Citation: Caroni, C. Regression Models for Lifetime Data: An Overview. Stats 2022, 5, 1294–1304. https://doi.org/10.3390/ stats5040078 Academic Editor: Wei Zhu Received: 6 November 2022 Accepted: 3 December 2022 Published: 7 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Regression Models for Lifetime Data: An Overview Chrys Caroni Department of Mathematics, National Technical University of Athens, 157 80 Athens, Greece; ccar@math.ntua.gr Abstract: Two methods dominate the regression analysis of time-to-event data: the accelerated failure time model and the proportional hazards model. Broadly speaking, these predominate in reliability modelling and biomedical applications, respectively. However, many other methods have been proposed, including proportional odds, proportional mean residual life and several other “proportional” models. This paper presents an overview of the field and the concept behind each of these ideas. Multi-parameter modelling is also discussed, in which (in contrast to, say, the proportional hazards model) more than one parameter of the lifetime distribution may depend on covariates. This includes first hitting time (or threshold) regression based on an underlying latent stochastic process. Many of the methods that have been proposed have seen little or no practical use. Lack of user-friendly software is certainly a factor in this. Diagnostic methods are also lacking for most methods. Keywords: lifetime data; regression; proportional hazards; proportional odds; mean residual life; median residual life; proportional reversed hazards; accelerated failure time; first hitting time 1. Introduction The purpose of the present paper is to give an overview of the several forms of regression models that have been proposed for use when the dependent variable is the time until the occurrence of an event, with simple examples being the death of a patient (survival analysis) and the failure of a machine (reliability modelling). A basic form of statistical model regresses the value y of a continuous dependent (response) random variable Y on the values of a vector of covariates (predictors or explanatory variables) x recorded for the same statistical unit. The standard example of regression is of course the general linear model Y = β ′ x + ǫ, (1) where β is a vector of regression coefficients and the random error term ǫ in the stan- dard model follows the Normal distribution with zero mean and constant variance σ 2 , ǫ ∼ N(0, σ 2 ). Taking, as usual, the values of x as fixed (not random), this implies that the conditional distribution of Y given x is also normal, Y|x ∼ N( β ′ x, σ 2 ). (2) This model is not suitable for application to lifetime data; the fact that the dependent variable—usually time or a proxy for time, such as the distance run by a vehicle—is non- negative demands a special approach. The same models also apply to other cases of non-negative dependent variables besides time, such as the load that can be applied to a sample of material before it breaks. The regression models examined here are all intended for use with dependent variables of these types. There will be no attempt to review each model in the sense of covering all the developments, as they are far too numerous to permit this. For example, it will be assumed here that the covariates x are measured at the baseline (time origin of the study), whereas a necessary practical extension of any regression model is also to allow for time-varying covariates. However, this extension does not alter the Stats 2022, 5, 1294–1304. https://doi.org/10.3390/stats5040078 https://www.mdpi.com/journal/stats