. Cambridge University Press, New York (1990)
Manda S.O.M., Meyer R.: Age at first marriage in Malawi: a Bayesian multilevel analysis using a discrete time-to-event model. J. R. Stat. Soc. A 168, 439–455 (2005)
Martin T.: Women’s education and fertility: results from 26 demographic and health surveys. Stud. Family Plan. 26, 187–202 (1995)
Montgomery, M.R., Cohen, B. (eds): From Death to Birth: Mortality Decline and Reproductive Change. National Academy Press, Washington D.C. (1998)
National Council for Population and Development (NCPD) [Kenya], Institute for Resource Development/Macro Systems Inc. (IRD): Kenya demographic and health survey 1989. NCPD and IRD, Columbia, Maryland (1989)
National Council for Population and Development (NCPD) [Kenya], Central Bureau of Statistics (CBS), Macro International Inc. (MI): Kenya demographic and health survey 1998. NCPD, CBS, and MI: Calverton, Maryland (1999)
Palloni A., Rafalimanana H.: The effects of infant mortality on fertility revisited: new evidence from Latin America. Demography 36(1), 41–58 (1999)
Richards, T.: Comparative Analysis of Fertility, Breastfeeding and Contraception: A Dynamic Model. Report 22. Panel on Fertility Determinants, National Academy of Sciences, Washington (1983)
Sastry N.: A nested frailty model for survival data, with an application to the study of child survival in Northeast Brazil. J. Am. Stat. Assoc. 92, 426–435 (1997)
StataCorp.: Stata Statistical Software: Release 9.0. Stata, College Station, TX (2005)
Trussell, J., Richards, T.: Correcting for unmeasured heterogeneity in hazard models using Heckman-Singer procedure. In: Tuma, N.B. (eds.) Sociological Methodology. Jossey-Bass Publishers, San Francisco (1985)
Trussell, J., Rodriguez, G.: Heterogeneity in demographic research. In: J. Adams, et al. (eds.) Convergent Issues in Genetics and Demography. Oxford University Press, New York (1990)
Trussell J., Martin L., Feldman R., Palmore J.A.: Determinants of birth interval length in the Philippines: a hazard model analysis. Demography 22(2), 145–168 (1985)
Vanden Berg G.J.: Duration models: specification, identification and multiple durations. In: Heckman, J.J., Leamer, E. (eds) Handbook of Econometrics, vol 5, North-Holland, Amsterdam (2001)
Vaupel J., Manton K., Stallard E.: Impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16(3), 439–454 (1979)
Westoff C.F.: Age at Marriage, Age at first Birth and Fertility in Africa, World Bank Technical Paper Number 169. The World Bank, Washington D.C. (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gyimah, S.O., Ezeh, A. & Fotso, J.C. Frailty models with applications to the study of infant deaths on birth timing in Ghana and Kenya. Qual Quant 46, 1505–1521 (2012). https://doi.org/10.1007/s11135-011-9464-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11135-011-9464-7