Survival Analysis generally uses the median survival time as a common summary statistic. While the median possesses the desirable characteristic of being unbiased, there are times when it is not the best statistic to describe the data at hand. Royston and Parmar (2011) provide an argument that the restricted mean survival time should be the summary statistic used when the proportional hazards assumption is in doubt. Work in Restricted Means dates back to 1949 when J.O. Irwin developed a calculation for the standard error of the restricted mean using Greenwood’s formula. Since then the development of the restricted mean has been thorough in the literature, but its use in practical analyses is still limited. One area that is not well developed in the literature is the choice of the time point to which the mean is restricted. The aim of this dissertation is to develop a data driven method that allows the user to find a cut-point to use to restrict the mean. Three methods are developed. The first is a simple method that locates the time at which the maximum distance between two curves exists. The second is a method adapted from a Renyi-type test, typically used when proportional hazards assumptions are not met, where the Renyi statistics are plotted and piecewise regression model is fit. The join point of the two pieces is where the meant will be restricted. Third is a method that applies a nonlinear model fit to the hazard estimates at each event time, the model allows for the hazards between the two groups to be different up until a certain time, after which the groups hazards are the same. The time point where the two groups’ hazards become the same is the time to which the mean is restricted. The methods are evaluated using MSE and bias calculations, and bootstrap techniques to estimate the variance.

<p>The aim of this study is to illustrate, adapt and develop methods of survival analysis in analysing and estimating customer survival time in subscription-based businesses. Two particular objectives are studied. The rst objective is to redene the existing survival analysis techniques in business terms and to discuss their uses in order to understand various issues related to the customer-rm relationship.</p>

<p>The aim of this study is to illustrate, adapt and develop methods of survival analysis in analysing and estimating customer survival time in subscription-based businesses. Two particular objectives are studied. The rst objective is to redene the existing survival analysis techniques in business terms and to discuss their uses in order to understand various issues related to the customer-rm relationship.</p>

Philosophiae Doctor - PhD / Subscription-based industries have seen a massive expansion in recent decades. In this type of industry the customer has to subscribe to be able to enjoy the service; there-fore, well-de ned start and end points of the customer relationship with the service provider are known. The length of this relationship, that is the time from subscription to service cancellation, is de ned as customer survival time. Unlike transaction-based businesses, where the emphasis is on the quality of a product and customer acquisition, subscription-based businesses focus on the customer and customer retention. A customer focus requires a new approach: managing according to customer equity (the value of a rm's customers) rather than brand equity (the value of a rm's brands). The concept of customer equity is attractive and straightforward, but the implementation and management of the customer equity approach do present some challenges. Amongst these challenges is that customer asset metric - customer lifetime value (the present value of all future pro ts generated from a customer) - depends upon assumptions about the expected survival time of the customer (Bell et al., 2002; Gupta and Lehmann, 2003). In addition, managing and valuing customers as an asset require extensive data and complex modelling. The aim of this study is to illustrate, adapt and develop methods of survival analysis in analysing and estimating customer survival time in subscription-based businesses. Two particular objectives are studied. The fi rst objective is to rede ne the existing survival analysis techniques in business terms and to discuss their uses in order to understand various issues related to the customer-fi rm relationship. The lesson to be learnt here is the ability of survival analysis techniques to extract important information on customers with regard to their loyalties, risk of cancellation of the service, and lifetime value. The ultimate outcome of this process of studying customer survival time will be to understand the dynamics and behaviour of customers with respect to their risk of cancellation, survival probability and lifetime value. The results of the estimates of customer mean survival time obtained from different nonparametric and parametric approaches; namely, the Kaplan-Meier method as well as exponential, Weibull and gamma regression models were found to vary greatly showing the importance of the assumption imposed on the distribution of the survival time. The second objective is to extrapolate the customer survival curve beyond the empirical distribution. The practical motivation for extrapolating the survival curve beyond the empirical distribution originates from two issues; that of calculating survival probabilities (retention rate) beyond the empirical data and of calculating the conditional survival probability and conditional mean survival time at a speci c point in time and for a speci c time window in the future. The survival probabilties are the main components needed to calculate customer lifetime value and thereafter customer equity. In this regard, we propose a survivor function that can be used to extrapolate the survival probabilities beyond the last observed failure time; the estimation of parameters of the newly proposed extrapolation function is based completely on the Kaplan-Meier estimate of the survival probabilities. The proposed function has shown a good mathematical accuracy. Furthermore, the standard error of the estimate of the extrapolation survival function has been derived. The function is ready to be used by business managers where the objective is to enhance customer retention and to emphasise a customer-centric approach. The extrapolation function can be applied and used beyond the customer survival time data to cover clinical trial applications. In general the survival analysis techniques were found to be valuable in understanding and managing a customer- rm relationship; yet, much still needs to be done in this area of research to make these techniques that are traditionally used in medical studies more useful and applicable in business settings. / South Africa

The cause and effect paradigm underlying medical research has led to an enhanced etiological understanding of many diseases and the development of many lifesaving drugs, but the paradigm does not always include an understanding of the pathways involved. Causal mediation analysis extends the cause and effect relationship to the cause and effect through a mediator, an intermediate variable on the causal pathway. The total effect of an exposure on an outcome is decomposed into two parts: 1) the indirect effect of the exposure on the outcome through the mediator and 2) the direct effect of the exposure on the outcome through all other pathways. In this dissertation, I describe various counterfactual causal mediation frameworks with identifiability assumptions that all lead to the Mediation Formula. The indirect and direct effects can be estimated from observable data using a semi-parametric algorithm derived from the Mediation Formula that I generalize to different types of mediators and outcomes. With an increased interest in causal mediation analysis, thoughtful consideration is necessary in the application of the Mediation Formula to real-world data challenges. Here, I consider three motivating causal mediation questions in the areas of HIV curative research and cardio-respiratory fitness. HIV curative treatments typically target the viral reservoir, cells infected with latent HIV. Quantifying the effect of an HIV curative treatment on viral rebound over a set time horizon mediated by reductions in the viral reservoir can inform future directions for improving curative treatments. In cardiorespiratory fitness research, metabolites, molecules involved with cellular respiration, are believed to mediate the effect of physical activity on cardiorespiratory fitness. I propose three novel adaptations to the semi-parametric estimation algorithm to address three data challenges: 1) Numerical integration and optimization of the observed data likelihood for mediators with an assay lower limit (left-censored mediators); 2) Pseudo-value approach for time-to-event outcomes on a restricted mean survival time scale; 3) Elastic net regression for high-dimensional mediators. My novel approaches provide estimation frameworks that can be applied to a broad spectrum of research questions. I provide simulation studies to assess the properties of the estimators and applications of the methodologies to the motivating data. / 2025-06-16T00:00:00Z

Le bénéfice de survie restreint à un horizon temporel donné a été proposé comme mesure alternative aux mesures relatives de l’effet d’un traitement notamment dans le cas de non proportionnalité des risques de décès. Le bénéfice de survie restreint correspond à la différence des survies moyennes entre deux bras de traitement et s’exprime en nombre d’années de vie gagnées. Dans la littérature, cette mesure est présentée comme plus intuitive que le hazard ratio et plusieurs auteurs ont proposé son utilisation pour le design et l’analyse d’un essai clinique. Toutefois, ce n’est pas actuellement la mesure qui est utilisée de manière courante dans les essais randomisés. Cette mesure s’applique quelle que soit la distribution des temps de survie et est adaptée si l’hypothèse des risques proportionnels n’est pas respectée. De plus, le bénéfice de survie restreint peut être utilisé en évaluation médico-économique où la mesure d’un effet absolu est nécessaire (nombre d’années de vie gagnées pondérées ou non par la qualité de vie). Si l’on souhaite estimer le bénéfice de survie restreint à partir d’une méta-analyse sur données individuelles, se pose alors la question de prendre en compte l’effet essai dû à la structure hiérarchique des données. L’objectif de cette thèse était de comparer des méthodes statistiques d’estimation du bénéfice de survie restreint à partir de données individuelles d’une méta-analyse d’essais cliniques randomisés. Le point de départ a été une étude de cas (étude coût-efficacité) réalisée à partir des données de la Meta-Analysis of Radiotherapy in Lung Cancer. Cette étude a montré que les cinq méthodes d’estimation étudiées conduisaient à des estimations différentes du bénéfice de survie et de son intervalle de confiance. Le choix de la méthode d’estimation avait également un impact sur les résultats de l’analyse coût-efficacité. Un second travail a consisté à mener une étude de simulation pour mieux comprendre les propriétés des méthodes d’estimation considérées en termes de biais moyen et d’erreur-type. Enfin, la dernière partie de la thèse a mis en application les enseignements de cette étude de simulation au travers de trois méta-analyses sur données individuelles dans le cancer du nasopharynx et dans le cancer du poumon à petites cellules. / The survival benefit restricted up to a certain time horizon has been suggested as an alternative measure to the common relative measures used to estimate the treatment effect, especially in case of non-proportional hazards of death. The restricted survival benefit corresponds to the difference of the two restricted mean survival times estimated for each treatment arm, and is expressed in terms of life years gained. In the literature, this measure is considered as more intuitive than the hazard ratio and many authors have suggested its use for the design and the analysis of clinical trials. However, it is not currently the most used measure in randomized trials. This measure is valid under any distribution of the survival times and is adapted if the proportional hazards assumption does not hold. In addition, the restricted survival benefit can be used in medico-economic evaluation where an absolute measure of the treatment effect is needed (number of [quality adjusted] life years gained). If one wants to estimate the restricted survival benefit from an individual participant data meta-analysis, there is a need to take into account the trial effect due to the hierarchical structure of the data. The aim of this thesis was to compare statistical methods to estimate the restricted survival benefit from an individual participant data meta-analysis of randomized trials. The starting point was a case study (cost-effectiveness analysis) using data from the Meta-Analysis of Radiotherapy in Lung Cancer. This study showed that the five investigated methods yielded different estimates for the restricted survival benefit and its confidence interval. The choice of a method to estimate the survival benefit also impacted on cost-effectiveness results. Our second project consisted in a simulation study to have a better understanding of the properties of the investigated methods in terms of bias and standard error. Finally, the last part of the thesis illustrated the lessons learned from the simulation study through three examples of individual participant data meta-analysis in nasopharynx cancer and in small cell lung cancer.