Conventional hazard regression analyses frequently assume constant regression coefficients and scalar covariates. However, some covariate effects may vary with time. Moreover, medical imaging has become an increasingly important tool in screening, diagnosis, and prognosis of various diseases, given its information visualization and quantitative assessment. This study considers an additive hazards model with time-varying coefficients and imaging predictors to examine the dynamic effects of potential scalar and imaging risk factors for the failure of interest. We develop a two-stage approach that comprises the high-dimensional functional principal component analysis technique in the first stage and the counting process-based estimating equation approach in the second stage. In addition, we construct the pointwise confidence intervals for the proposed estimators and provide a significance test for the effects of scalar and imaging covariates. Simulation studies demonstrate the satisfactory performance of the proposed method. An application to the Alzheimer’s disease neuroimaging initiative study further illustrates the utility of the methodology.
AalenOO. A linear regression model for the analysis of life times. Stat Med1989; 8: 907–925.
2.
HufferFWMcKeagueIW. Weighted least squares estimation for Aalen’s additive risk model. J Am Stat Assoc1991; 86: 114–129.
3.
LinDYingZ. Semiparametric analysis of the additive risk model. Biometrika1994; 81: 61–71.
4.
MartinussenTScheikeTH. Dynamic regression models for survival data. New York: Springer, 2006.
5.
JiangJHaiboZ. Additive hazard regression with auxiliary covariates. Biometrika2007; 94: 359–369.
6.
PanDHeHSongX, et al. Regression analysis of additive hazards model with latent variables. J Am Stat Assoc2015; 110: 1148–1159.
7.
QuLSongXSunL. Identification of local sparsity and variable selection for varying coefficient additive hazards models. Comput Stat Data Anal2018; 125: 119–135.
8.
BardJF. A reliability analysis of occupational exposure data using a family of proportional and additive hazard functions. J Environ Syst1986; 15: 293–306.
9.
ColoradoHAVelásquezEIGMonteiroSN. Sustainability of additive manufacturing: the circular economy of materials and environmental perspectives. J Mater Res Technol2020; 9: 8221–8234.
10.
KremerAWeißbachRLieseF. Maximum likelihood estimation for left-censored survival times in an additive hazard model. J Stat Plan Inference2014; 149: 33–45.
11.
YangQHeHLuB, et al. Mixture additive hazards cure model with latent variables: Application to corporate default data. Comput Stat Data Anal2022; 167: 107365.
12.
CoxDR. Regression models and life-tables. J R Stat Soc Ser B Methodol1972; 34: 187–202.
FlemingTRHarringtonDP. Counting processes and survival analysis. New York: Wiley, 2011.
15.
PanDKangKWangC, et al. Bayesian proportional hazards model with latent variables. Stat Methods Med Res2019; 28: 986–1002.
16.
BuckleyJ. Additive and multiplicative models for relative survival rates. Biometrics1984; 40: 51–62.
17.
McKeagueIW. Estimation for a semimartingale regression model using the method of sieves. Ann Stat1986; 4: 579–589.
18.
McKeagueIWSasieniPD. A partly parametric additive risk model. Biometrika1994; 81: 501–514.
19.
LiHYinGZhouY. Local likelihood with time-varying additive hazards model. Can J Stat2007; 35: 321–337.
20.
CaiZSunY. Local linear estimation for time-dependent coefficients in Cox’s regression models. Scand J Stat2003; 30: 93–111.
21.
YangQHeHSongX. Time-varying coefficient additive hazards model with latent variables. Stat Methods Med Res2022; 31(5): 928–946.
22.
RamsayJOSilvermanBW. Functional data analysis. New York: Springer-Verlag, 2005.
23.
FerratyFVieuP. Nonparametric functional data analysis: Theory and practice. New York: Springer, 2006.
24.
YaoF. Functional principal component analysis for longitudinal and survival data. Stat Sin2007; 17: 965–983.
25.
ZhuHFanJKongL. Spatially varying coefficient model for neuroimaging data with jump discontinuities. J Am Stat Assoc2014; 109: 1084–1098.
26.
MorrisJS. Functional regression. Annu Rev Stat Appl2015; 2: 321–359.
27.
WangXZhuHInitiativeADN. Generalized scalar-on-image regression models via total variation. J Am Stat Assoc2017; 112: 1156–1168.
28.
FengXLiTSongX, et al. Bayesian scalar on image regression with nonignorable nonresponse. J Am Stat Assoc2020; 115: 1574–1597.
29.
KangKSongXY. Joint modeling of longitudinal imaging and survival data. J Comput Graph Stat2022; 0: 1–11.
30.
ZipunnikovVGrevenSShouH, et al. Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis. Ann Appl Stat2014; 8: 2175.
31.
ZipunnikovVCaffoBYousemDM, et al. Functional principal component model for high-dimensional brain imaging. NeuroImage2011; 58: 772–784.
32.
Ramlau-HansenH. Smoothing counting process intensities by means of kernel functions. Ann Stat1983; 11: 453–466.
33.
KongDIbrahimJGLeeE, et al. FLCRM: Functional linear Cox regression model. Biometrics2018; 74: 109–117.
34.
HuangY. A-independent roles of apolipoprotein E4 in the pathogenesis of Alzheimer’s disease. Trends Mol Med2010; 16: 287–294.
35.
PetersenRCRobertsROKnopmanDSet al. Prevalence of mild cognitive impairment is higher in men: the Mayo clinic study of aging. Neurology2010; 75: 889–897.
TianLZuckerDWeiL. On the Cox model with time-varying regression coefficients. J Am Stat Assoc2005; 100: 172–183.
40.
ZhouXSongX. Mediation analysis for mixture Cox proportional hazards cure models. Stat Methods Med Res2021; 30: 1554–1572.
41.
SunRZhouXSongX. Bayesian causal mediation analysis with latent mediators and survival outcome. Struct Equ Modeling2021; 28: 778–790.
42.
HeHCaiJSongX, et al. Analysis of proportional mean residual life model with latent variables. Stat Med2017; 36: 813–826.
43.
HeHPanDSongX, et al. Additive mean residual life model with latent variables under right censoring. Stat Sin2019; 29: 47–66.
44.
OuyangMWangXWangC, et al. Bayesian semiparametric failure time models for multivariate censored data with latent variables. Stat Med2018; 37: 4279–4297.
45.
TangYSongXYiGY. Bayesian analysis under accelerated failure time models with error-prone time-to-event outcomes. Lifetime Data Anal2022; 28(1): 139–168.
46.
LipsitzSRIbrahimJG. Estimating equations with incomplete categorical covariates in the Cox model. Biometrics1998; 54: 1002–1013.
47.
ChenHYLittleRJ. Proportional hazards regression with missing covariates. J Am Stat Assoc1999; 94: 896–908.
48.
HerringAHIbrahimJG. Likelihood-based methods for missing covariates in the Cox proportional hazards model. J Am Stat Assoc2001; 96: 292–302.
49.
HerringAHIbrahimJGLipsitzSR. Non-ignorable missing covariate data in survival analysis: A case-study of an International Breast Cancer Study Group trial. J R Stat Soc Ser C Appl Stat2004; 53: 293–310.
50.
SongXWangCY. Time-varying coefficient proportional hazards model with missing covariates. Stat Med2013; 32: 2013–2030.
