Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Applied Statistics Unit (ASU-Kolkata)


Sengupta, Debasis (ASU-Kolkata; ISI)

Abstract (Summary of the Work)

The proportional hazards (PH) model, but more speciÖcally its special case the Cox regression model (Cox, 1972), plays an important role in the theory and practice of lifetime and duration data analysis. This is because the PH model (and the Cox regression model) provides a convenient way to evaluate the ináuence of one or several covariates on the probability of conclusion of lifetime or duration spells. However, the PH speciÖcation substantially restricts interdependence between the explanatory variables and the lifetime in determining the hazard. In particular, the Cox regression model model restricts the coe¢ cients of the regressors in the logarithm of the hazard function to be constant over the lifetime. This restriction may not hold in many situations, or may even be unreasonable from the point of view of relevant theory. Further, this and other kinds of misspeciÖcation often lead to misleading inferences about the e§ects of explanatory variables and the shape of the baseline hazard.Testing the Cox PH model, particularly against the omnibus alternative, has therefore been an area of active research. However, the omnibus tests do not o§er much clarity regarding the nature of departure from underlying assumptions. As a result, these tests do not provide useful inference for further modeling covariate e§ects when the Cox regression model does not hold. For example, it is often of interest to explore whether the hazard rate for one level of the covariate increases in lifetime relative to another level (i.e., the hazard ratio increases/decreases with lifetime). Ordered departures from proportionality of this and related types are useful in the two-sample (or binary covariate) setup for studying commonly observed features like crossing hazards. Similar situations also occur quite frequently in the k-sample setup and when the covariate is continuous. Throughout this thesis, we call such ordered departures generically as "order restrictions on covariate dependence", as distinct from "order restrictions on ageing" which refers to restrictions on the shape of the baseline hazard function (or, on duration dependence).The work included in this thesis develops analytical and graphical inference on covariate e§ects in situations when the Cox regression model, or more generally the PH model, may not hold. In particular, we develop methods to study covariate e§ects in the presence of potentially order restricted departures from proportionality. The thesis places emphasis on both theory and applications, and extends the literature along both these dimensions in several ways. In this sense, the work is Örmly set within the tradition of research in applied statistics and econometrics.In the following section (Section 1.1), we motivate our research on order restrictions on covariate dependence using a few real life examples, focusing on some useful ways in which order restrictions can be characterised and hazard regression models accommodating order restricted covariate e§ects. Next, in Section 1.2, we review recent research on hazard regression models, which are useful for modeling and estimation of covariate dependence under order restrictions, particularly when the covariate is continuous. The review is selective, focusing largely on order restrictions in these models and aimed at identifying gaps in the literature. As we proceed, we place the main contributions made in the thesis within the context of the literature. Finally, we outline the new research and describe the chapter scheme for the rest of the thesis (Section 1.3).


ProQuest Collection ID:

Control Number


Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Included in

Mathematics Commons