## Date of Submission

2-22-2008

## Date of Award

2-22-2009

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Sengupta, Debasis (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

The proportional hazards (PH) model, but more speciÃ–cally 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 inÃ¡uence of one or several covariates on the probability of conclusion of lifetime or duration spells. However, the PH speciÃ–cation substantially restricts interdependence between the explanatory variables and the lifetime in determining the hazard. In particular, the Cox regression model model restricts the coeÂ¢ 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 misspeciÃ–cation often lead to misleading inferences about the eÂ§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 oÂ§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 eÂ§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 eÂ§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 eÂ§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 Ã–rmly 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 eÂ§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).

## Control Number

ISILib-TH320

## Creative Commons License

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

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Bhattacharjee, Arnab Dr., "Order-Restricted Covariate Effects and Hazard Regression Models." (2009). *Doctoral Theses*. 51.

https://digitalcommons.isical.ac.in/doctoral-theses/51

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842827