Respuestas correspondientes al Grupo I

In this dissertation, we have developed practical and intuitive statistical methods for semipara- metric proportional and additive hazards modeling of the subdistribution of a competing risk in case-cohort studies. We also investigated tests to assess the proportionality assumption of the proportional hazards model for the subdistribution of a competing risk in case-cohort studies.

Since they were introduced by Ross L. Prentice in 1986, case-cohort designs have been widely used in large cohort studies especially when events are rare. A large number of statistical methods have been developed for various models with data from case-cohort studies since then. However, only a very few methods have been developed for competing risks from case-cohort studies.

A competing risk is an event whose occurrence either precludes the occurrence of the primary event or fundamentally alters the chance of the occurrence of this primary event. Unlike modeling the cause-specific hazard function of the different event types, which has been done in the past for case-cohort studies, we dealt with modeling the cumulative incidence function (CIF), also called the subdistribution, of an event of interest in the presence of competing risks in case-cohort studies. Modeling CIF enables us to directly assess the effect of a covariate on the marginal probability function, or the absolute risk, of the primary event.

In Chapter 2.5.2, we proposed estimating equations based on inverse probability weighting (IPW) methods for parameter estimation in a proportional hazards model for the subdistribution of a competing risk in case-cohort studies. In addition to the IPW for the censoring, we considered an appropriate inverse weighting technique to account for the over-representation of cases (i.e., the event of interest) in the case-cohort sample. Incorporating these weighting techniques, estimating equations were developed for parameter estimation of the proportional subdistribution hazards (PSH) model. In Chapter 3.8, we considered additive subdistribution hazards (ASH) model. Using similar weighting techniques, we constructed appropriate estimating equations for the additive subdistribution hazards (ASH) model for case-cohort data.

to establish the asymptotic properties of the estimators of the regression parameters and the cumulative baseline subdistribution hazard functions. The proposed estimators are consistent and asymptotically normally distributed. The proposed estimators for the PSH and ASH models reduce, respectively, to the maximum partial likelihood estimators for the Cox proportional hazards model and to the Lin-Ying estimators (Lin and Ying, 1994) for the additive hazards model if there were no competing events and with full cohort. The choice between PSH and ASH models depends on the scientific question of interest.

Recognizing the importance of assessing the proportionality assumption in the PSH models in case-cohort studies, in Chapter 4.8, we considered two goodness-of-fit tests of the PSH model by extending the correlation test (Schoenfeld, 1982) and the inclusion of time-varying regression coefficients followed by score test (Grambsch and Therneau, 1994; Zhou et al., 2013). While both proposed methods engage Schoenfeld-type residuals, i.e., weighted residuals adopted to the estimating functions of the PSH model of competing risks data in case-cohort studies, the second proposed method is applicable in situations where the nonproportionality is due to time-varying coefficients in the PSH model. The proposed tests are easily applicable since they require only Schoenfeld-type residuals along with the estimated regression coefficients and their covariance matrix from the ordinary PSH model with time-independent coefficients. That is, the tests do not require estimation under the alternative (i.e., nonproportional subdistribution hazards) model.

Extensive simulation studies were carried out for all of the proposed statistical methods in this dissertation in order to investigate the behavior of the proposed estimators (in Chapters 2.5.2 and 3.8) and statistical tests (in Chapter 4.8) in finite samples. The simulation results have shown that, under each of the assumed model, the proposed methods work adequately for practical sample sizes of case-cohort data. We observed that the power for testing proportional hazards assumption is positively associated with the event proportion.

There are a number of topics to which the proposed methods in this dissertation can be extended. First, in this dissertation, for the subcohort sampling, we considered simple random sampling. Future work could include extending the proposed methods to other sampling schemes such as stratified sampling (Samuelsen et al. 2007). For cohort studies where the event is not rare, the corresponding methods for generalized case-cohort sampling (Chen, 2001; Cai and Zeng, 2007), where cases are also sampled, are worthy of future development.

Another topic that can be explored in future work is related to interval censoring. Recently, Mao et al. (2017) and Li (2016) considered semiparametric analysis of competing risks data subject to interval censoring, mostly in proportional hazards modeling, in cohort studies. Extending the estimation methods we proposed here to interval censored data from case-cohort studies is worthy of future investigation.

In this dissertation, we considered using the Schoenfeld-type residuals of PSH model in case- cohort studies to assess the proportionality assumption of the model. In addition to testing the validity of the proportionality assumption, it would be important to examine other properties related to the adequacy of the PSH model, such as the functional forms of each covariate in the model, assessing leverage or influence of each subject in parameter estimation and the accuracy of the PSH model in predicting the outcome for a particular subject, as in the standard proportional hazards model (Lin et al., 1993; Therneau et al., 1990).

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