Abstract
Nested case–control sampled event time data under a highly stratified proportional hazards model, in which the number of strata increases proportional to sample size, is described and analyzed. The data can be characterized as stratified sampling from the event time risk sets and the analysis approach of Borgan et al. (Ann Stat 23:1749–1778, 1995) is adapted to accommodate both the stratification and case–control sampling from the stratified risk sets. Conditions for the consistency and asymptotic normality of the maximum partial likelihood estimator are provided and the results are used to compare the efficiency of the stratified analysis to an unstratified analysis when the baseline hazards can be semi-parametrically modeled in two special cases. Using the stratified sampling representation of the stratified analysis, methods for absolute risk estimation described by Borgan et al. (1995) for nested case–control data are used to develop methods for absolute risk estimation under the stratified model. The methods are illustrated by a year of birth stratified analysis of radon exposure and lung cancer mortality in a cohort of uranium miners from the Colorado Plateau.
Similar content being viewed by others
References
Aalen OO, Bjertness E, Sønju T (1995) Analysis of dependent survival data applied to lifetimes of amalgam fillings. Stat Med 14:1819–1829
Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120
Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York
Begun J, Hall W, Huang W, Wellner J (1983) Information and asymptotic efficiency in parametric-nonparametric models. Ann Stat 11:432–452
Borgan Ø (2002) Estimation of covariate-dependent Markov transition probabilities from nested case-control data. Stat Methods Med Res 11:183–202
Borgan Ø, Goldstein L, Langholz B (1995) Methods for the analysis of sampled cohort data in the Cox proportional hazards model. Ann Stat 23:1749–1778
Breslow N, Lubin J, Marek P, Langholz B (1983) Multiplicative models and the analysis of cohort data. J Am Stat Assoc 78:1–11
Cochran W (1963) Sampling techniques, 2nd edn. Wiley, New York
Cox DR (1972) Regression models and life-tables (with discussion). J R Stat Soc B 34:187–220
Gross ST, Huber C (1987) Matched pair experiments: Cox and maximum likelihood estimation. Scand J Stat 14:27–41
Heck J, Lombardi C, Meyers T, Cockburn M, Wilhelm M, Ritz B (2013a) Epidemiology of rhabdoid tumors in early childhood. Pediatr Blood Cancer 60:77–81
Heck J, Meyers T, Lombardi C, Park A, Cockburn M, Reynolds P, Ritz B (2013b) Case-control study of birth characteristics and the risk of hepatoblastoma. Cancer Epidemiol 37:390–395
Heck J, Wu J, Lombardi C, Qui J, Meyers T, Wilhelm M, Cockburn M, Ritz B (2013c) Childhood cancer and traffic-related air pollution exposure in pregnancy and early life. Environ Health Perspect 121:1385–1391
Hjort NL, Pollard D (1993) Asymptotics of minimizers of convex processes. Technical report 5/93, Department of Mathematics, University of Oslo, Oslo, Norway
Hornung R, Meinhardt T (1987) Quantitative risk assessment of lung cancer in US uranium miners. Health Phys 52:417–430
Langholz B, Borgan Ø (1995) Counter-matching: a stratified nested case-control sampling method. Biometrika 82:69–79
Langholz B, Borgan Ø (1997) Estimation of absolute risk from nested case-control data. Biometrics 53:767–774
Langholz B, Goldstein L (1996) Risk set sampling in epidemiologic cohort studies. Stat Sci 11:35–53
Langholz B, Tuomilehto-Wolf E, Thomas D, Pitkäniemi J, Tuomilehto J (1995) The DiMe Study Group (1995) variation in HLA-associated risks of childhood insulin dependent diabetes in the Finnish population: I. Allele effects at a, b, and dr loci. Genet Epidemiol 12:441–453
Langholz B, Thomas D, Stovall M, Smith S, Boice Jr J, Shore R, Bernstein L, Lynch C, Zhang X, Group TWS, Bernstein J (2009) Statistical methods for analysis of radiation effects with tumor and dose location-specific information with application to the wecare study of asynchronous contralateral breast cancer. Biometrics 65:599–608
Lubin J, Boice J, Edling C, Hornung R, Howe G, Kunz E, Kusiak R, Morrison H, Radford E, Samet J, Tirmarche M, Woodward A, Xiang Y, Pierce D (1994) Radon and lung cancer risk: a joint analysis of 11 underground miners studies. NIH Publication 94-3644, U.S. Department of Health and Human Services, Public Health Service, National Institutes of Health, Bethesda, MD
Lundin F, Wagoner J, Archer V (1971) Radon daughter exposure and respiratory cancer, quantitative and temporal aspects. Joint Monograph 1, U.S. Public Health Service, Washington, D.C
Struthers CA, Kalbfleisch JD (1986) Misspecified proportional hazard models. Biometrika 73:363–369
Tuomilehto J, Lounanmaa R, Tuomilehto-Wolf E, Reunanen A, Virtala E, Kaprio E, Åkerblom H, The DiMe Study Group (1992) Epidemiology of childhood diabetes mellitus in Finland: background of a nationwide study of Type 1 (insulin-dependent) diabetes mellitus. Diabetologia 35:70–76
Tuomilehto-Wolf E, Tuomilehto J, Cepaitis Z, Lounanmaa R, The DIME Study Group (1989) New susceptibility haplotype for type 1 diabetes. Lancet 2:299–302
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Goldstein, L., Langholz, B. Analysis and asymptotic theory for nested case–control designs under highly stratified proportional hazards models. Lifetime Data Anal 29, 342–371 (2023). https://doi.org/10.1007/s10985-022-09582-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10985-022-09582-4