CHAPTER 6: SUMMARY AND FUTURE RESEARCH
A.3 Appendix for Chapter 5
1. Asymptotic properties for γˆ0 and Λˆ0
First we present asymptotic properties for the partial likelihood estimator ˆγ0 and the
(2007).
We impose the following assumptions:
A1. Let τ be the endpoint of the study. We assume that Λ0(τ)<∞.
A2. Let Bi(t) be the at risk process forith subject, we assume that P(Bi(t) = 1) >0 for allt ∈(0, τ].
A3. The covariate space, X is a compact subset of Rp for somep≥1.
A4. Let Xi˜ = (Xi, Yi) be the vector of covariates included in the Cox model (5.3). Some further definitions are:
S(0)(γ, t) = 1 n0 X i∈V0 Bi(t) exp{γ 0 ˜ Xi}, S(1)(γ, t) = 1 n0 X i∈V0 ˜
XiBi(t) exp{γ0Xi},˜ S(2)(γ, t) = 1 n0 X i∈V0 ˜ Xi⊗2Bi(t) exp{γ 0 ˜ Xi}, E(γ, t) = S (1)(γ, t) S(0)(γ, t), V(γ, t) = S (2)(γ, t) S(0)(γ, t)−E(γ, t) ⊗2 , where X˜i⊗2 = ˜XiX˜ 0
i. Notice that all the quantities defined above are based on the simple random sample portion of the data. Let s(0), s(1), s(2) be the limiting values ofS(0), S(1), S(2),
and define e=s(1)/s(0), v =s(2)/s(0)−e⊗2.
A5. The matrix Σ1 =
Rτ
0 v(γ, t)s
(0)(γ, t)λ
0(t)dt is positive definite.
A6. The Lindeberg condition as Assumption C from Andersen and Gill (1982).
Using the above assumptions, it is easy to know that the partial likelihood estimator ˆγ0
has the following asymptotic normal distributions:
n10/2(ˆγ0−γ∗)
D
−→N(0,Σ−11),
function at time (t, s)is Z t 0 e(γ∗, u)λ0(u)du 0 Σ−11 Z s 0 e(γ∗, u)λ0(u)du + Z min(t,s) 0 λ0(u) s(0)(γ∗, u)du. 2. Consistency of ξˆP
Recall that the log-likelihood function is
lN(ξ,Λ0, G) = X i∈V logfξ,Λ0(Ti,∆i, Yi|Xi)+ X i∈V logg(Xi)+ X j∈V¯ logn Z x fξ,Λ0(Tj,∆j, Yj|x)dG(x) o .
The profile likelihood function is
plN(ξ) = max
G∈G lN(ξ, G,Λ0).
The estimated log-likelihood ˆlN(ξ, G) is the same as lN(ξ,Λ0, G)except that Λ0 is replaced byΛˆ0. In addition,plbN(ξ) = maxG∈GˆlN(ξ, G).
In order to prove the consistency and asymptotic normality property of our proposed estimator ξˆP, we need to impose additional assumptions:
A7. The parameter space Ξ is a compact subspace ofRq, and that the true underlying valueξ∗ lies in the interior of the parameter space.
A8. f(T,∆, Y | X;ξ) is continuous in both(T, Y) and ξ and is strictly positive for all
(T,∆, Y, X) and ξ ∈ Ξ. Furthermore, the partial derivatives ∂f(T,∆, Y | X;ξ)/∂ξi and ∂2f(T,∆, Y | X;ξ)/∂ξi∂ξj, for i, j = 1, ..., q, exist and are continuous for all ξ ∈ Ξ. The same condition is also assumed for f(T,∆, Y).
A9. The class of functions F ≡n ∂ ∂ξlogfξ,Λ0(T,∆, Y |X) + ∂ ∂ξ log Z x fξ,Λ0(T,∆, Y |x)g(x)dx:ξ∈Ξ, t ∈[0, τ] o
is P-Donsker with square-integrable envelop function. A10. The class of functions
G ≡n ∂ 2 ∂ξ∂ξT logfξ,Λ0(T,∆, Y |X)+ ∂2 ∂ξ∂ξT log Z x fξ,Λ0(T,∆, Y |x)g(x)dx:ξ ∈Ξ, t∈[0, τ] o
is Glivenko-Cantelli and bounded in L1(P).
A11. The estimator ∂ξ∂ logfξ,Λ0ˆ (T,∆, Y |X) +∂ξ∂ log
R
xfξ,Λ0ˆ (T,∆, Y |x)g(x)dxbelongs to
F. The estimator ∂ξ∂ξ∂2T logfξ,Λ0ˆ (T,∆, Y |X) + ∂ 2
∂ξ∂ξT log
R
xfξ,Λ0ˆ (T,∆, Y |x)g(x)dx belongs
toG.
A12. J(ξ) is finite and positive definite at ξ∗.
Using the fact that Λ0ˆ is a consistent estimator for Λ0, and −1
N ∂2pl N(ξ) ∂ξ∂ξT P −→J(ξ), it can be shown that − 1 N ∂2 b plN(ξ) ∂ξ∂ξT P −→J(ξ)
holds uniformly for ξ in parameter space. This comes naturally from Lemma 1of Yuan and Jennrich (2000). In addition, let Uˆ(ξ)be the score equation of the profile likelihood plbN(ξ). That is, Uˆ(ξ) =∂plbN(ξ)/∂ξ. It can be shown that
1 N ˆ U(ξ∗) P −→0.
Using Assumption A12, we know that J(ξ) is invertible. Hence, we can apply a result of Foutz (1977), which uses Inverse Function Theorem to prove that our proposed estimator is a consistent and unique solution to the likelihood equations.
3. Asymptotic normality of ξˆP
The smoothness conditions of Theorem 1in Murphy and Van der Vaart (2000) can be verified based on the assumptions A7 - A11. Using this theorem, we know that for any random sequence ξ˜−→P ξ∗, b plN( ˜ξ) =plbN(ξ∗) + ( ˜ξ−ξ∗)T ∂plbN(ξ∗) ∂ξ + 1 2( ˜ξ−ξ∗) T∂2plbN(ξ∗) ∂ξ∂ξT ( ˜ξ−ξ∗) +oP( √ N||ξ˜−ξ∗||+ 1)2.
Using the similar argument as Corollary 1from Murphy and Van der Vaart (2000), we know that our proposed estimator ξˆP is asymptotically normal, and has the asymptotic expansion √ N( ˆξP −ξ∗) = " −1 N ∂2plbN(ξ∗) ∂ξ∂ξT #−1 1 √ N ∂plbN(ξ∗) ∂ξ ! +op(1).
We already proved that −1
N ∂2plb
N(ξ)
∂ξ∂ξT
P
−→ J(ξ). Let Gˆξ be the maximizer of G 7→ ˆlN(ξ, G), then 1 √ N ∂plbN(ξ) ∂ξ = 1 √ N N X i=1 ( Ri· ∂fξˆ(Ti,∆i, Yi |Xi)/∂ξ ˆ fξ(Ti,∆i, Yi |Xi) +(1−Ri)· R x∂fξˆ(Ti,∆i, Yi |x)/∂ξdGξˆ (x) R xfˆξ(Ti,∆i, Yi |x)dGˆξ(x) ) .
Using Liapunov’s central limit theorem, we have
1 √ N ∂plbN(ξ) ∂ξ D →N(0,Σ(ξ)),
where Σ(ξ) =var{M(ξ)}. The definition of M(ξ) is given by (5.11). Finally, using Slutsky Theorem, we conclude that √N( ˆξP −ξ∗)
D
REFERENCES
Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: a large sample study. The annals of statisticspages 1100–1120.
Apelberg, B. J., Witter, F. R., Herbstman, J. B., Calafat, A. M., Halden, R. U., Needham, L. L., and Goldman, L. R. (2007). Cord serum concentrations of perfluorooctane sulfonate (pfos) and perfluorooctanoate (pfoa) in relation to weight and size at birth. Environmental health perspectives pages 1670–1676.
Barlow, W. E. (1994). Robust variance estimation for the case-cohort design. Biometrics
pages 1064–1072.
Bickel, P. J., Klaassen, C. A., Ritov, Y., and Wellner, J. A. (1993). Efficient and adaptive
estimation for semiparametric models. Johns Hopkins University Press; Baltimore.
Borgan, O., Langholz, B., Samuelsen, S. O., Goldstein, L., and Pogoda, J. (2000). Exposure stratified case-cohort designs. Lifetime data analysis 6(1), 39–58.
Breslow, N. and Cain, K. (1988). Logistic regression for two-stage case-control data.
Biometrika75(1), 11–20.
Breslow, N. and Powers, W. (1978). Are there two logistic regressions for retrospective studies? Biometrics pages 100–105.
Cai, J. and Zeng, D. (2007). Power calculation for case–cohort studies with nonrare events.
Biometrics63(4), 1288–1295.
Carroll, R. J. and Wand, M. P. (1991). Semiparametric estimation in logistic measurement error models. Journal of the Royal Statistical Society. Series B 53,573–585.
Chen, H. Y. and Little, R. J. (1999). Proportional hazards regression with missing covariates.
Journal of the American Statistical Association 94,896–908.
Chen, K. (2001). Generalized case–cohort sampling. Journal of the Royal Statistical Society:
Series B63(4),791–809.
Chen, K. and Lo, S.-H. (1999). Case-cohort and case-control analysis with cox’s model.
Biometrika86(4), 755–764.
Chen, Y.-H., Chatterjee, N., and Carroll, R. J. (2009). Shrinkage estimators for robust and efficient inference in haplotype-based case-control studies. Journal of the American
Statistical Association 104(485),220–233.
Cornfield, J. (1951). A method of estimating comparative rates from clinical data: Applications to cancer in the lung, breast, and cervix. Journal of the National Cancer Institute 11, 1269–1275.
Cornfield, J. (1956). A statistical problem arising from retrospective studies. In Proceedings of the third Berkeley symposium on mathematical statistics and probability, volume 4, pages 135–148. University of California Press Berkeley.
Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of the Royal Statistical Society34, 187–220.
Cox, D. R. (1975). Partial likelihood. Biometrika 62(2),269–276.
Dar, E., Kanarek, M. S., Anderson, H. A., and Sonzogni, W. C. (1992). Fish consumption and reproductive outcomes in green bay, wisconsin. Environmental Research 59,189–201. Ding, J., Zhou, H., Liu, Y., Cai, J., and Longnecker, M. P. (2014). Estimating effect of
environmental contaminants on women’s subfecundity for the Moba study data with an outcome-dependent sampling scheme. Biostatistics 15(4),636–650.
Fein, G. G., Jacobson, J. L., Jacobson, S. W., Schwartz, P. M., and Dowler, J. K. (1984). Prenatal exposure to polychlorinated biphenyls: effects on birth size and gestational age.
The Journal of pediatrics 105, 315–320.
Flanders, W. D. and Greenland, S. (1991). Analytic methods for two-stage case-control studies and other stratified designs. Statistics in Medicine 10(5),739–747.
Foutz, R. V. (1977). On the unique consistent solution to the likelihood equations. Journal
of the American Statistical Association 72(357),147–148.
Grandjean, P., Bjerve, K. S., Weihe, P., and Steuerwald, U. (2001). Birthweight in a fishing community: significance of essential fatty acids and marine food contaminants.
International journal of epidemiology 30,1272–1278.
Grice, M. M., Alexander, B. H., Hoffbeck, R., and Kampa, D. M. (2007). Self-reported medical conditions in perfluorooctanesulfonyl fluoride manufacturing workers. Journal of
Occupational and Environmental Medicine49, 722–729.
He, J., Li, H., Edmondson, A. C., Rader, D. J., and Li, M. (2011). A Gaussian copula approach for the analysis of secondary phenotypes in case–control genetic association studies. Biostatistics 13(3), 497–508.
Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American statistical Association47(260), 663–685. Inoue, K., Okada, F., Ito, R., Kato, S., Sasaki, S., Nakajima, S., Uno, A., Saijo, Y., Sata, F., Yoshimura, Y., et al. (2004). Perfluorooctane sulfonate (pfos) and related perfluorinated compounds in human maternal and cord blood samples: assessment of pfos exposure in a susceptible population during pregnancy. Environmental health perspectives pages 1204–1207.
Jiang, Y., Scott, A. J., and Wild, C. J. (2006). Secondary analysis of case-control data.
Statistics in medicine 25(8),1323–1339.
Joan, H., Lawless, J. F., et al. (1997). Pseudolikelihood estimation in a class of problems with response-related missing covariates. Canadian Journal of Statistics 25(2),125–142.
Kang, S. and Cai, J. (2009a). Marginal hazards model for case-cohort studies with multiple disease outcomes. Biometrika 96(4),887–901.
Kang, S. and Cai, J. (2009b). Marginal hazards model for case-cohort studies with multiple disease outcomes. Biometrika 96,887.
Karmaus, W. and Zhu, X. (2004). Maternal concentration of polychlorinated biphenyls and dichlorodiphenyl dichlorethylene and birth weight in michigan fish eaters: a cohort study.
Environmental Health3, 1.
Kim, S., Cai, J., and Lu, W. (2013). More efficient estimators for case-cohort studies.
Biometrikapage ast018.
Kim, S., Sandler, D. P., Carswell, G., De Roo, L. A., Parks, C. G., Cawthon, R., Weinberg, C. R., and Taylor, J. A. (2011). Telomere length in peripheral blood and breast cancer risk in a prospective case-cohort analysis: results from the sister study. Cancer Causes &
Control 22(7), 1061–1066.
Kim, S., Taylor, J. A., Milne, G. L., and Sandler, D. P. (2013). Association between urinary prostaglandin E2 metabolite and breast cancer risk: a prospective, case–cohort study of postmenopausal women. Cancer Prevention Research 6(6), 511–518.
Kolaczyk, E. D. (1994). Empirical likelihood for generalized linear models. Statistica Sinica
pages 199–218.
Kong, L., Cai, J., and Sen, P. K. (2004). Weighted estimating equations for semiparametric transformation models with censored data from a case-cohort design. Biometrika91(2), 305–319.
Kulich, M. and Lin, D. (2004). Improving the efficiency of relative-risk estimation in case- cohort studies. Journal of the American Statistical Association 99(467),832–844.
Lawless, J., Kalbfleisch, J., and Wild, C. (1999). Semiparametric methods for response- selective and missing data problems in regression. Journal of the Royal Statistical Society:
Series B61(2),413–438.
Lee, A., McMURCHY, L., and Scott, A. (1997). Re-using data from case-control studies.
Statistics in medicine 16, 1377–1389.
Lee, A., Scott, A., and Wild, C. (2010). Efficient estimation in multi-phase case-control studies. Biometrika 97(2),361–374.
Liang, K.-Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73(1),13–22.
Lin, D. (2007). On the breslow estimator. Lifetime data analysis 13, 471–480.
Lin, D. and Ying, Z. (1993). Cox regression with incomplete covariate measurements. Journal
Lin, D. and Zeng, D. (2009). Proper analysis of secondary phenotype data in case-control association studies. Genetic epidemiology 33(3),256–265.
Little, R. J. (1992). Regression with missing X’s: a review. Journal of the American Statistical
Association 87(420), 1227–1237.
Longnecker, M., Klebanoff, M., Zhou, H., Wilcox, A., Berendes, H., and Hoffman, H. (1997). Proposal to study in utero exposure to DDE and PCBs in relation to male birth defects and neurodevelopmental outcomes in the Collaborative Perinatal Project. Study Proposal,
National Institute of Environmental Health Sciences, Washington, DC.
Longnecker, M. P., Klebanoff, M. A., Brock, J. W., and Guo, X. (2005). Maternal levels of polychlorinated biphenyls in relation to preterm and small-for-gestational-age birth.
Epidemiology 16,641–647.
Ma, Y. (2010). A semiparametric efficient estimator in case-control studies. Bernoulli16(2), 585–603.
Ma, Y. and Carroll, R. J. (2016). Semiparametric estimation in the secondary analysis of case–control studies. Journal of the Royal Statistical Society: Series B 78(1), 127–151. Monroy, R., Morrison, K., Teo, K., Atkinson, S., Kubwabo, C., Stewart, B., and Foster, W. G.
(2008). Serum levels of perfluoroalkyl compounds in human maternal and umbilical cord blood samples. Environmental Research 108,56–62.
Monsees, G. M., Tamimi, R. M., and Kraft, P. (2009). Genome-wide association scans for secondary traits using case-control samples. Genetic epidemiology 33(8),717–728.
Morris, P. G., Hudis, C. A., Giri, D., Morrow, M., Falcone, D. J., Zhou, X. K., Du, B., Brogi, E., Crawford, C. B., Kopelovich, L., et al. (2011). Inflammation and increased aromatase expression occur in the breast tissue of obese women with breast cancer. Cancer prevention
research 4(7), 1021–1029.
Murphy, L. E., Gollenberg, A. L., Louis, G. M. B., Kostyniak, P. J., and Sundaram, R. (2010). Maternal serum preconception polychlorinated biphenyl concentrations and infant birth weight. Environmental health perspectives 118,297.
Murphy, S. A. and Van der Vaart, A. W. (2000). On profile likelihood. Journal of the
American Statistical Association 95, 449–465.
Niswander, K. R. and Gordon, M. J. (1972). The women and their pregnancies: the Collaborative Perinatal Study of the National Institute of Neurological Diseases and Stroke.
U.S. Department of Health, Education, and Welfare Publication (NIH) 73–379. U.S. Government Printing Office, Washington, D. C.
Nolan, L. A., Nolan, J. M., Shofer, F. S., Rodway, N. V., and Emmett, E. A. (2009). The relationship between birth weight, gestational age and perfluorooctanoic acid (pfoa)- contaminated public drinking water. Reproductive Toxicology 27, 231–238.
Owen, A. (1990). Empirical likelihood ratio confidence regions. The Annals of Statistics
pages 90–120.
Owen, A. (1991). Empirical likelihood for linear models. The Annals of Statistics pages 1725–1747.
Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional.
Biometrika75(2), 237–249.
Pepe, M. S. and Fleming, T. R. (1991). A nonparametric method for dealing with mismeasured covariate data. Journal of the American Statistical Association 86(413),108–113.
Prentice, R. L. (1986). A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika73(1), 1–11.
Prentice, R. L. and Pyke, R. (1979). Logistic disease incidence models and case-control studies. Biometrika 66(3),403–411.
Qin, G. and Zhou, H. (2011). Partial linear inference for a 2-stage outcome-dependent sampling design with a continuous outcome. Biostatistics 12(3), 506–520.
Qin, J. (1993). Empirical likelihood in biased sample problems. The Annals of Statistics
pages 1182–1196.
Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. The Annals of Statistics pages 300–325.
Reilly, M. and Pepe, M. S. (1995). A mean score method for missing and auxiliary covariate data in regression models. Biometrika 82(2),299–314.
Richardson, D. B., Rzehak, P., Klenk, J., and Weiland, S. K. (2007). Analyses of case–control data for additional outcomes. Epidemiology 18(4), 441–445.
Robins, J. M., Rotnitzky, A., and Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. Journal of the American statistical Association
89(427),846–866.
Saarela, O., Kulathinal, S., Arjas, E., and Läärä, E. (2008). Nested case–control data utilized for multiple outcomes: a likelihood approach and alternatives. Statistics in medicine 27, 5991–6008.
Saarela, O., Kulathinal, S., and Karvanen, J. (2012). Secondary analysis under cohort sampling designs using conditional likelihood. Journal of Probability and Statistics2012,. Salim, A., Xiangmei, M., Jialiang, L., and Reilly, M. (2014). A maximum likelihood method
for secondary analysis of nested case-control data. Statistics in medicine 33, 1842–1852. Scheike, T. H. and Martinussen, T. (2004). Maximum likelihood estimation for cox’s regression
Scott, A. and Wild, C. (1991). Fitting logistic regression models in stratified case-control studies. Biometrics pages 497–510.
Self, S. G. and Prentice, R. L. (1988). Asymptotic distribution theory and efficiency results for case-cohort studies. The Annals of Statistics pages 64–81.
Shen, Y., Cai, T., Chen, Y., Yang, Y., and Chen, J. (2015). Retrospective likelihood-based methods for analyzing case-cohort genetic association studies. Biometrics71(4), 960–968. Song, R., Zhou, H., and Kosorok, M. R. (2009). A note on semiparametric efficient inference
for two-stage outcome-dependent sampling with a continuous outcome. Biometrika 96, 221–228.
Stein, C. R., Savitz, D. A., and Dougan, M. (2009). Serum levels of perfluorooctanoic acid and perfluorooctane sulfonate and pregnancy outcome. American journal of epidemiology
170,837–846.
Subbaramaiah, K., Howe, L. R., Bhardwaj, P., Du, B., Gravaghi, C., Yantiss, R. K., Zhou, X. K., Blaho, V. A., Hla, T., Yang, P., et al. (2011). Obesity is associated with inflammation and elevated aromatase expression in the mouse mammary gland. Cancer prevention research 4(3), 329–346.
Tsiatis, A. (2007). Semiparametric theory and missing data. Springer Science & Business Media.
Vardi, Y. (1985). Empirical distributions in selection bias models. The Annals of Statistics
pages 178–203.
Vartiainen, T., Jaakkola, J., Saarikoski, S., and Tuomisto, J. (1998). Birth weight and sex of children and the correlation to the body burden of pcdds/pcdfs and pcbs of the mother.
Environmental health perspectives106, 61.
Verner, M.-A., Loccisano, A. E., Morken, N.-H., Yoon, M., Wu, H., McDougall, R., Maisonet, M., Marcus, M., Kishi, R., Miyashita, C., et al. (2015). Associations of perfluoroalkyl substances (pfas) with lower birth weight: an evaluation of potential confounding by glomerular filtration rate using a physiologically based pharmacokinetic model (pbpk).
Environmental health perspectives123, 1317.
Wacholder, S., Gail, M. H., Pee, D., and Brookmeyer, R. (1989). Alternative variance and efficiency calculations for the case-cohort design. Biometrika 76(1),117–123.
Wang, X. and Zhou, H. (2006). A semiparametric empirical likelihood method for biased sampling schemes with auxiliary covariates. Biometrics 62(4),1149–1160.
Wang, X. and Zhou, H. (2010). Design and inference for cancer biomarker study with an outcome and auxiliary-dependent subsampling. Biometrics 66(2), 502–511.
Washino, N., Saijo, Y., Sasaki, S., Kato, S., Ban, S., Konishi, K., Ito, R., Nakata, A., Iwasaki, Y., Saito, K., et al. (2009). Correlations between prenatal exposure to perfluorinated chemicals and reduced fetal growth. Environmental Health Perspectives 117, 660.
Weaver, M. A. and Zhou, H. (2005). An estimated likelihood method for continuous outcome regression models with outcome-dependent sampling. Journal of the American Statistical
Association 100(470), 459–469.
Wei, J., Carroll, R. J., Müller, U. U., Keilegom, I. V., and Chatterjee, N. (2013). Robust estimation for homoscedastic regression in the secondary analysis of case–control data.
Journal of the Royal Statistical Society: Series B 75(1), 185–206.
White, J. E. (1982). A two stage design for the study of the relationship between a rare exposure and a rare disease. American Journal of Epidemiology 115(1), 119–128.
Whitworth, K. W., Haug, L. S., Baird, D. D., Becher, G., Hoppin, J. A., Skjaerven, R., Thomsen, C., Eggesbo, M., Travlos, G., Wilson, R., et al. (2012). Perfluorinated compounds and subfecundity in pregnant women. Epidemiology (Cambridge, Mass.)23(2),257. Yuan, K.-H. and Jennrich, R. I. (2000). Estimating equations with nuisance parameters:
Theory and applications. Annals of the Institute of Statistical Mathematics 52,343–350. Zeng, D. and Lin, D. (2014). Efficient estimation of semiparametric transformation models
for two-phase cohort studies. Journal of the American Statistical Association 109(505), 371–383.
Zhao, L. and Lipsitz, S. (1992). Designs and analysis of two-stage studies. Statistics in
medicine 11(6), 769–782.
Zhao, L. P., Lipsitz, S., and Lew, D. (1996). Regression analysis with missing covariate data