Nonparametric Tests For Current Status Data Under Stratified Cox Proportional Hazards Model Based With Information Observation Times

Xiaodong Fana, Qingchun Zhangb,* School of Science, Jilin Institute of Chemical Technology, Jilin City, Jilin Province, 132022, China

afanxd14@mails.jlu.edu.cn, b48341846@qq.com *corresponding author

Keywords: Current Status Data; Nonparametric Test; Stratum Effects; Information Observation Times

Abstract: This paper considered the test of stratum interaction of current status data following the Cox Proportional Hazards model with Information Observation Times. We proposed a test procedure and the asymptotic property of the test statistic is derived by the martingale theory.

1. Introduction

Applications utilizing current status data include epidemiological surveys, medical studies and demographical studies (Zhang et al. 2005; Chen 2012). Statistical methods to test the stratum effects of right censored data and interval-censored data are developed under the cox model (Sun and Yang 2000; Fan et al. 2019). In this paper, we developed a test procedure for stratum effects of current status failure time data based on the Cox model with information Observation Times (Ma et al. 2015; Li et al. 2017). The models, notation and test procedure are discussed in section 2. The test statistics are developed based on the estimator of the cumulative hazard functions, which is obtained using the idea discussed in Fan et al. (2019). And we obtained the asymptotic distribution of the test statistics proposed in this paper. In section 3, we given some remarks and discussion.

2. Methodology

In a survival study, there are n independent subjects from q strata. For subject i in the k th

stratum, suppose that kiT s′ and kiZ s′ are the failure time and vector of covariates, respectively,

1, 2, , , 1, 2, ,ki n k q= = , where 1

q

kk

n n=

=∑. Assume that the observation time are dependent on the

survival time, which is referred to as information censoring. Given kiZ , supposed that kiT follows the Cox Proportional hazards model

0( , ( ), ) ( ) exp( ( ) ( ))tki ki k ki kih t Z u u t h t Z t B tβ ′≤ = + (1)

where ( )kh t is an unknown baseline hazard function in the k th Stratum, 0β is a set of regression

parameters, and random effect ( )kiB t are arbitrary processes. For the failure times kiT s′ , assumed

that every individual is observed only once at time kiC , and so we obtain current status data denoted

by {( , ( ), ( ), ); 1, , , 1, , }ki ki ki ki ki ki kC I T C Z t t C i n k qδ = ≥ ≤ = = . In practice, kiC may be information

censored. For this, supposed that given { ( ), }kiZ s s t≤ , kiC follow the Cox proportional hazards model

0( , ( ), ) ( ) exp( ( ) ( ))c cki ki ki kih t Z s s t h t Z t B tβ ′≤ = + (2)

2020 International Conference on Social Sciences and Innovative Economy (SSIE2020)

DOI: 10.38007/Proceedings.0001572 ISBN: 978-1-80052-010-3-500-

where ( )ch t is an unknown baseline hazard function, 1β is a set of regression parameters as 0β ,

random effect ( )kiB t are definded in (1) . Furthermore, assume that ( )kiB t have the same effects for all the study individuals. As discussed in Sun and Yang (2000), to simplify the notation, we suppose that the covariates have the same effects in models (1) and (2) for the individuals in all strata and the test method proposed here are still valid if the covariates have different effects. In the following, we focus on the null hypothesis

0 1 2: ( ) ( ) ( )qH h t h t h t= = = For each ki , write ( ) ( min( , ))ki ki kiN t I C t T= ≤ and ( ) ( )ki kiY t I C t= ≥ . Then we can derive that ( )kiN t is a counting process and has the intensity process

0 ( ) ( )1 0 0 0

( , ( ), ) exp( ( ) ) ( ) exp( ( ) ) (exp( ( ) ))ki kit t tZ s B s dsc

ki ki ki k b kih t Z s s t Z t e ds h t h s ds E B t eββ ′′≤ = − − −∫ ∫ ∫ 0( ) ( )

10 0( ) (exp( ( ) )) exp( ( ) )ki ki

t tB s ds Z sk b ki kih t E B t e Z t e dsββ ′′= − −∫ ∫

(3)

(Feng et al. 2018 and Fan et al. 2019), where

~

0( ) ( ) exp( ( ) )

tck kh t h t h s ds= −∫ , ' 'bE means the

expectation, which is calculated with respect to the random effect ( )kiB t s′ . Note that ( )

0(exp( ( ) ))ki

t B s dsb kiE B t e− ∫ is equality for all subjects and equation (3) is the Cox proportional

hazards model. Define 1( ) (t)kn c

k kiiN t N

==∑ , 1

( ) (t)qkk

N t N=

=∑ , 0 ( )

0 1 1 0(t, , ) (t) exp( ( ) )ki

t Z sk ki kiY Y Z t e dsββ β β ′′= − ∫ and 1

(t) ( (t) 0)knk kii

J I Y=

= >∑ . Under 0H , Let 0 1( , )β β

denote the partial maximum likelihood estimator of the parameter 0 1( , )β β (Lin et al., 1998, Feng et al., 2015). Then motivated by the test statistic in Fan et al. (2019), we propose to use the

test statistic 0 1 1 0 1 0 1( , ) ( ( , ), , ( , ))qW W Wβ β β β β β ′=

. Here

0 1 0 1 0 1 0 0 10( , ) ( ) ( , , ){ ( , , ) ( ) ( , , )}k k k kW L t Y t d t J t d t

tβ β β β β β β β= L − L∫

, (4) Here ( )L t is weight function. In (4),

00 1 0 ( )

11 0

(s)dN (s)ˆ (t, , )(s) exp( ( ) )k ki

ct k kk tn Z s

ki kii

J

Y Z t e dsββ β

β ′′=

L =−

∫∑ ∫ (5)

is the cumulative hazard function for the k th stratum, and under 0H ,

00 0 1 0 ( )

11 1 0

(s)dN(s)ˆ (t, , )(s) exp( ( ) )k ki

t

tq n Z ski kik i

J

Y Z t e dsββ β

β ′′= =

L =−

∫∑ ∑ ∫ (6)

Under mild regularity conditions and 0H , as n →∞ and /k kn n q→ , 0 1( , )W β β

has an

asymptotic normal distribution 0 1(0, ( , ))N β βΣ

, which the covariate matrix

1 20 1 0 1( , ) ( ( , ))k k q qβ β σ β β ×Σ =

, and

1 1

1 2 1 1

00 1

0 1 (0)1 1 00 1

(t, , )ˆ ( , ) { (t) J (t)[ ] (t)

(t, , )lq n k k c

k k k k l lil i

n SL dN

nSt β β

σ β β ξβ β= =

= −∑ ∑ ∫

2 2

1 2

00 1

(0)00 1

(t, , )(t) J (t)[ ] (t)}

(t, , )k k c

k k l li

n SL dN

nSt β β

ξβ β

× −∫

, (7)

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0kq > and 11q

klq

==∑ . In the above formula,

0 ( )00 1 1 0

1

1(t, , ) (t) exp( ( ) )k

ki

n t Z sk ki ki

ik

S Y Z t e dsn

ββ β β ′′

=

= −∑ ∫

, (8) and

0 ( )(0)0 1 1 0

1 1

1(t, , ) (t) exp( ( ) )l

ki

nq t Z ski ki

l iS Y Z t e ds

nββ β β ′′

= =

= −∑∑ ∫

. (9)

Let 0 0 1( , )W β β

denote 0 1( , )W β β

for the first 1k − components, and 0 1( , )D β β

the matrix 0 1( , )β βΣ

with the last row and column removed. Then the hypothesis 0H can be test by the test

statistic2 10 0 0 1 0 1 0 0 1( , ) ( , ) ( , )TW D Wχ β β β β β β−=

, which has asymptotically2χ -distribution with 1q −

degrees of freedom. This is because 0 11( , ) 0q

kkW β β

==∑

. To employ the above test procedure, it is needed to choose the weight process ( )L t . For this, there are usually three type of weight processes,

1( ) 1L t = , 1

2 1 1( ) ( )kq n

kik iL t n Y t−

= == ∑ ∑ and 3 2( ) 1 ( )L t L t= − . One can see Sun and Yang (2000) and

Andersen et al. (1993) to learn more comments on the weight function and the suggestions to choose the weight function.

3. Conclusion

In this paper, we discussed the test of the stratum effect of current status data based on the stratified Cox proportional hazards model with Information Observation Times. There are lots of literature for regression analysis and nonparametric comparison of survival functions of current status data. However, there seems no literature discussing the problem in this paper. For the problem, a kind of nonparametric test procedure is obtained and the asymptotic properties of the test procedures are derived.

References

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