Research ArticleAdditive Hazard Regression for the Analysis of Clustered SurvivalData from Case-Cohort Studies

June Liu 1 and Yi Zhang2

1School of Economics and Management Huaibei Normal University Huaibei China2School of Insurance Shanghai Lixin University of Accounting and Finance Shanghai China

Correspondence should be addressed to June Liu xun4025126com

Received 6 March 2020 Revised 11 June 2020 Accepted 23 June 2020 Published 16 July 2020

Academic Editor Antonio Di Crescenzo

Copyright copy 2020 June Liu and Yi Zhang is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

e case-cohort design is an effective and economical method in large cohort studies especially when the disease rate is low Case-cohort design in most of the existing literature is mainly used to analyze the univariate failure time data But in practicemultivariate failure time data are commonly encountered in biomedical research In this paper we will propose methods based onestimating equation method for case-cohort designs for clustered survival data By introducing the event failure rate threedifferent weight functions are constructed en three estimating equations and parameter estimators are presented Fur-thermore consistency and asymptotic normality of the proposed estimators are established Finally the simulation results showthat the proposed estimation procedure has reasonable finite sample behaviors

1 Introduction

In many failure time studies there is a natural clustering ofstudy subjects such that failure times within the same clustermay be correlated One example of clinical trial is that thefailure times of patients within a famlily may be correlatedbecause they have common genetic characteristics and envi-ronmental factors Another example is the time to onset ofblindness in the left and right eyes of the patients with diabeticretinopathy So in this case the focus is on the potentialclustering in the data Moreover the data structure is that theobservations from different clusters are independent whileobservations within a cluster may be correlated Interestedreaders can refer to the papers [1ndash4] and the references therein

In epidemiological research the collection of detailedfollow-up information is costly and time consuming especiallywhen the incidence of disease is low One may consider thecase-cohort design proposed by Prentice [5] which is widelyused for the large cohort studies It entails collecting covariatedata for all subjects who experienced the event of interest in thefull cohort and for a random sample from the entire cohort Ithas the same goals in studying risk factor as collecting the full

dataerefore many studies about case-cohort data have beenyielded For example it has been applied to the Cox pro-portional risk model [6] additive risk regression model [7]semiparametric transfer model [8] accelerated failure timemodel [9] and additive multiple risk regression model [10 11]

However there is little research on group failure timedata under case-cohort design For example Lu and Shih[12] applied the proportional risk model to clustered failuretime data under case cohort and proposed three designmethods of case cohort to extract subcolumns Under dif-ferent sampling mechanisms the estimation equations areestablished and then the large sample nature of the obtainedestimation is proved Unfortunately the risk set consideredby Lu and Shih [12] only contains the individuals in thesubcolumns Furthermore Zhang et al [13] added thefailure information out of the subcolumns to the propor-tional risk model and proposed three kinds of differentestimation equations and parameter estimates e nu-merical simulation results show that the proposed model iseffective Note that the above literature studied the clustereddata from case cohort based on the marginal proportionalhazard models In some practical cases the effect of

HindawiJournal of MathematicsVolume 2020 Article ID 9587870 10 pageshttpsdoiorg10115520209587870

covariates may be additive en the proportional riskmodel is no longer applicable erefore in this paper anadditive risk model will be applied to clustered data in case-cohort design Moreover we will construct the weightedestimation equation propose the estimation of regressionparameters and prove the asymptotic properties of theestimators

e remainder of the paper is organized as follows InSection 2 we describe the proposed estimation proceduresen we establish the consistency and asymptotic normalityof the resulting estimator in Section 3 Numerical evidencein support of the theory is presented in Section 4 Finally inSection 5 some conclusions are drawn

2 Model and Estimation Procedures

Suppose the full cohort consists of n independent clustersand the i-th cluster (i 1 n) has mi correlated subjectsWe assume that subjects within the same cluster are ex-changeable In advance of follow-up a random sample of theentire cohort called the subcohort is selected Covariate dataare then collected from individuals in the subcohort as well asthose observed to fail in the entire cohort Now we considerthe following three designs to obtain the subcohort [13]

Design A randomly sample individuals from eachcluster with Bernoulli sampling In other words eachindividual in each cluster has an independent fixedprobability of being selected to the subcohortDesign B randomly sample clusters from the full co-hort with Bernoulli samplingDesign C randomly sample clusters from the full co-hort with Bernoulli sampling and then randomlysample subjects with Bernoulli sampling from the se-lected clusters

Note that Design A and Design B are special cases ofDesign C

Let Tij and Cij denote the potential failure time and thepotent censoring time for the j-th subject in the i-th clusterrespectively Let Zij(t) represent a possibly time-dependentp times 1 vector of covariates restricted to be external Weassume that Tij and Cij are independent conditional on thegiven Zij(middot) e observed time is Xij min(Yij Cij) Letδij I(Tij leCij) be the indicator of failure Nij(t)

I(Xij le t δij 1) and Yij(t) I(Xij ge t)Let Hi denote the indicator for the i-th subject being

selected into the subcohort and Hij denote the indicator forthe jth subject in the i-th cluster being selected into thesample Both Hi and Hij are the independent Bernoullivariables where E(Hi) cE(Hij) θ Under Design AHi 1 for i 1 n Under Design B Hij 1 fori 1 n j 1 mi

Suppose that the marginal hazard function λij(t) isassociated with Zij(t) as follows

λij t | Zij(t)1113966 1113967 λ0(t) + βT0 Zij(t) (1)

where λ0(middot) is an unspecified baseline hazard function and β0is a p times 1 vector of regression parameters

If the full cohort data are available the estimate of thetrue regression parameter β0 in (1) could be obtained bysolving the following estimating function [14]

UH(β) 1113944n

i11113944

mi

j11113946τ

0Zij(t) minus ZH(t)1113966 1113967 dNij(t) minus Yij(t)βT

Zij(t)dt1113966 1113967

(2)

where ZH(t) (1113936ni11113936

mi

j1Yij(t)Zij(t)1113936ni11113936

mi

j1Yij(t))en the estimator of β0 in (1) can be estimated by 1113954βH

which is the solution to the estimating equations UH(β) 0at is

1113954βH 1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2⎡⎢⎢⎣ ⎤⎥⎥⎦

minus 1

middot 1113944n

i11113944

mi

j11113946τ

0Zij(t) minus Z(t)1113966 1113967dNij(t)⎡⎢⎢⎣ ⎤⎥⎥⎦

(3)

where τ ltinfin aotimes2 aaTFor clustered failure time data from case-cohort studies

we consider the observed-event probability Pr(δij 1)which we denote by p0 [15] We propose three procedures toestimate β0 which are the same designs proposed by Zhanget al [13] except that we consider the additive hazard re-gression model We then develop a weighted estimatingequation as follows

U(βp) 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967 dNij(t) minus Yij(t)βT

Zij(t)dt1113966 1113967

(4)

where Z(t) (1113936ni11113936

mi

j1ρijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ρij (p(N1N))δij + ((1 minus p)(n0N))(1 minus δij)HiHij withN 1113936

ni1miN1 1113936

ni11113936

mi

j1δij andn0 1113936ni11113936

mi

j1(1 minus δ)HiHijBy solving the estimation equation U(β p0) 0 we

could obtain 1113954β to estimate β0

1113954β 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967

otimes2dt⎡⎢⎢⎣ ⎤⎥⎥⎦

minus 1

middot 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967dNij(t)⎡⎢⎢⎣ ⎤⎥⎥⎦

(5)

However p0 is usually not known in most situations Incase where the study cohort is well defined we can use 1113954pwthe full cohort case proportion to estimate p When thestudy cohort is less well defined 1113954ps the subcohort caseproportion is a suitable alternative e correspondingestimator of β0 is written as 1113954βw or 1113954βs

e cumulative baseline hazard functionΛ0(t) 1113938

t

0 λ0(u)du can be consistently estimated by

1113954Λ0(t 1113954β 1113954p) 1113946t

0

1113936ni11113936

mi

j1ρij dNij(u) minus Yij(u)βTZij(u)du1113966 1113967

1113936ni11113936

mi

j1ρijYij(u)

(6)

In (4) and (6) either 1113954ps 1113954pw or p0 could be used toreplace the parameter 1113954p

2 Journal of Mathematics

3 Asymptotic Properties

To derive the asymptotic properties of the proposed esti-mates we impose the following assumptions

(C1) (Ti Ci Zi(middot) i 1 n) are independently andidentically distributed where Ti (Ti1 Timi

)T

Ci (Ci1 Cimi)T and Zi (Zi1(middot) Zimi

(middot))T

(C2) Pr Yij(τ)gt 01113966 1113967gt 0 and Λ0(τ)ltinfin for i 1 n

j 1 mi(C3) e matrix E W1(β0 p0)

otimes21113966 1113967 is positive definite

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G2i(p)

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(7)

(C4) ere exists a constant Cz such that |Zij(0) +

1113938τ0 dZij(u)|ltCz ltinfin holds almost everywhere

Theorem 1 Under the regularity conditions (C1)ndash(C4)nminus (12)U(β0 p0) converges to a mean zero normal distribu-tion with covariance 1113936(β0 p0) E W1(β0 p0)

otimes21113966 1113967 with

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i(p)

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

D1(β) E 1113944

m1

j1

δ1j

μ1113946τ

0Z1j(t) minus e(t)1113966 1113967dM1j(t)

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

D2(β) E 1113944

m1

j1

1μcθ

1 minus δ1j1113872 1113873H1H1j 1113946τ

0Z1j(t) minus e(t)1113966 1113967dM1j(t)

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

G1i(p) nminus 1

1113944

n

i1mi minus μ⎛⎝ ⎞⎠

G2i(p) nminus 1

1113944

mi

j1δij minus μp⎛⎝ ⎞⎠

G3i(p) nminus 1

1113944

mi

j11 minus δij1113872 1113873HiHij minus μcθ(1 minus p)

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

e(t) E Y11(t)Z11(t)1113864 1113865

E Z11(t)1113864 1113865

(8)

Theorem 2 Under conditions (C1)ndash(C4) 1113954βt converges inprobability to β0 and each of n12(1113954βt minus β0) converges indistribution to a mean zero normal distribution with co-variance matrix

Theorem 3 Under conditions (C1)ndash(C4) both 1113954βs and 1113954βw

converge in probability to β0 and each of n12(1113954βs minus β0) andn12(1113954βw minus β0) converges in distribution to a zero meannormal with covariance matrices A(β0)

minus 1Ωs(β0 p0)A(β0)minus 1

Journal of Mathematics 3

and A(β0)minus 1Ωw(β0 p0)A(β0)

minus 1 respectively where fora s or w Ωa(β) E ψ1(β p0)

otimes21113966 1113967 ψi(β p) Wi(β p) +

B(β)Qai (p) Ωa(β) E ψ1(β p0)

otimes21113966 1113967 ψi(β p) Wi(β p) +

B(β)Qai (p) Qs

i (p) (1μcθ)1113936mi

j1HiHij(δij minus p) Qwi (p)

(1μ)1113936mi

j1(δij minus p) B(β) 1113938τ0 ηijYij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

and ηij (1(N1N))δij minus (1(n0N))(1 minus δij)HiHij

Theorem 4 Under the regularity conditions (C1)ndash(C4)for each k 1 n 1113954Λ0(1113954β 1113954p t) converges in probability toΛ0(t) uniformly in t isin [0 τ] In addition n12 1113954Λ0(1113954β 1113954p t)minus1113966

Λ0(t) converges weakly to a Gaussian process with zero meanand covariance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t)

4 Numerical Studies

Simulation studies are conducted to examine the finitesample properties of the proposed estimators In the studywe generate the clustered failure time data from n 100clusters mi are simulated from a binomial (50 08) distri-bution for i 1 n with u E(mi) 40 e totalnumber in the whole queue is about 4000 e covariate Zij

takes values 1 and 0 with probabilities 05 and 05 re-spectively e failure time Ti1 Ti2 Timi

in the i-thcluster is simulated by the joint survival function (for detailssee [16])

S ti1 timi

11138681113868111386811138681113868 zi1 zimi1113874 1113875

1113944

mi

j1exp

1113938tij

0 2t + βTzij1113872 1113873dt

κ⎡⎢⎢⎣ ⎤⎥⎥⎦ minus mi minus 1( 1113857⎛⎝ ⎞⎠

minus κ

(9)

where κgt 0 is a parameter representing the degree of de-pendence among variables Smaller value of κ representsstronger correlation

In our experiments we choose κ 05 or 2 and β0

log(05) or 0e censoring times Cij are constant and equalto 1 e observed-event probabilities of p are equal to 045or 063

For each data generation for Design A individualswithin each cluster are selected into the subcohort byBernoulli sampling with equal probability 02 or 015 ForDesign B we select clusters by Bernoulli sampling withprobability 02 or 015 For Design C we first sample clustersby Bernoulli sampling with probability 04 or 03 and thensample individuals from those selected clusters by Bernoullisampling with probability 05 erefore for each design wewould expect approximately 800 or 600 individuals in thesubcohort

To assess the performance of the proposed estimator wecalculate the estimated standard error based on the boot-strap method Because the failure clusters inside and outsidethe subcohort have different structures we use the followingmethod to conduct bootstrap sampling Suppose the failureclusters in the subcohort contain M failures and Q non-failures whereas the failure clusters outside the subcohortcontain D failures only For each bootstrap sampling wepropose to obtain a bootstrap sample by sampling D failuresfrom the original D failures outside the subcohort withreplacement and then samplingM failures and Q nonfailuresfrom the original M failures and Q nonfailures in thesubcohort respectively e simulation results are based on1000 replications

Tables 1 and 2 report the simulation results of ourproposed estimators For simplicity the notation Bias de-notes the difference of the average of the estimates and the

Table 1 Simulation results for the estimation of β0 log(05)

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC 00010 00568 00574 00500 0948 minus 00021 00677 00635 00678 0944WC minus 00026 00526 00576 00550 0969 minus 00023 00603 00646 00630 0966T minus 00021 00533 00525 00496 0953 minus 00015 00594 00597 00642 0956

2SC 00015 00573 00569 00506 0948 minus 00020 00668 00656 00672 0942WC minus 00031 00531 00571 00560 0953 minus 00031 00625 00648 00602 0953T minus 00016 00529 00536 00562 0942 minus 00010 00584 00603 00618 0947

B

05SC minus 00035 00578 00569 00460 0937 minus 00012 00683 00644 00766 0920WC 00023 00593 00612 00609 0943 00004 00642 00640 00635 0938T minus 00030 00534 00511 00605 0934 minus 00039 00629 00613 00614 0936

2SC minus 00012 00568 00573 00589 0941 minus 00009 00679 00654 00685 0920WC 00027 00578 00581 00513 0939 00015 00653 00637 00591 0938T minus 00019 00534 00526 00480 0940 minus 00026 00629 00613 00557 0936

C

05SC minus 00009 00576 00570 00551 0948 minus 00019 00685 00642 00651 0929WC minus 00032 00543 00571 00571 0958 minus 00039 00605 00650 00632 0963T minus 00015 00540 00518 00481 0941 minus 00047 00605 00591 00679 0948

2SC minus 00021 00576 00570 00515 0948 minus 00024 00679 00653 00614 0934WC minus 00017 00547 00562 00565 0942 minus 00026 00614 00647 00644 0956T minus 00010 00549 00523 00510 0937 minus 00038 00587 00604 00613 0943

Estimate of β0 from five methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw T using true value p0

4 Journal of Mathematics

true values SE denotes the average of standard errors SEEdenotes the average asymptotic standard error BSE denotesthe estimated standard error based on the bootstrap methodand CP denotes the coverage probability of the nominal 95confidence interval

It can be seen from Tables 1 and 2 that the estimates ofboth regression parameters β0 log(05) and β0 0 areunbiased and both the estimated standard deviation and theempirical standard deviation are also close e coverageprobability of the nominal 95 confidence intervals seems tobe very reasonable Furthermore when ns 600 for DesignB in Tables 1 and 2 we can obtain that the values of SEE andCP are slightly underestimated It is due to the small numberof clusters in the subcohort In addition with the increase ofthe capacity of the subcolumns ie when ns is increased to800 the effect of the estimation is obviously improvedAmong the above three designs Design A and Design C aremore effective than Design B e conclusion is consistentwith that in [13] By comparing SE SEE and BSE SE andSEE agree with the BSE Furthermore the bootstrap stan-dard errors and the simulated standard errors are similare proposed semiparametric estimator works wellerefore one may use the bootstrap variance estimate forstatistical inference in practice

5 Conclusion

is paper proposed methods of fitting additive hazardregression models for clustered survival data from case-cohort studies Risk difference can provide information

value to medical research Specifically risk differences canprovide information regarding the reduction in the numberof cases developing a certain disease due to a decrease in aparticular exposure An advantage of the additive hazardmodels is that risk difference between different exposuregroups can be readily derived from the coefficients in theadditive hazard models With respect to differing observed-event probability we propose three procedures for pa-rameter estimator Consistency and asymptotic normality ofthe proposed estimators are established e simulationresults show that Design A results are more efficient esti-mators than Design C and Design C has greater efficiencythan Design B when subcohort size is approximately equal Itcan be attributed to differences in the number of sampledclusters in the subcohort

Appendix

Proof of Theorems

Proof of =eorem 1 Evaluated at the true values one canwrite

U β0 p0( 1113857 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A1)

where ρij (p0(N1N))δij + ((1 minus p0)(n0N))(1 minus δij)

HiHij

Table 2 Simulation results for the estimation of β0 0

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC minus 00023 00730 00723 00748 0945 minus 00018 00839 00823 00841 0950WC minus 00012 00739 00722 00717 0940 00001 00841 00823 00791 0943T 00002 00716 00723 00745 0746 00013 00825 00820 00868 0949

2SC minus 00010 00725 00736 00664 0947 minus 00022 00825 00831 00808 0947WC minus 00017 00732 00729 00683 0932 00012 00837 00830 00709 0937T 00021 00721 00730 00710 0944 00031 00819 00827 00806 0948

B

05SC minus 00013 00741 00714 00708 0938 minus 00057 00857 00800 00684 0924WC minus 00002 00736 00716 00765 0931 00053 00855 00799 00783 0925T 00021 00734 00709 00619 0927 minus 00030 00842 00806 00756 0933

2SC minus 00023 00738 00725 00625 0938 minus 00042 00849 00813 00878 0928WC minus 00015 00724 00721 00697 0934 00019 00858 00824 00774 0930T 00032 00741 00714 00725 0931 minus 00028 00836 00813 00833 0940

C

05SC 00005 00732 00715 00708 0949 00007 00843 00811 00746 0937WC 00002 00737 00718 00766 0948 minus 00022 00814 00858 00878 0937T minus 00007 00707 00721 00744 0948 minus 00035 00809 00813 00836 0941

2SC 00014 00739 00723 00726 0941 00014 00838 00820 00806 0941WC 00010 00741 00730 00685 0938 minus 00022 00823 00800 00763 0935T minus 00022 00712 00731 00709 0948 minus 00015 00821 00817 00746 0944

Estimate of β0 from four methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw Tusing true value p0

Journal of Mathematics 5

en we have

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i11113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11113946τ

0e(t) minus Z(t)1113864 1113865dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0e(t) minus Z(t)1113864 1113865dMij(t)

(A2)

e second term on the right-hand side of (A2) can beshown to converge to zero Specially for fixed t

E dMij(t)1113966 1113967 0 e third term on the right-hand side of(A2) can be written as

nminus (12)

1113944

n

i11113944

mi

j1δij

p0

N1N( 1113857minus 11113896 1113897 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11 minus δij1113872 1113873

1 minus p0( 1113857HiHij

n0N( 1113857minus 11113896 1113897 1113946

t

0Zij(t) minus e(t)1113966 1113967dMij(t)

(A3)

By functional Taylor expansion we can obtain that thefirst term of (A3) is written as

nminus (12)

1113944

n

i1G1i p0( 1113857 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G2i p0( 1113857 1113944

n

i11113944

mi

j1

δij

μp01113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A4)

Similarly the second term of (A3) is written as

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

G1i p0( 1113857 times 1113944n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G3i p0( 1113857 times 1113944

n

i11113944

mi

j1

1μ(cθ)2 1 minus p0( 1113857

1 minus δij1113872 1113873HiHij

times 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A5)

6 Journal of Mathematics

Combining (A4) and (A5) it follows that

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i1Wi β p0( 1113857 + op(1) (A6)

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i p0( 1113857

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(A7)

Since Mij(t) Nij(t) minus 1113938t

0 Yij(u)[λ0(t) + β0Zij]du is azero mean process E 1 minus (cθ)minus 1HiHij1113966 1113967 0 E dMij(t)1113966 1113967

0 E G1i1113864 1113865 0 E G2i1113864 1113865 0 and E G3i1113864 1113865 0 we have

E Wi β0 p0( 11138571113864 1113865 0 (A8)

for i 1 nUnder the general regular condition Wi(β p0)1113864 1113865

n

i1 is anindependent and identically distributed random variablewith a mean value of zero and E W1(β0 p0)

otimes21113966 1113967 It follows

from the multidimensional central limit theorem thatnminus (12)U(β0 p0)⟶D N(0 1113936(β0 p0))

Proof of =eorem 2 To prove the consistency of 1113954β we onlyuse the inverse function theorem [17] by verifying the fol-lowing conditions

(i) (zU(β p0)zβT) exists and is continuous in an open

neighborhood B of β0

(ii) (minus nminus 1zU(β p0)zβT)|ββ0 is positive definite with

probability going to one as n⟶infin(iii) (minus nminus 1zU(β p0)zβ

T) converges in probability toA(β) uniformly in an open neighborhood B of β0

(iv) Asymptotic unbiasedness of the estimating func-tions minus nminus 1U(β0 p0)⟶ p0

One can write

minusznminus 1U βp0( 1113857

zβT n

minus 11113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A9)

By the continuity of each component and (A9) (i)is clearly satisfied Now (A9) can be decomposed asfollows

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

nminus 1

1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

+ nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Yij(t) Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 times 1113944

n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

(A10)

Note that Z(t) uniformly converges toe(t) (E Y11(t)Z11(t)1113864 1113865E Z11(t)1113864 1113865) in t en it followsthat the first term on the right-hand side of (A10) convergesto A(β0 p0) in probability as n⟶infin Each of the

remaining terms on the right-hand side of (A10) can beshown to converge to zero as n⟶infin Hence we have

minusznminus 1U β p0( 1113857

zβT⟶ p

A β0 p0( 1113857 (A11)

Journal of Mathematics 7

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

3 Asymptotic Properties

To derive the asymptotic properties of the proposed esti-mates we impose the following assumptions

(C1) (Ti Ci Zi(middot) i 1 n) are independently andidentically distributed where Ti (Ti1 Timi

)T

Ci (Ci1 Cimi)T and Zi (Zi1(middot) Zimi

(middot))T

(C2) Pr Yij(τ)gt 01113966 1113967gt 0 and Λ0(τ)ltinfin for i 1 n

j 1 mi(C3) e matrix E W1(β0 p0)

otimes21113966 1113967 is positive definite

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G2i(p)

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(7)

(C4) ere exists a constant Cz such that |Zij(0) +

1113938τ0 dZij(u)|ltCz ltinfin holds almost everywhere

Theorem 1 Under the regularity conditions (C1)ndash(C4)nminus (12)U(β0 p0) converges to a mean zero normal distribu-tion with covariance 1113936(β0 p0) E W1(β0 p0)

otimes21113966 1113967 with

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i(p)

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

D1(β) E 1113944

m1

j1

δ1j

μ1113946τ

0Z1j(t) minus e(t)1113966 1113967dM1j(t)

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

D2(β) E 1113944

m1

j1

1μcθ

1 minus δ1j1113872 1113873H1H1j 1113946τ

0Z1j(t) minus e(t)1113966 1113967dM1j(t)

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

G1i(p) nminus 1

1113944

n

i1mi minus μ⎛⎝ ⎞⎠

G2i(p) nminus 1

1113944

mi

j1δij minus μp⎛⎝ ⎞⎠

G3i(p) nminus 1

1113944

mi

j11 minus δij1113872 1113873HiHij minus μcθ(1 minus p)

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

e(t) E Y11(t)Z11(t)1113864 1113865

E Z11(t)1113864 1113865

(8)

Theorem 2 Under conditions (C1)ndash(C4) 1113954βt converges inprobability to β0 and each of n12(1113954βt minus β0) converges indistribution to a mean zero normal distribution with co-variance matrix

Theorem 3 Under conditions (C1)ndash(C4) both 1113954βs and 1113954βw

converge in probability to β0 and each of n12(1113954βs minus β0) andn12(1113954βw minus β0) converges in distribution to a zero meannormal with covariance matrices A(β0)

minus 1Ωs(β0 p0)A(β0)minus 1

Journal of Mathematics 3

and A(β0)minus 1Ωw(β0 p0)A(β0)

minus 1 respectively where fora s or w Ωa(β) E ψ1(β p0)

otimes21113966 1113967 ψi(β p) Wi(β p) +

B(β)Qai (p) Ωa(β) E ψ1(β p0)

otimes21113966 1113967 ψi(β p) Wi(β p) +

B(β)Qai (p) Qs

i (p) (1μcθ)1113936mi

j1HiHij(δij minus p) Qwi (p)

(1μ)1113936mi

j1(δij minus p) B(β) 1113938τ0 ηijYij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

and ηij (1(N1N))δij minus (1(n0N))(1 minus δij)HiHij

Theorem 4 Under the regularity conditions (C1)ndash(C4)for each k 1 n 1113954Λ0(1113954β 1113954p t) converges in probability toΛ0(t) uniformly in t isin [0 τ] In addition n12 1113954Λ0(1113954β 1113954p t)minus1113966

Λ0(t) converges weakly to a Gaussian process with zero meanand covariance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t)

4 Numerical Studies

Simulation studies are conducted to examine the finitesample properties of the proposed estimators In the studywe generate the clustered failure time data from n 100clusters mi are simulated from a binomial (50 08) distri-bution for i 1 n with u E(mi) 40 e totalnumber in the whole queue is about 4000 e covariate Zij

takes values 1 and 0 with probabilities 05 and 05 re-spectively e failure time Ti1 Ti2 Timi

in the i-thcluster is simulated by the joint survival function (for detailssee [16])

S ti1 timi

11138681113868111386811138681113868 zi1 zimi1113874 1113875

1113944

mi

j1exp

1113938tij

0 2t + βTzij1113872 1113873dt

κ⎡⎢⎢⎣ ⎤⎥⎥⎦ minus mi minus 1( 1113857⎛⎝ ⎞⎠

minus κ

(9)

where κgt 0 is a parameter representing the degree of de-pendence among variables Smaller value of κ representsstronger correlation

In our experiments we choose κ 05 or 2 and β0

log(05) or 0e censoring times Cij are constant and equalto 1 e observed-event probabilities of p are equal to 045or 063

For each data generation for Design A individualswithin each cluster are selected into the subcohort byBernoulli sampling with equal probability 02 or 015 ForDesign B we select clusters by Bernoulli sampling withprobability 02 or 015 For Design C we first sample clustersby Bernoulli sampling with probability 04 or 03 and thensample individuals from those selected clusters by Bernoullisampling with probability 05 erefore for each design wewould expect approximately 800 or 600 individuals in thesubcohort

To assess the performance of the proposed estimator wecalculate the estimated standard error based on the boot-strap method Because the failure clusters inside and outsidethe subcohort have different structures we use the followingmethod to conduct bootstrap sampling Suppose the failureclusters in the subcohort contain M failures and Q non-failures whereas the failure clusters outside the subcohortcontain D failures only For each bootstrap sampling wepropose to obtain a bootstrap sample by sampling D failuresfrom the original D failures outside the subcohort withreplacement and then samplingM failures and Q nonfailuresfrom the original M failures and Q nonfailures in thesubcohort respectively e simulation results are based on1000 replications

Tables 1 and 2 report the simulation results of ourproposed estimators For simplicity the notation Bias de-notes the difference of the average of the estimates and the

Table 1 Simulation results for the estimation of β0 log(05)

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC 00010 00568 00574 00500 0948 minus 00021 00677 00635 00678 0944WC minus 00026 00526 00576 00550 0969 minus 00023 00603 00646 00630 0966T minus 00021 00533 00525 00496 0953 minus 00015 00594 00597 00642 0956

2SC 00015 00573 00569 00506 0948 minus 00020 00668 00656 00672 0942WC minus 00031 00531 00571 00560 0953 minus 00031 00625 00648 00602 0953T minus 00016 00529 00536 00562 0942 minus 00010 00584 00603 00618 0947

B

05SC minus 00035 00578 00569 00460 0937 minus 00012 00683 00644 00766 0920WC 00023 00593 00612 00609 0943 00004 00642 00640 00635 0938T minus 00030 00534 00511 00605 0934 minus 00039 00629 00613 00614 0936

2SC minus 00012 00568 00573 00589 0941 minus 00009 00679 00654 00685 0920WC 00027 00578 00581 00513 0939 00015 00653 00637 00591 0938T minus 00019 00534 00526 00480 0940 minus 00026 00629 00613 00557 0936

C

05SC minus 00009 00576 00570 00551 0948 minus 00019 00685 00642 00651 0929WC minus 00032 00543 00571 00571 0958 minus 00039 00605 00650 00632 0963T minus 00015 00540 00518 00481 0941 minus 00047 00605 00591 00679 0948

2SC minus 00021 00576 00570 00515 0948 minus 00024 00679 00653 00614 0934WC minus 00017 00547 00562 00565 0942 minus 00026 00614 00647 00644 0956T minus 00010 00549 00523 00510 0937 minus 00038 00587 00604 00613 0943

Estimate of β0 from five methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw T using true value p0

4 Journal of Mathematics

true values SE denotes the average of standard errors SEEdenotes the average asymptotic standard error BSE denotesthe estimated standard error based on the bootstrap methodand CP denotes the coverage probability of the nominal 95confidence interval

It can be seen from Tables 1 and 2 that the estimates ofboth regression parameters β0 log(05) and β0 0 areunbiased and both the estimated standard deviation and theempirical standard deviation are also close e coverageprobability of the nominal 95 confidence intervals seems tobe very reasonable Furthermore when ns 600 for DesignB in Tables 1 and 2 we can obtain that the values of SEE andCP are slightly underestimated It is due to the small numberof clusters in the subcohort In addition with the increase ofthe capacity of the subcolumns ie when ns is increased to800 the effect of the estimation is obviously improvedAmong the above three designs Design A and Design C aremore effective than Design B e conclusion is consistentwith that in [13] By comparing SE SEE and BSE SE andSEE agree with the BSE Furthermore the bootstrap stan-dard errors and the simulated standard errors are similare proposed semiparametric estimator works wellerefore one may use the bootstrap variance estimate forstatistical inference in practice

5 Conclusion

is paper proposed methods of fitting additive hazardregression models for clustered survival data from case-cohort studies Risk difference can provide information

value to medical research Specifically risk differences canprovide information regarding the reduction in the numberof cases developing a certain disease due to a decrease in aparticular exposure An advantage of the additive hazardmodels is that risk difference between different exposuregroups can be readily derived from the coefficients in theadditive hazard models With respect to differing observed-event probability we propose three procedures for pa-rameter estimator Consistency and asymptotic normality ofthe proposed estimators are established e simulationresults show that Design A results are more efficient esti-mators than Design C and Design C has greater efficiencythan Design B when subcohort size is approximately equal Itcan be attributed to differences in the number of sampledclusters in the subcohort

Appendix

Proof of Theorems

Proof of =eorem 1 Evaluated at the true values one canwrite

U β0 p0( 1113857 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A1)

where ρij (p0(N1N))δij + ((1 minus p0)(n0N))(1 minus δij)

HiHij

Table 2 Simulation results for the estimation of β0 0

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC minus 00023 00730 00723 00748 0945 minus 00018 00839 00823 00841 0950WC minus 00012 00739 00722 00717 0940 00001 00841 00823 00791 0943T 00002 00716 00723 00745 0746 00013 00825 00820 00868 0949

2SC minus 00010 00725 00736 00664 0947 minus 00022 00825 00831 00808 0947WC minus 00017 00732 00729 00683 0932 00012 00837 00830 00709 0937T 00021 00721 00730 00710 0944 00031 00819 00827 00806 0948

B

05SC minus 00013 00741 00714 00708 0938 minus 00057 00857 00800 00684 0924WC minus 00002 00736 00716 00765 0931 00053 00855 00799 00783 0925T 00021 00734 00709 00619 0927 minus 00030 00842 00806 00756 0933

2SC minus 00023 00738 00725 00625 0938 minus 00042 00849 00813 00878 0928WC minus 00015 00724 00721 00697 0934 00019 00858 00824 00774 0930T 00032 00741 00714 00725 0931 minus 00028 00836 00813 00833 0940

C

05SC 00005 00732 00715 00708 0949 00007 00843 00811 00746 0937WC 00002 00737 00718 00766 0948 minus 00022 00814 00858 00878 0937T minus 00007 00707 00721 00744 0948 minus 00035 00809 00813 00836 0941

2SC 00014 00739 00723 00726 0941 00014 00838 00820 00806 0941WC 00010 00741 00730 00685 0938 minus 00022 00823 00800 00763 0935T minus 00022 00712 00731 00709 0948 minus 00015 00821 00817 00746 0944

Estimate of β0 from four methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw Tusing true value p0

Journal of Mathematics 5

en we have

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i11113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11113946τ

0e(t) minus Z(t)1113864 1113865dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0e(t) minus Z(t)1113864 1113865dMij(t)

(A2)

e second term on the right-hand side of (A2) can beshown to converge to zero Specially for fixed t

E dMij(t)1113966 1113967 0 e third term on the right-hand side of(A2) can be written as

nminus (12)

1113944

n

i11113944

mi

j1δij

p0

N1N( 1113857minus 11113896 1113897 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11 minus δij1113872 1113873

1 minus p0( 1113857HiHij

n0N( 1113857minus 11113896 1113897 1113946

t

0Zij(t) minus e(t)1113966 1113967dMij(t)

(A3)

By functional Taylor expansion we can obtain that thefirst term of (A3) is written as

nminus (12)

1113944

n

i1G1i p0( 1113857 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G2i p0( 1113857 1113944

n

i11113944

mi

j1

δij

μp01113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A4)

Similarly the second term of (A3) is written as

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

G1i p0( 1113857 times 1113944n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G3i p0( 1113857 times 1113944

n

i11113944

mi

j1

1μ(cθ)2 1 minus p0( 1113857

1 minus δij1113872 1113873HiHij

times 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A5)

6 Journal of Mathematics

Combining (A4) and (A5) it follows that

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i1Wi β p0( 1113857 + op(1) (A6)

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i p0( 1113857

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(A7)

Since Mij(t) Nij(t) minus 1113938t

0 Yij(u)[λ0(t) + β0Zij]du is azero mean process E 1 minus (cθ)minus 1HiHij1113966 1113967 0 E dMij(t)1113966 1113967

0 E G1i1113864 1113865 0 E G2i1113864 1113865 0 and E G3i1113864 1113865 0 we have

E Wi β0 p0( 11138571113864 1113865 0 (A8)

for i 1 nUnder the general regular condition Wi(β p0)1113864 1113865

n

i1 is anindependent and identically distributed random variablewith a mean value of zero and E W1(β0 p0)

otimes21113966 1113967 It follows

from the multidimensional central limit theorem thatnminus (12)U(β0 p0)⟶D N(0 1113936(β0 p0))

Proof of =eorem 2 To prove the consistency of 1113954β we onlyuse the inverse function theorem [17] by verifying the fol-lowing conditions

(i) (zU(β p0)zβT) exists and is continuous in an open

neighborhood B of β0

(ii) (minus nminus 1zU(β p0)zβT)|ββ0 is positive definite with

probability going to one as n⟶infin(iii) (minus nminus 1zU(β p0)zβ

T) converges in probability toA(β) uniformly in an open neighborhood B of β0

(iv) Asymptotic unbiasedness of the estimating func-tions minus nminus 1U(β0 p0)⟶ p0

One can write

minusznminus 1U βp0( 1113857

zβT n

minus 11113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A9)

By the continuity of each component and (A9) (i)is clearly satisfied Now (A9) can be decomposed asfollows

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

nminus 1

1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

+ nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Yij(t) Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 times 1113944

n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

(A10)

Note that Z(t) uniformly converges toe(t) (E Y11(t)Z11(t)1113864 1113865E Z11(t)1113864 1113865) in t en it followsthat the first term on the right-hand side of (A10) convergesto A(β0 p0) in probability as n⟶infin Each of the

remaining terms on the right-hand side of (A10) can beshown to converge to zero as n⟶infin Hence we have

minusznminus 1U β p0( 1113857

zβT⟶ p

A β0 p0( 1113857 (A11)

Journal of Mathematics 7

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

and A(β0)minus 1Ωw(β0 p0)A(β0)

minus 1 respectively where fora s or w Ωa(β) E ψ1(β p0)

otimes21113966 1113967 ψi(β p) Wi(β p) +

B(β)Qai (p) Ωa(β) E ψ1(β p0)

otimes21113966 1113967 ψi(β p) Wi(β p) +

B(β)Qai (p) Qs

i (p) (1μcθ)1113936mi

j1HiHij(δij minus p) Qwi (p)

(1μ)1113936mi

j1(δij minus p) B(β) 1113938τ0 ηijYij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

and ηij (1(N1N))δij minus (1(n0N))(1 minus δij)HiHij

Theorem 4 Under the regularity conditions (C1)ndash(C4)for each k 1 n 1113954Λ0(1113954β 1113954p t) converges in probability toΛ0(t) uniformly in t isin [0 τ] In addition n12 1113954Λ0(1113954β 1113954p t)minus1113966

Λ0(t) converges weakly to a Gaussian process with zero meanand covariance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t)

4 Numerical Studies

Simulation studies are conducted to examine the finitesample properties of the proposed estimators In the studywe generate the clustered failure time data from n 100clusters mi are simulated from a binomial (50 08) distri-bution for i 1 n with u E(mi) 40 e totalnumber in the whole queue is about 4000 e covariate Zij

takes values 1 and 0 with probabilities 05 and 05 re-spectively e failure time Ti1 Ti2 Timi

in the i-thcluster is simulated by the joint survival function (for detailssee [16])

S ti1 timi

11138681113868111386811138681113868 zi1 zimi1113874 1113875

1113944

mi

j1exp

1113938tij

0 2t + βTzij1113872 1113873dt

κ⎡⎢⎢⎣ ⎤⎥⎥⎦ minus mi minus 1( 1113857⎛⎝ ⎞⎠

minus κ

(9)

where κgt 0 is a parameter representing the degree of de-pendence among variables Smaller value of κ representsstronger correlation

In our experiments we choose κ 05 or 2 and β0

log(05) or 0e censoring times Cij are constant and equalto 1 e observed-event probabilities of p are equal to 045or 063

For each data generation for Design A individualswithin each cluster are selected into the subcohort byBernoulli sampling with equal probability 02 or 015 ForDesign B we select clusters by Bernoulli sampling withprobability 02 or 015 For Design C we first sample clustersby Bernoulli sampling with probability 04 or 03 and thensample individuals from those selected clusters by Bernoullisampling with probability 05 erefore for each design wewould expect approximately 800 or 600 individuals in thesubcohort

To assess the performance of the proposed estimator wecalculate the estimated standard error based on the boot-strap method Because the failure clusters inside and outsidethe subcohort have different structures we use the followingmethod to conduct bootstrap sampling Suppose the failureclusters in the subcohort contain M failures and Q non-failures whereas the failure clusters outside the subcohortcontain D failures only For each bootstrap sampling wepropose to obtain a bootstrap sample by sampling D failuresfrom the original D failures outside the subcohort withreplacement and then samplingM failures and Q nonfailuresfrom the original M failures and Q nonfailures in thesubcohort respectively e simulation results are based on1000 replications

Tables 1 and 2 report the simulation results of ourproposed estimators For simplicity the notation Bias de-notes the difference of the average of the estimates and the

Table 1 Simulation results for the estimation of β0 log(05)

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC 00010 00568 00574 00500 0948 minus 00021 00677 00635 00678 0944WC minus 00026 00526 00576 00550 0969 minus 00023 00603 00646 00630 0966T minus 00021 00533 00525 00496 0953 minus 00015 00594 00597 00642 0956

2SC 00015 00573 00569 00506 0948 minus 00020 00668 00656 00672 0942WC minus 00031 00531 00571 00560 0953 minus 00031 00625 00648 00602 0953T minus 00016 00529 00536 00562 0942 minus 00010 00584 00603 00618 0947

B

05SC minus 00035 00578 00569 00460 0937 minus 00012 00683 00644 00766 0920WC 00023 00593 00612 00609 0943 00004 00642 00640 00635 0938T minus 00030 00534 00511 00605 0934 minus 00039 00629 00613 00614 0936

2SC minus 00012 00568 00573 00589 0941 minus 00009 00679 00654 00685 0920WC 00027 00578 00581 00513 0939 00015 00653 00637 00591 0938T minus 00019 00534 00526 00480 0940 minus 00026 00629 00613 00557 0936

C

05SC minus 00009 00576 00570 00551 0948 minus 00019 00685 00642 00651 0929WC minus 00032 00543 00571 00571 0958 minus 00039 00605 00650 00632 0963T minus 00015 00540 00518 00481 0941 minus 00047 00605 00591 00679 0948

2SC minus 00021 00576 00570 00515 0948 minus 00024 00679 00653 00614 0934WC minus 00017 00547 00562 00565 0942 minus 00026 00614 00647 00644 0956T minus 00010 00549 00523 00510 0937 minus 00038 00587 00604 00613 0943

Estimate of β0 from five methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw T using true value p0

4 Journal of Mathematics

true values SE denotes the average of standard errors SEEdenotes the average asymptotic standard error BSE denotesthe estimated standard error based on the bootstrap methodand CP denotes the coverage probability of the nominal 95confidence interval

It can be seen from Tables 1 and 2 that the estimates ofboth regression parameters β0 log(05) and β0 0 areunbiased and both the estimated standard deviation and theempirical standard deviation are also close e coverageprobability of the nominal 95 confidence intervals seems tobe very reasonable Furthermore when ns 600 for DesignB in Tables 1 and 2 we can obtain that the values of SEE andCP are slightly underestimated It is due to the small numberof clusters in the subcohort In addition with the increase ofthe capacity of the subcolumns ie when ns is increased to800 the effect of the estimation is obviously improvedAmong the above three designs Design A and Design C aremore effective than Design B e conclusion is consistentwith that in [13] By comparing SE SEE and BSE SE andSEE agree with the BSE Furthermore the bootstrap stan-dard errors and the simulated standard errors are similare proposed semiparametric estimator works wellerefore one may use the bootstrap variance estimate forstatistical inference in practice

5 Conclusion

is paper proposed methods of fitting additive hazardregression models for clustered survival data from case-cohort studies Risk difference can provide information

value to medical research Specifically risk differences canprovide information regarding the reduction in the numberof cases developing a certain disease due to a decrease in aparticular exposure An advantage of the additive hazardmodels is that risk difference between different exposuregroups can be readily derived from the coefficients in theadditive hazard models With respect to differing observed-event probability we propose three procedures for pa-rameter estimator Consistency and asymptotic normality ofthe proposed estimators are established e simulationresults show that Design A results are more efficient esti-mators than Design C and Design C has greater efficiencythan Design B when subcohort size is approximately equal Itcan be attributed to differences in the number of sampledclusters in the subcohort

Appendix

Proof of Theorems

Proof of =eorem 1 Evaluated at the true values one canwrite

U β0 p0( 1113857 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A1)

where ρij (p0(N1N))δij + ((1 minus p0)(n0N))(1 minus δij)

HiHij

Table 2 Simulation results for the estimation of β0 0

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC minus 00023 00730 00723 00748 0945 minus 00018 00839 00823 00841 0950WC minus 00012 00739 00722 00717 0940 00001 00841 00823 00791 0943T 00002 00716 00723 00745 0746 00013 00825 00820 00868 0949

2SC minus 00010 00725 00736 00664 0947 minus 00022 00825 00831 00808 0947WC minus 00017 00732 00729 00683 0932 00012 00837 00830 00709 0937T 00021 00721 00730 00710 0944 00031 00819 00827 00806 0948

B

05SC minus 00013 00741 00714 00708 0938 minus 00057 00857 00800 00684 0924WC minus 00002 00736 00716 00765 0931 00053 00855 00799 00783 0925T 00021 00734 00709 00619 0927 minus 00030 00842 00806 00756 0933

2SC minus 00023 00738 00725 00625 0938 minus 00042 00849 00813 00878 0928WC minus 00015 00724 00721 00697 0934 00019 00858 00824 00774 0930T 00032 00741 00714 00725 0931 minus 00028 00836 00813 00833 0940

C

05SC 00005 00732 00715 00708 0949 00007 00843 00811 00746 0937WC 00002 00737 00718 00766 0948 minus 00022 00814 00858 00878 0937T minus 00007 00707 00721 00744 0948 minus 00035 00809 00813 00836 0941

2SC 00014 00739 00723 00726 0941 00014 00838 00820 00806 0941WC 00010 00741 00730 00685 0938 minus 00022 00823 00800 00763 0935T minus 00022 00712 00731 00709 0948 minus 00015 00821 00817 00746 0944

Estimate of β0 from four methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw Tusing true value p0

Journal of Mathematics 5

en we have

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i11113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11113946τ

0e(t) minus Z(t)1113864 1113865dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0e(t) minus Z(t)1113864 1113865dMij(t)

(A2)

e second term on the right-hand side of (A2) can beshown to converge to zero Specially for fixed t

E dMij(t)1113966 1113967 0 e third term on the right-hand side of(A2) can be written as

nminus (12)

1113944

n

i11113944

mi

j1δij

p0

N1N( 1113857minus 11113896 1113897 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11 minus δij1113872 1113873

1 minus p0( 1113857HiHij

n0N( 1113857minus 11113896 1113897 1113946

t

0Zij(t) minus e(t)1113966 1113967dMij(t)

(A3)

By functional Taylor expansion we can obtain that thefirst term of (A3) is written as

nminus (12)

1113944

n

i1G1i p0( 1113857 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G2i p0( 1113857 1113944

n

i11113944

mi

j1

δij

μp01113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A4)

Similarly the second term of (A3) is written as

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

G1i p0( 1113857 times 1113944n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G3i p0( 1113857 times 1113944

n

i11113944

mi

j1

1μ(cθ)2 1 minus p0( 1113857

1 minus δij1113872 1113873HiHij

times 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A5)

6 Journal of Mathematics

Combining (A4) and (A5) it follows that

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i1Wi β p0( 1113857 + op(1) (A6)

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i p0( 1113857

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(A7)

Since Mij(t) Nij(t) minus 1113938t

0 Yij(u)[λ0(t) + β0Zij]du is azero mean process E 1 minus (cθ)minus 1HiHij1113966 1113967 0 E dMij(t)1113966 1113967

0 E G1i1113864 1113865 0 E G2i1113864 1113865 0 and E G3i1113864 1113865 0 we have

E Wi β0 p0( 11138571113864 1113865 0 (A8)

for i 1 nUnder the general regular condition Wi(β p0)1113864 1113865

n

i1 is anindependent and identically distributed random variablewith a mean value of zero and E W1(β0 p0)

otimes21113966 1113967 It follows

from the multidimensional central limit theorem thatnminus (12)U(β0 p0)⟶D N(0 1113936(β0 p0))

Proof of =eorem 2 To prove the consistency of 1113954β we onlyuse the inverse function theorem [17] by verifying the fol-lowing conditions

(i) (zU(β p0)zβT) exists and is continuous in an open

neighborhood B of β0

(ii) (minus nminus 1zU(β p0)zβT)|ββ0 is positive definite with

probability going to one as n⟶infin(iii) (minus nminus 1zU(β p0)zβ

T) converges in probability toA(β) uniformly in an open neighborhood B of β0

(iv) Asymptotic unbiasedness of the estimating func-tions minus nminus 1U(β0 p0)⟶ p0

One can write

minusznminus 1U βp0( 1113857

zβT n

minus 11113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A9)

By the continuity of each component and (A9) (i)is clearly satisfied Now (A9) can be decomposed asfollows

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

nminus 1

1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

+ nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Yij(t) Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 times 1113944

n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

(A10)

Note that Z(t) uniformly converges toe(t) (E Y11(t)Z11(t)1113864 1113865E Z11(t)1113864 1113865) in t en it followsthat the first term on the right-hand side of (A10) convergesto A(β0 p0) in probability as n⟶infin Each of the

remaining terms on the right-hand side of (A10) can beshown to converge to zero as n⟶infin Hence we have

minusznminus 1U β p0( 1113857

zβT⟶ p

A β0 p0( 1113857 (A11)

Journal of Mathematics 7

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

true values SE denotes the average of standard errors SEEdenotes the average asymptotic standard error BSE denotesthe estimated standard error based on the bootstrap methodand CP denotes the coverage probability of the nominal 95confidence interval

It can be seen from Tables 1 and 2 that the estimates ofboth regression parameters β0 log(05) and β0 0 areunbiased and both the estimated standard deviation and theempirical standard deviation are also close e coverageprobability of the nominal 95 confidence intervals seems tobe very reasonable Furthermore when ns 600 for DesignB in Tables 1 and 2 we can obtain that the values of SEE andCP are slightly underestimated It is due to the small numberof clusters in the subcohort In addition with the increase ofthe capacity of the subcolumns ie when ns is increased to800 the effect of the estimation is obviously improvedAmong the above three designs Design A and Design C aremore effective than Design B e conclusion is consistentwith that in [13] By comparing SE SEE and BSE SE andSEE agree with the BSE Furthermore the bootstrap stan-dard errors and the simulated standard errors are similare proposed semiparametric estimator works wellerefore one may use the bootstrap variance estimate forstatistical inference in practice

5 Conclusion

is paper proposed methods of fitting additive hazardregression models for clustered survival data from case-cohort studies Risk difference can provide information

value to medical research Specifically risk differences canprovide information regarding the reduction in the numberof cases developing a certain disease due to a decrease in aparticular exposure An advantage of the additive hazardmodels is that risk difference between different exposuregroups can be readily derived from the coefficients in theadditive hazard models With respect to differing observed-event probability we propose three procedures for pa-rameter estimator Consistency and asymptotic normality ofthe proposed estimators are established e simulationresults show that Design A results are more efficient esti-mators than Design C and Design C has greater efficiencythan Design B when subcohort size is approximately equal Itcan be attributed to differences in the number of sampledclusters in the subcohort

Appendix

Proof of Theorems

Proof of =eorem 1 Evaluated at the true values one canwrite

U β0 p0( 1113857 1113944n

i11113944

mi

j1ρij 1113946

τ

0Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A1)

where ρij (p0(N1N))δij + ((1 minus p0)(n0N))(1 minus δij)

HiHij

Table 2 Simulation results for the estimation of β0 0

Design κ Methodn 100 ns 800 n 100 ns 600

Bias SE SEE BSE CP Bias SE SEE BSE CP

A

05SC minus 00023 00730 00723 00748 0945 minus 00018 00839 00823 00841 0950WC minus 00012 00739 00722 00717 0940 00001 00841 00823 00791 0943T 00002 00716 00723 00745 0746 00013 00825 00820 00868 0949

2SC minus 00010 00725 00736 00664 0947 minus 00022 00825 00831 00808 0947WC minus 00017 00732 00729 00683 0932 00012 00837 00830 00709 0937T 00021 00721 00730 00710 0944 00031 00819 00827 00806 0948

B

05SC minus 00013 00741 00714 00708 0938 minus 00057 00857 00800 00684 0924WC minus 00002 00736 00716 00765 0931 00053 00855 00799 00783 0925T 00021 00734 00709 00619 0927 minus 00030 00842 00806 00756 0933

2SC minus 00023 00738 00725 00625 0938 minus 00042 00849 00813 00878 0928WC minus 00015 00724 00721 00697 0934 00019 00858 00824 00774 0930T 00032 00741 00714 00725 0931 minus 00028 00836 00813 00833 0940

C

05SC 00005 00732 00715 00708 0949 00007 00843 00811 00746 0937WC 00002 00737 00718 00766 0948 minus 00022 00814 00858 00878 0937T minus 00007 00707 00721 00744 0948 minus 00035 00809 00813 00836 0941

2SC 00014 00739 00723 00726 0941 00014 00838 00820 00806 0941WC 00010 00741 00730 00685 0938 minus 00022 00823 00800 00763 0935T minus 00022 00712 00731 00709 0948 minus 00015 00821 00817 00746 0944

Estimate of β0 from four methods SC estimating p0 using the subcohort 1113954ps WC estimating p0 using the whole cohort 1113954pw Tusing true value p0

Journal of Mathematics 5

en we have

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i11113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11113946τ

0e(t) minus Z(t)1113864 1113865dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0e(t) minus Z(t)1113864 1113865dMij(t)

(A2)

e second term on the right-hand side of (A2) can beshown to converge to zero Specially for fixed t

E dMij(t)1113966 1113967 0 e third term on the right-hand side of(A2) can be written as

nminus (12)

1113944

n

i11113944

mi

j1δij

p0

N1N( 1113857minus 11113896 1113897 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11 minus δij1113872 1113873

1 minus p0( 1113857HiHij

n0N( 1113857minus 11113896 1113897 1113946

t

0Zij(t) minus e(t)1113966 1113967dMij(t)

(A3)

By functional Taylor expansion we can obtain that thefirst term of (A3) is written as

nminus (12)

1113944

n

i1G1i p0( 1113857 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G2i p0( 1113857 1113944

n

i11113944

mi

j1

δij

μp01113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A4)

Similarly the second term of (A3) is written as

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

G1i p0( 1113857 times 1113944n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G3i p0( 1113857 times 1113944

n

i11113944

mi

j1

1μ(cθ)2 1 minus p0( 1113857

1 minus δij1113872 1113873HiHij

times 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A5)

6 Journal of Mathematics

Combining (A4) and (A5) it follows that

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i1Wi β p0( 1113857 + op(1) (A6)

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i p0( 1113857

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(A7)

Since Mij(t) Nij(t) minus 1113938t

0 Yij(u)[λ0(t) + β0Zij]du is azero mean process E 1 minus (cθ)minus 1HiHij1113966 1113967 0 E dMij(t)1113966 1113967

0 E G1i1113864 1113865 0 E G2i1113864 1113865 0 and E G3i1113864 1113865 0 we have

E Wi β0 p0( 11138571113864 1113865 0 (A8)

for i 1 nUnder the general regular condition Wi(β p0)1113864 1113865

n

i1 is anindependent and identically distributed random variablewith a mean value of zero and E W1(β0 p0)

otimes21113966 1113967 It follows

from the multidimensional central limit theorem thatnminus (12)U(β0 p0)⟶D N(0 1113936(β0 p0))

Proof of =eorem 2 To prove the consistency of 1113954β we onlyuse the inverse function theorem [17] by verifying the fol-lowing conditions

(i) (zU(β p0)zβT) exists and is continuous in an open

neighborhood B of β0

(ii) (minus nminus 1zU(β p0)zβT)|ββ0 is positive definite with

probability going to one as n⟶infin(iii) (minus nminus 1zU(β p0)zβ

T) converges in probability toA(β) uniformly in an open neighborhood B of β0

(iv) Asymptotic unbiasedness of the estimating func-tions minus nminus 1U(β0 p0)⟶ p0

One can write

minusznminus 1U βp0( 1113857

zβT n

minus 11113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A9)

By the continuity of each component and (A9) (i)is clearly satisfied Now (A9) can be decomposed asfollows

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

nminus 1

1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

+ nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Yij(t) Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 times 1113944

n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

(A10)

Note that Z(t) uniformly converges toe(t) (E Y11(t)Z11(t)1113864 1113865E Z11(t)1113864 1113865) in t en it followsthat the first term on the right-hand side of (A10) convergesto A(β0 p0) in probability as n⟶infin Each of the

remaining terms on the right-hand side of (A10) can beshown to converge to zero as n⟶infin Hence we have

minusznminus 1U β p0( 1113857

zβT⟶ p

A β0 p0( 1113857 (A11)

Journal of Mathematics 7

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

en we have

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i11113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11113946τ

0e(t) minus Z(t)1113864 1113865dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j1ρij minus 11113872 1113873 1113946

τ

0e(t) minus Z(t)1113864 1113865dMij(t)

(A2)

e second term on the right-hand side of (A2) can beshown to converge to zero Specially for fixed t

E dMij(t)1113966 1113967 0 e third term on the right-hand side of(A2) can be written as

nminus (12)

1113944

n

i11113944

mi

j1δij

p0

N1N( 1113857minus 11113896 1113897 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

1113944

n

i11113944

mi

j11 minus δij1113872 1113873

1 minus p0( 1113857HiHij

n0N( 1113857minus 11113896 1113897 1113946

t

0Zij(t) minus e(t)1113966 1113967dMij(t)

(A3)

By functional Taylor expansion we can obtain that thefirst term of (A3) is written as

nminus (12)

1113944

n

i1G1i p0( 1113857 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G2i p0( 1113857 1113944

n

i11113944

mi

j1

δij

μp01113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A4)

Similarly the second term of (A3) is written as

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

+ nminus (12)

G1i p0( 1113857 times 1113944n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t)

minus nminus (12)

1113944

n

i1G3i p0( 1113857 times 1113944

n

i11113944

mi

j1

1μ(cθ)2 1 minus p0( 1113857

1 minus δij1113872 1113873HiHij

times 1113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) + op(1)

(A5)

6 Journal of Mathematics

Combining (A4) and (A5) it follows that

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i1Wi β p0( 1113857 + op(1) (A6)

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i p0( 1113857

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(A7)

Since Mij(t) Nij(t) minus 1113938t

0 Yij(u)[λ0(t) + β0Zij]du is azero mean process E 1 minus (cθ)minus 1HiHij1113966 1113967 0 E dMij(t)1113966 1113967

0 E G1i1113864 1113865 0 E G2i1113864 1113865 0 and E G3i1113864 1113865 0 we have

E Wi β0 p0( 11138571113864 1113865 0 (A8)

for i 1 nUnder the general regular condition Wi(β p0)1113864 1113865

n

i1 is anindependent and identically distributed random variablewith a mean value of zero and E W1(β0 p0)

otimes21113966 1113967 It follows

from the multidimensional central limit theorem thatnminus (12)U(β0 p0)⟶D N(0 1113936(β0 p0))

Proof of =eorem 2 To prove the consistency of 1113954β we onlyuse the inverse function theorem [17] by verifying the fol-lowing conditions

(i) (zU(β p0)zβT) exists and is continuous in an open

neighborhood B of β0

(ii) (minus nminus 1zU(β p0)zβT)|ββ0 is positive definite with

probability going to one as n⟶infin(iii) (minus nminus 1zU(β p0)zβ

T) converges in probability toA(β) uniformly in an open neighborhood B of β0

(iv) Asymptotic unbiasedness of the estimating func-tions minus nminus 1U(β0 p0)⟶ p0

One can write

minusznminus 1U βp0( 1113857

zβT n

minus 11113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A9)

By the continuity of each component and (A9) (i)is clearly satisfied Now (A9) can be decomposed asfollows

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

nminus 1

1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

+ nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Yij(t) Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 times 1113944

n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

(A10)

Note that Z(t) uniformly converges toe(t) (E Y11(t)Z11(t)1113864 1113865E Z11(t)1113864 1113865) in t en it followsthat the first term on the right-hand side of (A10) convergesto A(β0 p0) in probability as n⟶infin Each of the

remaining terms on the right-hand side of (A10) can beshown to converge to zero as n⟶infin Hence we have

minusznminus 1U β p0( 1113857

zβT⟶ p

A β0 p0( 1113857 (A11)

Journal of Mathematics 7

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

Combining (A4) and (A5) it follows that

nminus (12)

U β p0( 1113857 nminus (12)

1113944

n

i1Wi β p0( 1113857 + op(1) (A6)

where

Wi(β p) 1113944

mi

j11113946τ

0Zij(t) minus e(t)1113966 1113967dMij(t) +

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967dMij(t)1113896 1113897

+ D1(β)G1i(p) minus G1i p0( 1113857

p1113896 1113897 + D2(β)

G1i(p) minus G3i(p)

cθ1113896 1113897

(A7)

Since Mij(t) Nij(t) minus 1113938t

0 Yij(u)[λ0(t) + β0Zij]du is azero mean process E 1 minus (cθ)minus 1HiHij1113966 1113967 0 E dMij(t)1113966 1113967

0 E G1i1113864 1113865 0 E G2i1113864 1113865 0 and E G3i1113864 1113865 0 we have

E Wi β0 p0( 11138571113864 1113865 0 (A8)

for i 1 nUnder the general regular condition Wi(β p0)1113864 1113865

n

i1 is anindependent and identically distributed random variablewith a mean value of zero and E W1(β0 p0)

otimes21113966 1113967 It follows

from the multidimensional central limit theorem thatnminus (12)U(β0 p0)⟶D N(0 1113936(β0 p0))

Proof of =eorem 2 To prove the consistency of 1113954β we onlyuse the inverse function theorem [17] by verifying the fol-lowing conditions

(i) (zU(β p0)zβT) exists and is continuous in an open

neighborhood B of β0

(ii) (minus nminus 1zU(β p0)zβT)|ββ0 is positive definite with

probability going to one as n⟶infin(iii) (minus nminus 1zU(β p0)zβ

T) converges in probability toA(β) uniformly in an open neighborhood B of β0

(iv) Asymptotic unbiasedness of the estimating func-tions minus nminus 1U(β0 p0)⟶ p0

One can write

minusznminus 1U βp0( 1113857

zβT n

minus 11113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A9)

By the continuity of each component and (A9) (i)is clearly satisfied Now (A9) can be decomposed asfollows

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

nminus 1

1113944

n

i11113944

mi

j11113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

+ nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Yij(t) Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897 times 1113944

n

i11113944

mi

j1

δij

μ1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

+ nminus 1

1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 times 1113944

n

i11113944

mi

j1

1μcθ

1 minus δij1113872 1113873HiHij 1113946τ

0Zij(t) minus e(t)1113966 1113967

otimes2dt

(A10)

Note that Z(t) uniformly converges toe(t) (E Y11(t)Z11(t)1113864 1113865E Z11(t)1113864 1113865) in t en it followsthat the first term on the right-hand side of (A10) convergesto A(β0 p0) in probability as n⟶infin Each of the

remaining terms on the right-hand side of (A10) can beshown to converge to zero as n⟶infin Hence we have

minusznminus 1U β p0( 1113857

zβT⟶ p

A β0 p0( 1113857 (A11)

Journal of Mathematics 7

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

Proof of =eorem 3 Here we prove results of 1113954βs since resultsof 1113954βw can be proved similarly By two Taylor expansions thescore function can be decomposition as follows

nminus (12)

U 1113954βs 1113954ps1113872 1113873 nminus (12)

U β0 1113954ps( 1113857 minus 1113954A βlowast 1113954ps( 1113857 1113954βs minus β01113872 1113873

+ op(1)

(A12)

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

U β0 p0( 1113857 + 1113954B β0 plowast( 1113857 1113954ps minus p0( 1113857

+ op(1)

(A13)

where βlowast is on the line segment between 1113954βs and β0 plowast is onthe line segment between 1113954p and p0 and

1113954A(β p) minus nminus 1zU(β p)

zβT

nminus 1

1113944

n

i11113944

mi

j1ρij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt

(A14)

1113954B(β p) nminus 1zU(β p)

zp

1113944n

i11113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944n

i11113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A15)

where η(t) (1113936ni11113936

mi

j1ηijYij(t)Zij(t)1113936ni11113936

mi

j1ρijYij(t))ηij (zρijzp) (1(N1N))δij minus (1(n0N))(1 minus δij)HiHijand W(t) (1113936

ni11113936

mi

j1ηijYij(t)1113936ni11113936

mi

j1ρijYij(t))Since 1113954βs⟶ pβ0 and βlowast minus β0 le βs minus β0 we can obtain

that βlowast ⟶ pβ0 Using the fact that 1113954ps⟶ pp0 we have

1113954A βlowast ps( 1113857 1113946τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

otimes2dt (A16)

Since ps⟶ pp0 and plowast minus p0 le 1113954ps minus p0 we obtainthat plowast ⟶ pp0 en by continuous mapping it followsthat1113954B β0 plowast( 1113857

⟶ p1113944

mi

j1ηij 1113946

τ

0Yij(t) Zij(t) minus Z(t)1113966 1113967

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

minus 1113944

mi

j1ρij 1113946

τ

0η(t) minus Z(t)W(t)1113864 1113865

middot dNij(t) minus Yij(t)βT0 Zij(t)dt1113966 1113967

(A17)

Using the fact that

1113954ps minus p0 1113936

ni11113936

mi

j1HiHijδij

1113936ni11113936

mi

j1HiHij

minus p0

1113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1HiHij

nminus 11113936

ni11113936

mi

j1HiHij δij minus p01113872 1113873

1113936ni11113936

mi

j1 HiHijn1113872 1113873

(A18)

we have

n12 1113954ps minus p0( 1113857 n

minus (12)1113944

n

i1Qi p0( 1113857 + op(1) (A19)

Note that E HiHij(δij minus p0)1113966 1113967 0 then E Qi(p0)1113864 1113865 0erefore using (A13) and (A15)ndash(A19) we have

nminus (12)

U β0 1113954ps( 1113857 nminus (12)

1113944

n

i1ψi β0 p0( 1113857 + op(1) (A20)

It follows from E ψi(β0 p0)1113864 1113865 0 and the multidimen-sional central limit theorem that

nminus (12)

U β0 1113954ps( 1113857⟶D

N 0Ω β0( 1113857( 1113857 (A21)

where Ω(β0) E ψi(β0 p0)otimes2

1113966 1113967Since U(1113954βs 1113954ps) 0 using (A12) and (A21) we have

n12 1113954βs minus β01113872 1113873 1113954A βlowast 1113954ps( 1113857

minus 1times U β0 1113954ps( 1113857 (A22)

Note that 1113954A(βlowast 1113954ps)⟶ PA(β0) By Slutskyrsquos theoremwe have

n12 1113954βs minus β01113872 1113873⟶

DN 0 A β0( 1113857

minus 1Ω β0( 1113857A β0( 1113857minus 1

1113872 1113873 (A23)

Proof of =eorem 4 One can make decomposition

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 n

12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967

+ n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967

+ n12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

(A24)

By the Taylor expansion the first term of (A24) can bewritten as

n12 1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 11138731113966 1113967 n

12Γ(t) 1113954p minus p0( 1113857

(A25)

where

Γ(t) 1113946t

0

1113936ni11113936

mi

j1 ηij minus ρijηij1113966 1113967 dNij(t) minus Yij(u)1113954βTZij(u)du1113876 1113877

1113936ni11113936

mi

j1ρijYij(u)

(A26)

Since p converges to in probability to p0 we have

1113954Λ0(1113954β 1113954p t) minus 1113954Λ0 1113954β p0 t1113872 1113873⟶P

0 (A27)

8 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

By computing the remaining terms of (A24) can bewritten as

n12 1113954Λ0 1113954β p0 t1113872 1113873 minus 1113954Λ0 β0 p0 t( 11138571113966 1113967 + n

12 1113954Λ0 β0 p0 t( 1113857 minus Λ0(t)1113966 1113967

1113946t

0

1113936ni11113936

mi

j1ρij1113954β minus β01113872 1113873

TYij(u)Zij(u)du

1113936ni11113936

mi

j1ρijYij(u)+ 1113946

t

0

1113936ni11113936

mi

j1dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

+ 1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

(A28)

Z(t) uniformly converges to e(t) in te second term of(A28) follows immediately that

1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

n

i11113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ op(1)

(A29)

For the last term of (A28) using eorem 1 we canobtain that

1113946t

0

1113936ni11113936

mi

j1 ρij minus 11113872 1113873dMij(u)

1113936ni11113936

mi

j1ρijYij(u)

nminus (12)

1113944

n

i11113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12)1113944

n

i1

G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897 + op(1)

(A30)

Combining (A28)ndash(A30) and n12(1113954βs minus β0) 1113954A(βlowast 1113954ps)

minus 1 times U(β0 1113954ps) we have that

n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 1U β0 1113954ps( 1113857 + n

minus (12)1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)Γ(t) 1113944

n

i1Qi p0( 1113857 + n

minus (12)rk β0 t( 1113857

TA β0( 1113857

minus 11113944

n

i1ψi β0 p0( 1113857

+ nminus (12)

1113944

n

i1ϑi β0 p0 t( 1113857 + op(1)

nminus (12)

1113944

n

i1ϕi β0 p0 t( 1113857 + op(1)

(A31)

Journal of Mathematics 9

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics

where

ϑi β0 p0 t( 1113857 1113946t

0

1E 1113936

mi

j1Y1j(u)1113960 1113961d n

minus (12)1113944

mi

j1Mij β0 u( 1113857

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

+ nminus (12)

1113944

mi

j1

1 minus δij

cθHiHij minus cθ1113872 1113873 1113946

τ

0Zij(t) minus e(t)1113966 1113967

dMij(t)

E 1113936mi

j1Y1j(u)1113960 1113961

+D1(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G2i p0( 1113857

p01113896 1113897

+D2(β)

E 1113936mi

j1Y1j(u)1113960 1113961times n

minus (12) G1i p0( 1113857 minus G3i p0( 1113857

cθ1113896 1113897

(A32)

It follows immediately from the multidimensionalcentral limit theorem that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 con-verges to a multivariate normal with zero mean and co-variance function at (s t) given by E ϕ1(β0 p0 s)1113864

ϕ1(β0 p0 t) It also proves that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 istight

Using the functional central limit theorem we canobtain that n12 1113954Λ0(1113954β 1113954p t) minus Λ0(t)1113966 1113967 converges weakly to azero mean Gaussian process with covariance function at(s t) given by E ϕ1(β0 p0 s)ϕ1(β0 p0 t)1113864 1113865

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Liursquos work presented in this paper was supported by theNatural Science Research Project of Universities of AnhuiProvince (no KJ2018A0390) and Zhangrsquos work was sup-ported by the Xulun Scholar Program of Shanghai LixinUniversity of Accounting and Finance

References

[1] D Zheng and D Lin ldquoEfficient estimation for the acceleratedfailure time modelrdquo Journal of the American Statistical As-sociation vol 102 no 480 pp 1387ndash1396 2007

[2] D Zeng D Y Lin and X Lin ldquoSemiparametric transfor-mation models with random effects for clustered failure timedatardquo Statistica Sinica vol 18 no 1 pp 355ndash377 2008

[3] D Zeng and J Cai ldquoAdditive transformation models forclustered failure time datardquo Lifetime Data Analysis vol 16no 3 pp 333ndash352 2010

[4] B Liu W Lu and J Zhang ldquoKernel smoothed profile like-lihood estimation in the accelerated failure time frailty model

for clustered survival datardquo Biometrika vol 100 no 3pp 741ndash755 2013

[5] R L Prentice ldquoA case-cohort design for epidemiologic cohortstudies and disease prevention trialsrdquo Biometrika vol 73no 1 pp 1ndash11 1986

[6] S G Self and R L Prentice ldquoAsymptotic distribution theoryand efficiency results for case-cohort studiesrdquo =e Annals ofStatistics vol 16 no 1 pp 64ndash81 1988

[7] M Kulich and D Y Lin ldquoAdditive hazards regression forcase-cohort studiesrdquo Biometrika vol 87 no 1 pp 73ndash872000

[8] K Chen L Sun and X Tong ldquoAnalysis of cohort survival datawith transformation modelrdquo Statistica Sinica vol 22pp 489ndash508 2012

[9] S H Chiou S Kang and J Yan ldquoFast accelerated failure timemodeling for case-cohort datardquo Statistics and Computingvol 24 no 4 pp 559ndash568 2014

[10] Y Sun W Yu and M Zheng ldquoCase-cohort analysis withgeneral additive-multiplicative hazard modelsrdquo Acta Math-ematicae Applicatae Sinica vol 32 no 4 pp 851ndash866 2016

[11] J-e Liu and J Zhou ldquoAdditive-multiplicative hazards modelfor case-cohort studies with multiple disease outcomesrdquo ActaMathematicae Applicatae Sinica English Series vol 33 no 1pp 183ndash192 2017

[12] S-E Lu and J H Shih ldquoCase-cohort designs and analysis forclustered failure time datardquo Biometrics vol 62 no 4pp 1138ndash1148 2006

[13] H Zhang D E Schaubel and J D Kalbfleisch ldquoProportionalhazards regression for the analysis of clustered survival datafrom case-cohort studiesrdquo Biometrics vol 67 no 1 pp 18ndash282011

[14] G Yin and J Cai ldquoAdditive hazards model with multivariatefailure time datardquo Biometrika vol 91 no 4 pp 801ndash818 2004

[15] K Chen and S H Lo ldquoCase-cohort and case-control analysiswith Coxrsquos modelrdquo Biometrika vol 86 no 4 pp 755ndash7641999

[16] D Clayton and J Cuzick ldquoMultivariate generalizations of theproportional hazards modelrdquo Journal of the Royal StatisticalSociety Series A (General) vol 148 no 2 pp 82ndash117 1985

[17] R V Foutz ldquoOn the unique consistent solution to the like-lihood equationsrdquo Journal of the American Statistical Asso-ciation vol 72 no 357 pp 147-148 1977

10 Journal of Mathematics